A177349 Primes p for which no m! has a prime power factorization of the form 2^p*...*p^1*...
2, 5, 13, 17, 29, 37, 43, 59, 61, 83, 103, 107, 139, 151, 157, 163, 167, 179, 199, 211, 223, 227, 233, 241, 251, 257, 269, 283, 307, 313, 317, 331, 347, 373, 379, 409, 419, 433, 443, 457, 503, 509, 523, 541, 547, 563, 569, 571, 587, 601, 607, 617, 619, 643
Offset: 1
Keywords
Examples
p=109 is not in the sequence because for m=112 we have 112! = 2^109*3^54*...*109 which is of the form 2^p*...*p*.. [_R. J. Mathar_, Oct 29 2010]
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126 (2007) 195-236.
Crossrefs
Cf. A000142.
Programs
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Maple
isA177349 := proc(p) if isprime(p) then pid := numtheory[pi](p) ; for m from 1 do h2 := A115627(m,1) ; if h2 > p then return true; elif h2 = p then if A115627(m,pid) = 1 then return false; end if; end if; end do; else false; fi ; end proc: for i from 1 to 120 do p := ithprime(i) ; if isA177349(p) then printf("%d,",p); end if; end do: # R. J. Mathar, Oct 29 2010
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Mathematica
c[n_, p_] := Sum[IntegerExponent[k, p], {k, 2, n}]; m = 120; v = Table[0, {m}]; s = 0; n = 2; While[s <= Prime[m], s += IntegerExponent[n, 2]; If[ PrimeQ[s] && (i = PrimePi[s]) <= m && c[n, s] == 1, v[[i]] = 1]; n++]; t = Prime /@ (Position[v, ?(# == 0 &)] // Flatten); t (* _Amiram Eldar, Sep 13 2019 *)
Extensions
2 added, 109 replaced by 107, sequence extended beyond 199 by R. J. Mathar, Oct 29 2010
Comments