A075254 a(n) = n + (sum of prime factors of n taken with repetition).

1, 4, 6, 8, 10, 11, 14, 14, 15, 17, 22, 19, 26, 23, 23, 24, 34, 26, 38, 29, 31, 35, 46, 33, 35, 41, 36, 39, 58, 40, 62, 42, 47, 53, 47, 46, 74, 59, 55, 51, 82, 54, 86, 59, 56, 71, 94, 59, 63, 62, 71, 69, 106, 65, 71, 69, 79, 89, 118, 72, 122, 95, 76, 76, 83, 82, 134, 89, 95, 84, 142

Offset: 1

Author

Zak Seidov, Sep 10 2002

Keywords

• nonn
• look

Comments

a(n) = n + A001414(n).

Product of prime factors plus sum of prime factors of n. For minus instead of plus we have A075255, zeros A175787. - Gus Wiseman, Jan 26 2025

Examples

			a(6)=11 because 6=2*3, sopfr(6)=2+3=5 and 6+5=11.
		

Links

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

Crossrefs

A000027 gives product of prime factors, indices A003963.

A000040 lists the primes, differences A001223.

A001414 gives sum of prime factors, indices A056239.

A027746 lists prime factors, indices A112798, count A001222.

A075255 gives product of prime factors minus sum of prime factors.

Cf. A000720, A001055, A001222, A008472, A090340, A096461 (iteration), A175508, A319000, A325036, A325037, A325038, A379681, A379682.

Programs

Haskell

a075254 n = n + a001414 n  -- Reinhard Zumkeller, Feb 27 2012

Magma

[n eq 1 select 1 else (&+[p[1]*p[2]: p in Factorization(n)]) + n: n in [1..80]]; // G. C. Greubel, Jan 10 2019

Maple

A075254 := proc(n)
    n+A001414(n) ;
end proc: # R. J. Mathar, Jul 27 2015

Mathematica

Table[If[n==1,1, n +Plus@@Times@@@FactorInteger@n], {n, 80}] (* G. C. Greubel, Jan 10 2019 *)

PARI

a(n) = my(f = factor(n)); n + sum(k=1, #f~, f[k,1]*f[k,2]); \\ Michel Marcus, Feb 22 2017

Sage

[n + sum(factor(n)[j][0]*factor(n)[j][1] for j in range(0, len(factor(n)))) for n in range(1, 80)] # G. C. Greubel, Jan 10 2019

Formula

From Gus Wiseman, Jan 26 2025: (Start)

First differences are 1 - A090340(n).

a(n) = 2*n - A075255(n).

a(n) = 2*A001414(n) + A075255(n).

(End)