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A342662
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a(n) = sigma(n) * A064989(n), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma is the sum of the divisors of n.
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7
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1, 3, 8, 7, 18, 24, 40, 15, 52, 54, 84, 56, 154, 120, 144, 31, 234, 156, 340, 126, 320, 252, 456, 120, 279, 462, 320, 280, 690, 432, 928, 63, 672, 702, 720, 364, 1178, 1020, 1232, 270, 1554, 960, 1804, 588, 936, 1368, 2064, 248, 1425, 837, 1872, 1078, 2538, 960, 1512, 600, 2720, 2070, 3180, 1008, 3658, 2784, 2080, 127
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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) = q^e * (p^(e+1)-1)/(p-1), where q = 1 for p = 2, and for odd primes p, q = A151799(p), i.e., the previous prime.
Sum_{k=1..n} a(k) ~ c * n^3, where c = (16/63) * Product_{p prime > 2} (p^4*(p-1)/((p^3-prevprime(p))*(p^2-prevprime(p))) = 0.1935405..., where prevprime is A151799. - Amiram Eldar, Dec 24 2022
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MATHEMATICA
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f[p_, e_] := If[p == 2, 1, NextPrime[p, -1]^e]*(p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)
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PROG
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(PARI)
A064989(n) = { my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f) };
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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