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A361683
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a(n) is the least k such that tau(k) divides sigma_n(k) but not sigma(k), or -1 if no such k exists.
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0
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4, 64, 4, 7168, 4, 606528, 4, 64, 4, 4194304, 4
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OFFSET
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2,1
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COMMENTS
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a(17) <= 15211807202738752817960438464512 and a(19) <= 2^190*11.
Conjecture: a(n) is of the form 2^b*p1^c*p2^d*...*pk^j with b > 0 and A020639(n) divides b*(c+1)*(d+1)*...*(j+1). (p1, p2, ..., pk are distinct odd prime numbers). (End)
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LINKS
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FORMULA
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a(2*m) = 4 for m >= 1.
a(6*m-3) = 64 for m >= 1.
Conjecture: For primes q > p, a(q) > a(p). If true, we could replace "<=" with "=" in the above formula. (End)
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MATHEMATICA
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a[n_] := Module[{k = 1, d}, While[Divisible[DivisorSigma[1, k], (d = DivisorSigma[0, k])] || !Divisible[DivisorSigma[n, k], d], k++]; k]; Array[a, 11, 2] (* Amiram Eldar, Mar 20 2023 *)
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PROG
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(PARI) isok(k, n) = my(f=factor(k), nd=numdiv(f)); (sigma(f) % nd) && !(sigma(f, n) % nd);
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Mar 20 2023
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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