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%I #8 Mar 24 2017 00:47:52
%S 9,22,23,22,42,37,40,90,63,96,147,120,111,134,237,166,219,304,214,279,
%T 254,252,369,484,399,520,429,270,519,481,709,426,793,581,611,734,661,
%U 691,1003,615,1087,914,1129,647,707,1094,1339,1130,1032,1423,915,1140
%N 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 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 For n=2 the corresponding value is not known and >=25; moreover, we do not know if this value is finite.
%C 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 V. Shevelev, <a href="http://journals.impan.gov.pl/aa/Inf/126-3-1.html">Compact integers and factorials</a>, Acta Arithmetica 126 (2007), no. 3, 195-236.
%e For n=3, the 9 values of m are 7, 8, 9, 10, 11, 12, 13, 14, and 20.
%e m=6, for example, is not counted because 6!=2^4*3^2*5 does not contain prime(4)=7.
%e 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 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 Cf. A000142, A177355, A177349, A177378, A177436.
%K nonn
%O 3,1
%A _Vladimir Shevelev_, May 09 2010, May 10 2010
%E Edited, example and relation to A115627 added, terms after 120 added - _R. J. Mathar_, Oct 29 2010
%E Extended by T. D. Noe, Apr 10 2012
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