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A338038 a(n) is the sum of the primes and exponents in the prime factorization of n, but ignoring 1-exponents. 6

%I #16 Oct 28 2021 02:00:01

%S 0,2,3,4,5,5,7,5,5,7,11,7,13,9,8,6,17,7,19,9,10,13,23,8,7,15,6,11,29,

%T 10,31,7,14,19,12,9,37,21,16,10,41,12,43,15,10,25,47,9,9,9,20,17,53,8,

%U 16,12,22,31,59,12,61,33,12,8,18,16,67,21,26,14,71,10

%N a(n) is the sum of the primes and exponents in the prime factorization of n, but ignoring 1-exponents.

%C First differs from A106492 for n=64.

%H Robert Israel, <a href="/A338038/b338038.txt">Table of n, a(n) for n = 1..10000</a>

%H Daniel Tsai, <a href="https://arxiv.org/abs/2010.03151">A recurring pattern in natural numbers of a certain property</a>, arXiv:2010.03151 [math.NT], 2020.

%H Daniel Tsai, <a href="http://math.colgate.edu/~integers/v32/v32.mail.html">A recurring pattern in natural numbers of a certain property</a>, Integers (2021) Vol. 21, Article #A32.

%F a(n) = A008474(n) for powerful numbers (A001694).

%e For n = 18 = 2*3^2, a(18) = 2 + (3+2) = 7.

%p f:= proc(n) local t;

%p add(t[1]+t[2],t=subs(1=0,ifactors(n)[2]));

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Oct 13 2020

%t a[1] = 0; a[n_] := Plus @@ First /@ (f = FactorInteger[n]) + Plus @@ Select[Last /@ f, # > 1 &]; Array[a, 100] (* _Amiram Eldar_, Oct 08 2020 *)

%o (PARI) a(n) = my(f=factor(n)); vecsum(f[,1]) + sum(k=1, #f~, if (f[k,2]!=1, f[k,2]));

%Y Cf. A008474, A001694, A106492, A338039.

%K nonn

%O 1,2

%A _Michel Marcus_, Oct 08 2020

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)