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A177458
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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.
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5
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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
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OFFSET
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3,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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