cp's OEIS Frontend

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

%I A075254 #35 Jul 20 2025 03:26:15
%S A075254 1,4,6,8,10,11,14,14,15,17,22,19,26,23,23,24,34,26,38,29,31,35,46,33,
%T A075254 35,41,36,39,58,40,62,42,47,53,47,46,74,59,55,51,82,54,86,59,56,71,94,
%U A075254 59,63,62,71,69,106,65,71,69,79,89,118,72,122,95,76,76,83,82,134,89,95,84,142
%N A075254 a(n) = n + (sum of prime factors of n taken with repetition).
%C A075254 a(n) = n + A001414(n).
%C A075254 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
%H A075254 Reinhard Zumkeller, <a href="/A075254/b075254.txt">Table of n, a(n) for n = 1..10000</a>
%F A075254 From _Gus Wiseman_, Jan 26 2025: (Start)
%F A075254 First differences are 1 - A090340(n).
%F A075254 a(n) = 2*n - A075255(n).
%F A075254 a(n) = 2*A001414(n) + A075255(n).
%F A075254 (End)
%e A075254 a(6)=11 because 6=2*3, sopfr(6)=2+3=5 and 6+5=11.
%p A075254 A075254 := proc(n)
%p A075254     n+A001414(n) ;
%p A075254 end proc: # _R. J. Mathar_, Jul 27 2015
%t A075254 Table[If[n==1,1, n +Plus@@Times@@@FactorInteger@n], {n, 80}] (* _G. C. Greubel_, Jan 10 2019 *)
%o A075254 (Haskell)
%o A075254 a075254 n = n + a001414 n  -- _Reinhard Zumkeller_, Feb 27 2012
%o A075254 (PARI) a(n) = my(f = factor(n)); n + sum(k=1, #f~, f[k,1]*f[k,2]); \\ _Michel Marcus_, Feb 22 2017
%o A075254 (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
%o A075254 (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
%Y A075254 A000027 gives product of prime factors, indices A003963.
%Y A075254 A000040 lists the primes, differences A001223.
%Y A075254 A001414 gives sum of prime factors, indices A056239.
%Y A075254 A027746 lists prime factors, indices A112798, count A001222.
%Y A075254 A075255 gives product of prime factors minus sum of prime factors.
%Y A075254 Cf. A000720, A001055, A001222, A008472, A090340, A096461 (iteration), A175508, A319000, A325036, A325037, A325038, A379681, A379682.
%K A075254 nonn,look
%O A075254 1,2
%A A075254 _Zak Seidov_, Sep 10 2002