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

1,1

Comments

One can prove that for p>=5 the number of such m is 0 or 2.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126 (2007) 195-236.

Example

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]

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

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 *)

Crossrefs

Cf. A000142.

Sequence in context: A182198 A291275 A291278 * A160215 A068486 A099332

Adjacent sequences: A177346 A177347 A177348 * A177350 A177351 A177352

Keyword

nonn

Author

Vladimir Shevelev, May 07 2010

Extensions

2 added, 109 replaced by 107, sequence extended beyond 199 by R. J. Mathar, Oct 29 2010

Status

approved