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A309891
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a(n) is the total number of trailing zeros in the representations of n over all bases b >= 2.
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4
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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, 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, 1, 6, 3, 7, 1, 15, 1, 3, 6, 6, 3, 7, 1, 13, 8, 3, 1, 12
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OFFSET
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1,4
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COMMENTS
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a(n) depends only on the prime signature of n.
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LINKS
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FORMULA
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a(n) = Sum_{d|n, d>1} A286561(n,d), where A286561 gives the d-valuation of n.
a(p) = 1 for any prime number p.
a(p^k) = A006218(k) for any k >= 0 and any prime number p.
a(n) = 2^A001221(n) - 1 for any squarefree number n.
a(n) = 3 for any semiprime number n.
a(m*n) >= a(m) + a(n).
G.f.: Sum_{k>=2, j>=1} x^(k^j)/(1-x^(k^j)).
Dirichlet g.f.: zeta(s) * Sum_{k>=1} (zeta(k*s) - 1).
Sum_{n>=1} a(n)/n^2 = Pi^2/8. (End)
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EXAMPLE
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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.
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MATHEMATICA
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Table[DivisorSum[n, IntegerExponent[n, #] &, # > 1 &], {n, 84}] (* Jon Maiga, Aug 25 2019 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, if (d>1, valuation(n, d), 0))
(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
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CROSSREFS
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Cf. A001221, A006218, A007814, A007949, A033093, A056239, A112765, A122841, A169594, A214411, A235127, A244413, A286561, A293514, A369180.
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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