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A347236
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a(n) = Sum_{d|n} A061019(d) * A003961(n/d), where A061019 negates the primes in the prime factorization, while A003961 shifts the factorization one step towards larger primes.
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7
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1, 1, 2, 7, 2, 2, 4, 13, 19, 2, 2, 14, 4, 4, 4, 55, 2, 19, 4, 14, 8, 2, 6, 26, 39, 4, 68, 28, 2, 4, 6, 133, 4, 2, 8, 133, 4, 4, 8, 26, 2, 8, 4, 14, 38, 6, 6, 110, 93, 39, 4, 28, 6, 68, 4, 52, 8, 2, 2, 28, 6, 6, 76, 463, 8, 4, 4, 14, 12, 8, 2, 247, 6, 4, 78, 28, 8, 8, 4, 110, 421, 2, 6, 56, 4, 4, 4, 26, 8, 38, 16
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OFFSET
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1,3
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COMMENTS
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Dirichlet convolution of A003961 and A061019.
Dirichlet convolution of A003973 and A158523.
Multiplicative because A003961 and A061019 are.
All terms are positive because all terms of A347237 are nonnegative and A347237(1) = 1.
Union of sequences A001359 and A108605 (= 2*A001359) seems to give the positions of 2's in this sequence.
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..16383
Index entries for primes, gaps between
Index entries for sequences computed from indices in prime factorization
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FORMULA
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a(n) = Sum_{d|n} A003961(n/d) * A061019(d).
a(n) = Sum_{d|n} A003973(n/d) * A158523(d).
a(n) = Sum_{d|n} A347237(d).
a(n) = A347239(n) - A347238(n).
For all n >= 1, a(A000040(n)) = A001223(n).
Multiplicative with a(p^e) = (A151800(p)^(e+1)-(-p)^(e+1))/(A151800(p)+p). - Sebastian Karlsson, Sep 02 2021
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MATHEMATICA
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f[p_, e_] := ((np = NextPrime[p])^(e + 1) - (-p)^(e + 1))/(np + p); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 02 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); };
A061019(n) = (((-1)^bigomega(n))*n);
A347236(n) = sumdiv(n, d, A061019(d)*A003961(n/d));
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CROSSREFS
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Cf. A000040, A001223, A001359, A003961, A003973, A061019, A108605, A158523, A347237 (Möbius transform), A347238 (Dirichlet inverse), A347239.
Cf. also A347136.
Cf. A151800.
Sequence in context: A082066 A179931 A130335 * A073246 A021790 A266390
Adjacent sequences: A347233 A347234 A347235 * A347237 A347238 A347239
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KEYWORD
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nonn,mult
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AUTHOR
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Antti Karttunen, Aug 24 2021
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STATUS
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approved
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