cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A347096 a(1) = 1; a(n) = -Sum_{d|n, d < n} A341512(n/d) * a(d), where A341512(n) = sigma(n)*A003961(n) - n*sigma(A003961(n)).

Original entry on oeis.org

1, -1, -2, -10, -2, -32, -4, -64, -42, -54, -2, -214, -4, -112, -112, -316, -2, -469, -4, -412, -232, -168, -6, -792, -90, -262, -612, -860, -2, -1208, -6, -1216, -340, -354, -320, -1655, -4, -484, -532, -1760, -2, -2528, -4, -1438, -1850, -732, -6, 160, -364, -1863, -712, -2210, -6, -4596, -384, -3696, -976, -942, -2
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2021

Keywords

Comments

Dirichlet inverse of the pointwise sum of A341512 and A063524 (1, 0, 0, 0, ...).

Links

Crossrefs

Cf. A000203, A001223, A003961, A063524, A341512, A347097.
Cf. also A346235, A346239.

Programs

  • PARI
    up_to = 16384;
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341512(n) = { my(u=A003961(n)); ((sigma(n)*u) - (n*sigma(u))); };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA341512(n));
v347096 = DirInverseCorrect(vector(up_to,n,Aux347096(n)));
A347096(n) = v347096[n];

Formula

a(1) = 1; and for n > 1, a(n) = -Sum_{d|n, d < n} A341512(n/d) * a(d).
For all n >= 1, a(A000040(n)) = -A001223(n).

A346240 Difference between A341512 and its Möbius transform.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 11, 2, 3, 0, 46, 0, 5, 4, 85, 0, 80, 0, 68, 6, 3, 0, 398, 2, 5, 46, 130, 0, 209, 0, 575, 4, 3, 6, 981, 0, 5, 6, 640, 0, 397, 0, 182, 164, 7, 0, 2830, 4, 150, 4, 280, 0, 1435, 4, 1250, 6, 3, 0, 2586, 0, 7, 292, 3661, 6, 551, 0, 368, 8, 507, 0, 7983, 0, 5, 212, 502, 6, 847, 0, 4700, 788, 3, 0, 5078, 4, 5, 4, 1894
Offset: 1

Views

Author

Antti Karttunen, Jul 13 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A008683, A341528, A341529, A341512, A346239.
Cf. also A347097.

Programs

Formula

a(n) = -Sum_{d|n, dA008683(n/d) * A341512(d).
a(n) = A341512(n) - A346239(n).

A347099 a(1) = 2; and for n > 1, a(n) = A336853(n) + A347098(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 9, 4, 4, 0, 32, 0, 8, 8, 49, 0, 56, 0, 36, 16, 4, 0, 153, 4, 8, 56, 66, 0, 96, 0, 207, 8, 4, 16, 295, 0, 8, 16, 187, 0, 168, 0, 48, 120, 12, 0, 553, 16, 80, 8, 78, 0, 444, 8, 323, 16, 4, 0, 480, 0, 12, 216, 745, 16, 144, 0, 60, 24, 200, 0, 1016, 0, 8, 152, 90, 16, 216, 0, 723, 472, 4, 0, 786, 8, 8, 8, 289
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2021

Keywords

Comments

Sum of {the pointwise sum of A336853 and A063524 (1, 0, 0, 0, ...)} and its Dirichlet inverse.
The first negative term is a(720) = -6306.

Links

Crossrefs

Cf. A001223, A001248, A003961, A063524, A336853, A347098.
Cf. also A346250, A347097.

Programs

  • PARI
    up_to = 16384;
A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
Aux347098(n) = if(1==n,n,A336853(n));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA347098(n) = v347098[n];
A347099(n) = if(1==n,2,A336853(n)+A347098(n));

Formula

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A336853(d) * A347098(n/d).
For all n >= 1, a(A001248(n)) = A001223(n)^2.
Showing 1-3 of 3 results.

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