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A341528
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a(n) = n * sigma(A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of the divisors of n.
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19
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1, 8, 18, 52, 40, 144, 84, 320, 279, 320, 154, 936, 234, 672, 720, 1936, 340, 2232, 456, 2080, 1512, 1232, 690, 5760, 1425, 1872, 4212, 4368, 928, 5760, 1178, 11648, 2772, 2720, 3360, 14508, 1554, 3648, 4212, 12800, 1804, 12096, 2064, 8008, 11160, 5520, 2538, 34848, 6517, 11400, 6120, 12168, 3180, 33696, 6160, 26880
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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) * (q^(e+1)-1)/(q-1) where q = nextPrime(p).
(End)
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} p^3/((p+1)*(p^2-nextprime(p))) = 2.26342530..., where nextprime is A151800. - Amiram Eldar, Dec 08 2022
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MATHEMATICA
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Array[#1 DivisorSigma[1, #2] & @@ {#, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &, 56] (* Michael De Vlieger, Feb 22 2021 *)
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PROG
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(PARI)
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
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CROSSREFS
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Cf. A000203, A003961, A003973, A016754 (positions of the odd terms), A151800, A341512, A341526, A341527, A341529, A341530, A342661, A342662, A342673.
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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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