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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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: A329007 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

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Last modified October 27 12:08 EDT 2021. Contains 348276 sequences. (Running on oeis4.)