A074583 Numbers k such that sopfr(k) = S(k), where sopfr = A001414 and S = A002034.

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Author

Jason Earls, Aug 24 2002

Keywords

• easy
• nonn

Comments

These are the prime powers p^e with e <= p. - Reinhard Zumkeller, Dec 15 2003

Complement to A192135 with respect to A000961;

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A000961; A000040, A000430, and A051674 are subsequences.

Cf. A095874, A192135, A192188, A343983.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a074583 n = a074583_list !! (n-1)
a074583_list = 1 : f (singleton 2) a000040_list where
  f s ps'@(p:p':ps)
    | m == p      = p : f (insert (p*p) $ insert p' s') (p':ps)
    | m < spf^spf = m : f (insert (m*spf) s') ps'
    | otherwise   = m : f s' ps'
      where spf = a020639 m  -- smallest prime factor of m, cf. A020639
            (m, s') = deleteFindMin s
-- Simpler version:
a074583_list = map a000961 a192188_list
-- Reinhard Zumkeller, Jun 05 2011, Jun 26 2011

Mathematica

sopfr[n_] := Total[Times @@@ FactorInteger[n]];
S[n_] := Module[{m = 1}, While[!IntegerQ[m!/n], m++]; m];
Select[Range[1000], sopfr[#] == S[#]&] (* Jean-François Alcover, Nov 09 2017 *)

PARI

isok(n) = my(f=factor(n)); n==1 || (#f~==1 && f[1, 1]>=f[1, 2]); \\ Seiichi Manyama, May 07 2021

Formula

a(n) = A000961(A192188(n)); A095874(a(n)) = A192188(n). - Reinhard Zumkeller, Jun 26 2011