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A340774
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Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s-1) * zeta(2*s-1).
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11
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1, 2, 3, 6, 5, 6, 7, 12, 12, 10, 11, 18, 13, 14, 15, 28, 17, 24, 19, 30, 21, 22, 23, 36, 30, 26, 36, 42, 29, 30, 31, 56, 33, 34, 35, 72, 37, 38, 39, 60, 41, 42, 43, 66, 60, 46, 47, 84, 56, 60, 51, 78, 53, 72, 55, 84, 57, 58, 59, 90, 61, 62, 84, 120, 65, 66, 67
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OFFSET
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1,2
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LINKS
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FORMULA
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Multiplicative with a(p^e) = (p^(e+1)-p^floor((e+1)/2))/(p-1).
G.f.: Sum_{k>=1} k * x^(k^2) / (1 - x^(k^2))^2. - Ilya Gutkovskiy, Aug 19 2021
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MAPLE
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a:= n-> mul((i[1]^(i[2]+1)-i[1]^iquo(i[2]+1, 2))/(i[1]-1), i=ifactors(n)[2]):
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MATHEMATICA
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f[p_, e_] := (p^(e + 1) - p^Floor[(e + 1)/2])/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 20 2021 *)
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PROG
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(PARI) A340774(n) = { my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); ((p^(e+1)-(p^((e+1)\2))) / (p-1))); }; \\ Antti Karttunen, Aug 19 2021
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CROSSREFS
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Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k =
0..10: A046951 (k=0), this sequence (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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