cp's OEIS Frontend

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.

%I A177458 #11 Jan 23 2025 10:23:32
%S A177458 9,22,23,22,42,37,40,90,63,96,147,120,111,134,237,166,219,304,214,279,
%T A177458 254,252,369,484,399,520,429,270,519,481,709,426,793,581,611,734,661,
%U A177458 691,1003,615,1087,914,1129,647,707,1094,1339,1130,1032,1423,915,1140
%N 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.
%C A177458 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,..}.
%C A177458 For n=2 the corresponding value is not known and >=25; moreover, we do not know if this value is finite.
%C A177458 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.
%H A177458 Vladimir Shevelev, <a href="https://eudml.org/doc/277854">Compact integers and factorials</a>, Acta Arithmetica 126 (2007), no. 3, 195-236.
%e A177458 For n=3, the 9 values of m are 7, 8, 9, 10, 11, 12, 13, 14, and 20.
%e A177458 m=6, for example, is not counted because 6!=2^4*3^2*5 does not contain prime(4)=7.
%e A177458 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.
%t A177458 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 *)
%Y A177458 Cf. A000142, A177355, A177349, A177378, A177436.
%K A177458 nonn
%O A177458 3,1
%A A177458 _Vladimir Shevelev_, May 09 2010, May 10 2010
%E A177458 Edited, example and relation to A115627 added, terms after 120 added by _R. J. Mathar_, Oct 29 2010
%E A177458 Extended by _T. D. Noe_, Apr 10 2012