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A354827
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Numerators of Dirichlet inverse of fraction A003961(n) / sigma(n).
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4
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1, -1, -5, -2, -7, 5, -11, -8, -75, 7, -13, 5, -17, 11, 35, -1648, -19, 75, -23, 1, 55, 13, -29, 2, -245, 17, -225, 11, -31, -35, -37, -1664, 65, 19, 77, 75, -41, 23, 85, 4, -43, -55, -47, 13, 175, 29, -53, 412, -847, 245, 95, 17, -59, 225, 91, 11, 23, 31, -61, -5, -67, 37, 825, -7662464, 17, -65, -71, 19, 145, -77
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OFFSET
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1,3
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COMMENTS
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Because the ratio A003961(n) / A000203(n) is multiplicative, so is also its Dirichlet inverse (which also is a sequence of rational numbers). This sequence gives the numerators when presented in its lowest terms, while A354828 gives the denominators. See the examples.
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LINKS
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EXAMPLE
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The ratio a(n)/A354828(n) for n = 1..21: 1, -1, -5/4, -2/7, -7/6, 5/4, -11/8, -8/35, -75/208, 7/6, -13/12, 5/14, -17/14, 11/8, 35/24, -1648/7595, -19/18, 75/208, -23/20, 1/3, 55/32.
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PROG
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(PARI)
up_to = 65537;
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
AuxA354827(n) = (A003961(n)/sigma(n));
vDirInv = DirInverseCorrect(vector(up_to, n, AuxA354827(n)));
A354827(n) = numerator(vDirInv[n]);
A354828(n) = denominator(vDirInv[n]);
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CROSSREFS
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KEYWORD
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sign,frac
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AUTHOR
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STATUS
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approved
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