A001405 -id:A001405 - cp's OEIS frontend

A161247 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 8 times.

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 2, 4, 4, 5, 4, 7, 5, 8, 8, 9, 9, 12, 10, 13, 13, 15, 14, 18, 16, 19, 19, 21, 20, 26, 23, 27, 28, 31, 31, 38, 36, 42, 44, 48, 49, 57, 56, 63, 66, 72, 73, 83, 82, 91, 94, 102, 103, 117, 115, 127, 131, 141, 142, 160, 157, 174, 178, 190

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A001405.

A161248 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 9 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 3, 3, 5, 4, 5, 6, 6, 6, 9, 8, 9, 11, 11, 11, 14, 13, 14, 17, 16, 17, 20, 19, 20, 23, 22, 23, 28, 26, 28, 32, 32, 34, 40, 39, 42, 48, 49, 52, 59, 59, 63, 70, 72, 76, 85, 85, 91, 98, 102, 106, 117, 117, 125, 134, 139, 144, 158, 158

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A001405.

A161249 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 10 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 2, 4, 4, 5, 4, 7, 5, 7, 7, 9, 8, 11, 10, 12, 12, 14, 13, 16, 15, 18, 17, 20, 19, 22, 21, 24, 23, 26, 25, 30, 28, 32, 32, 36, 36, 42, 41, 46, 47, 53, 54, 61, 61, 67, 69, 76, 78, 86, 87, 95, 97, 106, 108, 118, 119, 129, 131, 142

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A001405.

A161250 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 11 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 3, 3, 5, 4, 5, 6, 6, 6, 8, 7, 9, 10, 10, 11, 13, 12, 14, 15, 15, 16, 18, 18, 19, 21, 21, 22, 24, 24, 25, 27, 27, 28, 32, 31, 33, 36, 37, 39, 44, 44, 47, 51, 53, 57, 63, 64, 68, 73, 76, 81, 88, 90, 95, 101, 106, 111, 120

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A001405.

A161251 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 12 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 2, 4, 4, 5, 4, 7, 5, 7, 7, 8, 7, 11, 9, 11, 12, 13, 12, 16, 14, 16, 17, 18, 17, 22, 19, 22, 23, 24, 23, 28, 25, 28, 29, 30, 29, 36, 32, 36, 38, 40, 40, 48, 45, 50, 53, 56, 57, 67, 65, 71, 75, 79, 81, 92, 91, 98, 103

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A001405.

A161252 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 13 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 3, 3, 5, 4, 5, 6, 6, 6, 8, 7, 8, 9, 10, 10, 12, 12, 13, 14, 15, 15, 17, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 36, 36, 38, 40, 42, 44, 48, 49, 52, 55, 58, 61, 66, 69, 73, 77, 81, 85, 91

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A001405.

A161253 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 14 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 2, 4, 4, 5, 4, 7, 5, 7, 7, 8, 7, 10, 8, 11, 11, 12, 12, 15, 13, 16, 16, 17, 17, 20, 18, 21, 21, 23, 22, 26, 24, 27, 27, 29, 28, 32, 30, 33, 33, 35, 34, 40, 37, 41, 42, 45, 45, 52, 50, 55, 57, 61, 62, 70, 69, 76

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A001405.

A048683 Values of n for which the difference of maximal and central squarefree kernel numbers dividing values of {C(n,k)} or A001405(n) is zero.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 23, 24, 31, 32, 33, 35, 36, 40, 41, 42, 55, 56, 57, 59, 65, 71, 72, 73, 80, 84, 100, 108, 109, 112, 113, 114, 115, 131, 132, 133, 155, 160, 161, 162, 163, 168, 183, 184, 199, 200, 201, 203, 209, 220, 224, 256

Offset: 1

Labos Elemer

nonn

Indices of 0's in A048682. - Sean A. Irvine, Jun 26 2021

			For n=23 both the maximal and central largest-squarefree number dividing the corresponding {C(23,k)} values is 1352078=2*7*13*17*19*23=C(23,12) accidentally. The same 1352078 is the maximal-largest squarefree divisor for C(24,k) values but 1352078=C(24,12)/2. Thus both 23 and 24 are in this sequence.

Analogous cases for A001221, A001222 functions as applied to {C(n, k)} are given in A020731 and A048627.

Cf. A048682.

max{sqf kernel(C(n, k)} - sqf kernel(C(n, [ n/2 ])) = 0

More terms from Sean A. Irvine, Jun 26 2021

A056611 Quotient: squarefree kernel of A002944(n) divided by that of A001405.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 5, 5, 5, 7, 7, 7, 7, 21, 21, 15, 5, 55, 165, 33, 33, 143, 143, 1001, 1001, 1001, 1001, 91, 91, 221, 221, 221, 221, 323, 323, 323, 323, 2261, 2261, 24871, 24871, 572033, 572033, 81719, 81719, 24035, 24035, 312455, 312455, 85215

Offset: 1

Labos Elemer, Aug 07 2000

nonn

			n=14,LCM[1,14,....,14,1]=24024,its sqf kernel is 6006; binomial[14,7]=3432,its sqf kernel is 858.a(14)=6006/858=7

A002944, A001405, A007947,

a(n)=A007947[A002944(n)]/A007947[A001405(n)]

A056646 a(n) = A056622(A001405(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 2, 2, 1, 3, 3, 1, 2, 1, 2, 2, 1, 1, 2, 10, 5, 10, 5, 3, 12, 3, 3, 1, 2, 5, 10, 10, 5, 3, 6, 2, 1, 2, 1, 5, 20, 15, 30, 42, 21, 14, 7, 7, 28, 2, 1, 1, 4, 1, 4, 4, 2, 21, 21, 7, 14, 7, 14, 6, 3, 1, 2, 2, 1, 10, 5, 35, 140, 7, 14, 126, 63, 2, 1, 5, 20, 90, 45, 3, 12

Offset: 1

Labos Elemer, Aug 09 2000

nonn

Previous name "Square root of largest unitary square divisor of central binomial coefficient" was incorrect. See A376554 for the correct sequence with this name. - Amiram Eldar, Sep 28 2024

			a(28) = A056622(binomial(28,14)) = A056622(40116600) = 5.

Cf. A000188, A001405, A055229, A056056, A056059, A056622, A376554.

a(n) = A000188(A001405(n))/A055229(A001405(n)) = A056056(n)/A056059(n).

Incorrect name replaced with a formula by Amiram Eldar, Sep 28 2024

cp's OEIS Frontend

A161247 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 8 times.

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A161248 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 9 times.

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A161249 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 10 times.

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A161250 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 11 times.

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A161251 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 12 times.

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A161252 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 13 times.

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A161253 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 14 times.

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A048683 Values of n for which the difference of maximal and central squarefree kernel numbers dividing values of {C(n,k)} or A001405(n) is zero.

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A056611 Quotient: squarefree kernel of A002944(n) divided by that of A001405.

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A056646 a(n) = A056622(A001405(n)).

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