A343983 -id:A343983 - cp's OEIS frontend

A344029 Numbers included in A343983 but not in A074583.

72, 2025, 78447, 5922181, 84238825, 1141011175

Offset: 1

Seiichi Manyama, May 07 2021

			If n is in A074583, n can be expressed as n = p^e (p>=e) using the prime p.
On the other hand, the terms of this sequence are factorized as follows.
72 = 2^3 * 3^2.
2025 = 3^4 * 5^2.
78447 = 3 * 79 * 331.
5922181 = 71 * 239 * 349.
84238825 = 5^2 * 11 * 17 * 37 * 487.

Cf. A074583, A343983.

isok(n) = my(f=factor(n)); sumdiv(n, d, Mod(d, n)^d)==1 && n>1 && !(#f~==1 && f[1, 1]>=f[1, 2]);

a(6) from Seiichi Manyama, Aug 01 2023

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

Original entry on oeis.org

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

Jason Earls, Aug 24 2002

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

Complement to A192135 with respect to A000961;

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

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

Cf. A095874, A192135, A192188, A343983.

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

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 *)

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

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

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A344029 Numbers included in A343983 but not in A074583.

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

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