A175729 Numbers n such that the sum of the prime factors with multiplicity of n divides n-1.

6, 21, 45, 52, 225, 301, 344, 441, 697, 1225, 1333, 1540, 1625, 1680, 1695, 1909, 2025, 2041, 2145, 2295, 2466, 2601, 2926, 3051, 3104, 3146, 3400, 3510, 3738, 3888, 3901, 4030, 4186, 4251, 4375, 4641, 4675, 4693, 4930, 5005, 5085, 5244, 5425, 6025, 6105

Offset: 1

Author

K. T. Lee (7x3(AT)21cn.com), Aug 23 2010

Keywords

• easy
• nonn

Examples

			For example, 21=7x3, 7+3=10 which divides 21-1=20.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Disjoint from A130871 and A046346.

Cf. A001414.

Contains A164643.

Programs

Magma

[k:k in [2..6200]| IsIntegral((k-1)/( &+[m[1]*m[2]: m in Factorization(k)]))]; // Marius A. Burtea, Sep 16 2019

Maple

A001414 := proc(n) ifactors(n)[2] ; add( op(1,p)*op(2,p),p=%) ; end proc:
isA175729 := proc(n) if (n-1) mod A001414(n) = 0 then true; else false; end if; end proc:
for n from 2 to 10000 do if isA175729(n) then printf("%d,",n) ; end if; end do:
# R. J. Mathar, Aug 24 2010

Mathematica

fQ[n_] := Mod[n - 1, Plus @@ Flatten[ Table[ #1, {#2}] & @@@ FactorInteger@ n]] == 0; Select[ Range@ 6174, fQ] (* Robert G. Wilson v, Aug 25 2010 *)

Python

from sympy import factorint
def ok(n): return n>1 and (n-1)%sum(p*e for p, e in factorint(n).items())==0
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Sep 30 2022

Formula

{n : A001414(n) | (n-1)}. [R. J. Mathar, Aug 24 2010]

Extensions

Extended by R. J. Mathar and Robert G. Wilson v, Aug 24 2010