A061768 -id:A061768 - cp's OEIS frontend

A061770 Numbers m = a(n) > a(n-1) such that there exists a smallest integer k > 1 such that k!/(k+1)^m is an integer.

0, 1, 2, 5, 7, 8, 9, 10, 11, 14, 17, 19, 21, 28, 35, 44, 58, 88, 95, 103, 110, 178, 179, 185, 208, 222, 287, 313, 334, 358, 371, 419, 479, 502, 558, 629, 670, 718, 838, 1006, 1118, 1259, 1438

Offset: 0

Robert G. Wilson v, Jun 21 2001

nonn

Original name: The least exponent m = a(n) > a(n-1) for which k is the first number where k!/(k+1)^m is an integer.

			a(5) = 8 because the first integer k > 1 such that (k+1)^8 divides k! is k = 39, which is larger than the first integer k > 1 such that (k+1)^7 divides k! (k = 35).
6 is not in the sequence because the first integer k > 1 such that (k+1)^6 divides k! is k = 23, which is equal to the first integer k > 1 such that (k+1)^5 divides k!.

Locations of records in A061768.

l = 0; Do[k = Max[l - 1, 1]; While[ !IntegerQ[ k! / (k + 1)^n], k++ ]; If[ k > l, l = k; Print[n] ], {n, 0, 1500} ]

b(n)=k=2;while(k!%(k+1)^n,k++);k
print1(0,", ");for(n=1,100,if(b(n)>b(n-1),print1(n,", "))) \\ Derek Orr, Apr 16 2015

Name and example edited by Derek Orr, Apr 16 2015

A061764 Positive integers k such that k! is divisible by (k+1)^12.

Original entry on oeis.org

59, 71, 79, 83, 89, 95, 104, 107, 111, 119, 125, 127, 131, 134, 139, 143, 149, 153, 159, 161, 164, 167, 174, 175, 179, 181, 188, 191, 194, 195, 197, 199, 207, 209, 215, 219, 220, 223, 224, 230, 233, 237, 239, 242, 244, 246, 249

Offset: 1

Robert G. Wilson v, Jun 21 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A061743, A061751, A061752, A061753, A061754, A061755, A061756, A061757, A061758, A061759, A061768.

Select[Range[250], IntegerQ[ #!/(# + 1)^12] &]

A306772 a(n) is the least number k such that k! is divisible by (k+1)^n but not by (k+1)^(n+1).

Original entry on oeis.org

1, 5, 14, 17, 11, 31, 23, 35, 39, 44, 47, 99, 83, 59, 153, 164, 71, 95, 79, 125, 89, 134, 285, 199, 311, 263, 167, 119, 296, 188, 159, 329, 543, 209, 143, 223, 299, 384, 395, 323, 251, 679, 349, 179, 279, 747, 571, 485, 399, 404, 314, 527, 319, 335, 449, 511, 287, 239, 714

Offset: 0

Jinyuan Wang, Mar 09 2019

nonn

k+1 is not a prime.
a(n) + 1 is 17-smooth in DATA. - David A. Corneth, Mar 15 2019
But fails at n 99, 114, 125, 127, 130, 135, 143, 146, ... - Michel Marcus, Apr 30 2019

			For n = 1, 1! = 1 is not divisible by 2, 2! = 2 is not divisible by 3, 3! = 6 is not divisible by 4, 4! = 24 is not divisible by 5, and 5! = 120 is divisible by 6 but not 36. Therefore a(1) = 5. - _Michael B. Porter_, Apr 21 2019

Cf. A061768, A082482, A133481, A240751.

Array[Block[{k = 1}, While[Nand[Mod[k!, (k + 1)^#] == 0, Mod[k!, (k + 1)^(# + 1)] != 0], k++]; k] &, 58] (* Michael De Vlieger, Mar 11 2019 *)

a(n) = {my(k=1); while((k! % (k+1)^n) || !(k! % (k+1)^(n+1)), k++); k; }

a(n) = A133481(n+1) - 1.
a(n) >= A061768(n).
If n = floor((p^j-1)/(j*(p-1)))-1, a(n) <= p^j-1 for prime p. For example, (p = 2), a(n) <= 2^j-1 for n = floor((2^j-1)/j)-1 (A082482(j)-1).

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A061770 Numbers m = a(n) > a(n-1) such that there exists a smallest integer k > 1 such that k!/(k+1)^m is an integer.

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A061764 Positive integers k such that k! is divisible by (k+1)^12.

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A306772 a(n) is the least number k such that k! is divisible by (k+1)^n but not by (k+1)^(n+1).

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