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%I #35 Mar 23 2024 21:10:54
%S 0,1,1,3,1,3,1,5,3,3,1,6,1,3,3,8,1,6,1,6,3,3,1,9,3,3,5,6,1,7,1,10,3,3,
%T 3,11,1,3,3,9,1,7,1,6,6,3,1,13,3,6,3,6,1,9,3,9,3,3,1,12,1,3,6,14,3,7,
%U 1,6,3,7,1,15,1,3,6,6,3,7,1,13,8,3,1,12
%N a(n) is the total number of trailing zeros in the representations of n over all bases b >= 2.
%C a(n) depends only on the prime signature of n.
%H Antti Karttunen, <a href="/A309891/b309891.txt">Table of n, a(n) for n = 1..16384</a>
%H Antti Karttunen, <a href="/A309891/a309891_1.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F a(n) = Sum_{d|n, d>1} A286561(n,d), where A286561 gives the d-valuation of n.
%F a(p) = 1 for any prime number p.
%F a(p^k) = A006218(k) for any k >= 0 and any prime number p.
%F a(n) = 2^A001221(n) - 1 for any squarefree number n.
%F a(n) = 3 for any semiprime number n.
%F a(m*n) >= a(m) + a(n).
%F a(n) >= A007814(n) + A007949(n) + A235127(n) + A112765(n) + A122841(n) + A214411(n) + A244413(n).
%F a(n) = A056239(A293514(n)). - _Antti Karttunen_, Aug 22 2019
%F a(n) <= A033093(n). - _Michel Marcus_, Aug 22 2019
%F a(n) = A169594(n) - 1. - _Jon Maiga_, Aug 25 2019
%F From _Friedjof Tellkamp_, Feb 27 2024: (Start)
%F G.f.: Sum_{k>=2, j>=1} x^(k^j)/(1-x^(k^j)).
%F Dirichlet g.f.: zeta(s) * Sum_{k>=1} (zeta(k*s) - 1).
%F Sum_{n>=1} a(n)/n^2 = Pi^2/8. (End)
%e For n = 12: 12 has 2 trailing zeros in base 2 (1100), 1 trailing zero in bases 3, 4, 6 and 12 (110, 30, 20, 10) and no trailing zero in other bases, hence a(12) = 1*2 + 4*1 = 6.
%t Table[DivisorSum[n, IntegerExponent[n, #] &, # > 1 &], {n, 84}] (* _Jon Maiga_, Aug 25 2019 *)
%o (PARI) a(n) = sumdiv(n, d, if (d>1, valuation(n,d), 0))
%o (PARI) a(n) = {if(n == 1, return(0)); my(f = factor(n)[, 2], res = 0, t = 2, of = f, nf = f >> 1, nd(v) = prod(i = 1, #v, v[i] + 1)); while(Set(of) != [0], res += (nd(of) - nd(nf)) * (t-1); of = nf; t++; nf = f \ t); res} \\ _David A. Corneth_, Aug 22 2019
%Y Cf. A001221, A006218, A007814, A007949, A033093, A056239, A112765, A122841, A169594, A214411, A235127, A244413, A286561, A293514, A369180.
%K nonn,base
%O 1,4
%A _Rémy Sigrist_, Aug 21 2019
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