A177458 The number of positive integers m for which the exponents of prime(n) and prime(n+1) in the prime power factorization of m! are both powers of 2.

9, 22, 23, 22, 42, 37, 40, 90, 63, 96, 147, 120, 111, 134, 237, 166, 219, 304, 214, 279, 254, 252, 369, 484, 399, 520, 429, 270, 519, 481, 709, 426, 793, 581, 611, 734, 661, 691, 1003, 615, 1087, 914, 1129, 647, 707, 1094, 1339, 1130, 1032, 1423, 915, 1140

Offset: 3

Author

Vladimir Shevelev, May 09 2010, May 10 2010

Keywords

• nonn

Comments

This gives the number of rows in A115627 for which the n-th and (n+1)st column are both in {1,2,4,8,16,..}.

For n=2 the corresponding value is not known and >=25; moreover, we do not know if this value is finite.

A more general result concerning the cases for non-adjacent primes and a finite search interval for the values of m is in the 2007 publication.

Examples

			For n=3, the 9 values of m are 7, 8, 9, 10, 11, 12, 13, 14, and 20.
m=6, for example, is not counted because 6!=2^4*3^2*5 does not contain prime(4)=7.
m=15, for example, is not counted because 15!=2^11*3^6*5^3*7^2*11*13 contains a third power of prime(3)=5.
		

Links

• Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126 (2007), no. 3, 195-236.

Crossrefs

Cf. A000142, A177355, A177349, A177378, A177436.

Programs

Mathematica

tp[n_] := Flatten[Position[FoldList[Plus, 0, IntegerExponent[Range[100000], n]], ?(IntegerQ[Log[2, #]] &)]]; Table[s = Intersection[tp[Prime[n]], tp[Prime[n + 1]]] - 1; Length[s], {n, 3, 60}] (* _T. D. Noe, Apr 10 2012 *)

Extensions

Edited, example and relation to A115627 added, terms after 120 added by R. J. Mathar, Oct 29 2010

Extended by T. D. Noe, Apr 10 2012