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

3,1

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.

Links

Table of n, a(n) for n=3..54.

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A368120 A250729 A251292 * A228009 A330477 A295008

Adjacent sequences: A177455 A177456 A177457 * A177459 A177460 A177461

Keyword

nonn

Author

Vladimir Shevelev, May 09 2010, May 10 2010

Extensions

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

Extended by T. D. Noe, Apr 10 2012

Status

approved