A049042 -id:A049042 - cp's OEIS frontend

A049041 Least k > 0 such that A049042(n) | A003422(k-1).

2, 4, 6, 6, 5, 7, 7, 12, 22, 16, 55, 54, 42, 24, 25, 86, 97, 133, 64, 94, 72, 58, 49, 69, 19, 78, 14, 208, 167, 138, 80, 59, 63, 142, 41, 110, 22, 286, 39, 84, 215, 80, 14, 305, 188, 151, 53, 187, 180, 44, 32, 83, 92, 300, 16, 421, 146, 507, 28, 243, 119, 429, 239, 415

Offset: 1

David W. Wilson

nonn

F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House, 2000.

F. Smarandache, Definitions, Solved and Unsolved Problems, Conjectures, ...
Eric Weisstein's World of Mathematics, Smarandache-Kurepa Function.

Cf. A049042, A049043, A003422.

A049043 Primes not in domain of A049041.

Original entry on oeis.org

3, 13, 29, 43, 47, 53, 59, 67, 79, 83, 97, 101, 107, 109, 127, 137, 149, 151, 179, 181, 199, 223, 229, 241, 269, 271, 277, 281, 311, 347, 353, 359, 389, 421, 433, 439, 443, 449, 479, 499, 509, 563, 587, 593, 599, 607, 613, 631, 643, 653, 659, 683, 709, 739

Offset: 1

David W. Wilson

nonn

Primes not in A049042.

Cf. A003422, A049041.

A101548 Number of k such that prime(n) divides the left factorial !k = sum_{i=0..k-1} i!.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 3, 1, 0, 2, 1, 2, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 2, 1, 1, 2, 0, 0, 2, 2, 1, 0, 3, 0, 3, 0, 1, 1, 0, 1, 1, 2, 0, 0, 0, 0, 2, 2, 3, 0, 1, 1, 2, 1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 2, 4, 1, 2, 0, 1, 3, 0, 1, 0, 1, 1, 1, 1, 2, 0, 1, 2, 1

Offset: 2

T. D. Noe, Dec 06 2004

nonn

Note that 2 divides every left factorial !k for k>1. A result of Barsky and Benzaghou shows that there is no odd prime p such that p divides !p. Hence if an odd prime p divides !k then we must have k < p.

			a(8) = 3 because 19 divides !7, !12 and !16.

D. Barsky and B. Benzaghou, Nombres de Bell et somme de factorielles, Journal de Théorie des Nombres de Bordeaux, 16:1-17, 2004.
Bernd C. Kellner, Some remarks on Kurepa's left factorial, arXiv:math/0410477 [math.NT], 2004.

Cf. A003422 (left factorials), A049042 (primes dividing some left factorial), A049043 (primes not dividing any left factorial).

nn=1000; s=0; t=Table[s=s+n!, {n, 0, nn}]; Table[p=Prime[i]; Length[Position[t, _?(0==Mod[ #, p]&)]], {i, 2, PrimePi[nn]}]

A335869 Primitive terms of A275608.

Original entry on oeis.org

3, 8, 13, 20, 25, 28, 29, 35, 43, 44, 47, 49, 53, 55, 59, 67, 68, 76, 79, 83, 85, 92, 95, 97, 101, 107, 109, 115, 119, 121, 124, 127, 133, 137, 148, 149, 151, 155, 161, 164, 179, 181, 185, 187, 199, 205, 209, 217, 223, 229, 241, 244, 253, 259, 269, 271, 277

Offset: 1

J. Lowell, Jun 27 2020

nonn

			8 is in the sequence because 8 is in A275608, but none of 1, 2, and 4 are in A275608.

Cf. A003422, A049042, A049043, A275608.

cp's OEIS Frontend

A049041 Least k > 0 such that A049042(n) | A003422(k-1).

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A049043 Primes not in domain of A049041.

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A101548 Number of k such that prime(n) divides the left factorial !k = sum_{i=0..k-1} i!.

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A335869 Primitive terms of A275608.

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