Кіріспе
Фигуралы санның түрі
Type of figurate number
A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The formula for the nth hexagonal number
The first few hexagonal numbers are:
1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946
Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula
where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal. For example, the 2nd hexagonal number is 2×3 = 6; the 4th is 4×7 = 28; the 16th is 16×31 = 496; and the 64th is 64×127 = 8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130. Adrien Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. In addition, only two integers cannot be expressed using five hexagonal numbers (but can be with six), those being 11 and 26. Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".
A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The formula for the nth hexagonal number
The first few hexagonal numbers are:
1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946
Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula
where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal. For example, the 2nd hexagonal number is 2×3 = 6; the 4th is 4×7 = 28; the 16th is 16×31 = 496; and the 64th is 64×127 = 8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130. Adrien Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. In addition, only two integers cannot be expressed using five hexagonal numbers (but can be with six), those being 11 and 26. Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".
Алтыбұрышты сан – фигуралы сан. n-ші алтыбұрышты сан hn – бұл n нүктеге дейінгі қабырғалары бар тұрақты алтыбұрыштардың контурларынан тұратын нүктелер үлгісіндегі ерекше нүктелердің саны, ал алтыбұрыштар бір төбесімен ортақ болатындай етіп жабыстырылған. n-ші алтыбұрышты санның формуласы:
Type of figurate number
A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The formula for the nth hexagonal number
The first few hexagonal numbers are:
1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946
Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula
where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal. For example, the 2nd hexagonal number is 2×3 = 6; the 4th is 4×7 = 28; the 16th is 16×31 = 496; and the 64th is 64×127 = 8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130. Adrien Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. In addition, only two integers cannot be expressed using five hexagonal numbers (but can be with six), those being 11 and 26. Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".
A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The formula for the nth hexagonal number
The first few hexagonal numbers are:
1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946
Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula
where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal. For example, the 2nd hexagonal number is 2×3 = 6; the 4th is 4×7 = 28; the 16th is 16×31 = 496; and the 64th is 64×127 = 8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130. Adrien Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. In addition, only two integers cannot be expressed using five hexagonal numbers (but can be with six), those being 11 and 26. Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".
Алғашқы бірнеше алтыбұрышты сандар: 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946
Type of figurate number
A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The formula for the nth hexagonal number
The first few hexagonal numbers are:
1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946
Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula
where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal. For example, the 2nd hexagonal number is 2×3 = 6; the 4th is 4×7 = 28; the 16th is 16×31 = 496; and the 64th is 64×127 = 8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130. Adrien Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. In addition, only two integers cannot be expressed using five hexagonal numbers (but can be with six), those being 11 and 26. Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".
A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The formula for the nth hexagonal number
The first few hexagonal numbers are:
1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946
Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula
where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal. For example, the 2nd hexagonal number is 2×3 = 6; the 4th is 4×7 = 28; the 16th is 16×31 = 496; and the 64th is 64×127 = 8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130. Adrien Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. In addition, only two integers cannot be expressed using five hexagonal numbers (but can be with six), those being 11 and 26. Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".
Кез келген алтыбұрышты сан – үшбұрышты сан, бірақ тек әрбір екінші үшбұрышты сан (1-ші, 3-ші, 5-ші, 7-ші және т.б.) – алтыбұрышты сан. Үшбұрышты сан сияқты, алтыбұрышты санның 10-дық санау жүйесіндегі сандық түбірі тек 1, 3, 6 немесе 9 болуы мүмкін. Цифрлық түбір үлгісі, тоғыз сан сайын қайталанып, "1 6 6 1 9 3 1 3 9" болып табылады. Кез келген жұп толық сан алтыбұрышты болып табылады, ол мына формуламен берілген:
Type of figurate number
A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The formula for the nth hexagonal number
The first few hexagonal numbers are:
1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946
Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula
where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal. For example, the 2nd hexagonal number is 2×3 = 6; the 4th is 4×7 = 28; the 16th is 16×31 = 496; and the 64th is 64×127 = 8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130. Adrien Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. In addition, only two integers cannot be expressed using five hexagonal numbers (but can be with six), those being 11 and 26. Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".
A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The formula for the nth hexagonal number
The first few hexagonal numbers are:
1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946
Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula
where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal. For example, the 2nd hexagonal number is 2×3 = 6; the 4th is 4×7 = 28; the 16th is 16×31 = 496; and the 64th is 64×127 = 8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130. Adrien Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. In addition, only two integers cannot be expressed using five hexagonal numbers (but can be with six), those being 11 and 26. Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".
мұндағы Mp – Мерсенн жай саны. Жұп емес толық сандар белгісіз, сондықтан барлық белгілі толық сандар алтыбұрышты болып табылады. Мысалы, екінші алтыбұрышты сан 2×3 = 6; 4-ші – 4×7 = 28; 16-шы – 16×31 = 496; ал 64-ші – 64×127 = 8128. Ең көп дегенде төрт алтыбұрышты санның қосындысы түрінде жазылмайтын ең үлкен сан – 130. Адриен Мари Лежандр 1830 жылы 1791-ден үлкен кез келген бүтін санды осылай өрнектеуге болатынын дәлелдеді. Сонымен қатар, бес алтыбұрышты санмен өрнектеуге болмайтын (бірақ алтысымен өрнектеуге болатын) тек екі бүтін сан бар, олар – 11 және 26. Алтыбұрышты сандарды орталық алтыбұрышты сандармен шатастырмау керек, олар Вена шұжықтарының стандартты қаптамасын бейнелейді. Сандарды шатастырмау үшін алтыбұрышты сандар кейде "төбелік алтыбұрышты сандар" деп аталады.
Type of figurate number
A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The formula for the nth hexagonal number
The first few hexagonal numbers are:
1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946
Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula
where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal. For example, the 2nd hexagonal number is 2×3 = 6; the 4th is 4×7 = 28; the 16th is 16×31 = 496; and the 64th is 64×127 = 8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130. Adrien Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. In addition, only two integers cannot be expressed using five hexagonal numbers (but can be with six), those being 11 and 26. Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".
A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The formula for the nth hexagonal number
The first few hexagonal numbers are:
1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946
Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula
where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal. For example, the 2nd hexagonal number is 2×3 = 6; the 4th is 4×7 = 28; the 16th is 16×31 = 496; and the 64th is 64×127 = 8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130. Adrien Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. In addition, only two integers cannot be expressed using five hexagonal numbers (but can be with six), those being 11 and 26. Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".
Қос алтыбұрышты сандардың қосындысы
Қарсы шамадағы алтыбұрышты сандардың қосындысы 2ln(2) тең, мұнда ln – табиғи логарифмді көрсетеді.
The sum of the reciprocal hexagonal numbers is 2ln(2), where ln denotes natural logarithm.
Кейбір табиғи сандардың күштерінің бөлгіштерінің саны
n>0 үшін бөлгіштері бар. Сол сияқты, p және q екі түрлі жай сандар болатын кез келген натурал сан үшін, n>0-дың бөлгіштері бар. Дәлелдеме. k = 0-ден 2(n − 1)-ге дейін және l = 0-ден n − 1-ге дейін түріндегі бөлгіштері бар. k пен l-дың әр комбинациясы ерекше бөлгішті береді, сондықтан бөлгіштері бар, яғни бөлгіштер бар. ∎
for n>0 has divisors. Likewise, for any natural number of the form where p and q are distinct prime numbers, for n>0 has divisors. Proof. has divisors of the form , for k = 0 2(n − 1), l = 0 n − 1. Each combination of k and l yields a distinct divisor, so has divisors, i. e. divisors. ∎
Шексұмбақты квадрат сандар
Алтыбұрышты және толық квадрат сандар тізбегі 1, 1225, 1413721, ... басталады.
The sequence of numbers that are both hexagonal and perfect squares starts 1, 1225, 1413721, .