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 1-5 of 5 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).

A347121 a(n) = A347136(n) - 2*n.

Original entry on oeis.org

-1, 1, 2, 11, 2, 28, 4, 49, 31, 40, 2, 128, 4, 62, 66, 179, 2, 209, 4, 188, 102, 76, 6, 472, 59, 98, 218, 286, 2, 420, 6, 601, 126, 112, 146, 859, 4, 134, 162, 700, 2, 636, 4, 368, 498, 168, 6, 1592, 149, 445, 186, 466, 6, 1252, 178, 1058, 222, 184, 2, 1704, 6, 216, 756, 1931, 230, 828, 4, 548, 278, 940, 2, 3041, 6
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2021

Keywords

Links

Crossrefs

Cf. A000040, A001223, A003961, A347136, A347122 (Möbius transform).
Cf. also A341512, A346239, A347236.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A347121(n) = (sumdiv(n,d,d*A003961(n/d))-(2*n));

Formula

a(n) = A347136(n) - 2*n.
a(A000040(n)) = A001223(n).

A347124 Möbius transform of A341528, n * sigma(A003961(n)).

Original entry on oeis.org

1, 7, 17, 44, 39, 119, 83, 268, 261, 273, 153, 748, 233, 581, 663, 1616, 339, 1827, 455, 1716, 1411, 1071, 689, 4556, 1385, 1631, 3933, 3652, 927, 4641, 1177, 9712, 2601, 2373, 3237, 11484, 1553, 3185, 3961, 10452, 1803, 9877, 2063, 6732, 10179, 4823, 2537, 27472, 6433, 9695, 5763, 10252, 3179, 27531, 5967, 22244
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2021

Keywords

Comments

Multiplicative because A341528 is.

Links

Crossrefs

Cf. A002117, A003961, A003973, A008683, A151800, A341528, A341512, A346239, A347125.

Programs

  • Mathematica
    f[p_, e_] := Module[{q = NextPrime[p]}, p^(e-1) * (q^e * (p*q-1) - p + 1)/(q-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2023 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A003973(n) = sigma(A003961(n));
A341528(n) = (n*A003973(n));
A347124(n) = sumdiv(n,d,moebius(n/d)*A341528(d));

Formula

a(n) = Sum_{d|n} A008683(n/d) * A341528(d).
a(n) = A347125(n) - A346239(n).
Multiplicative with a(p^e) = p^(e-1)*(q^e*(p*q-1)-p+1)/(q-1), where q = A151800(p). - Sebastian Karlsson, Sep 02 2021
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = (1/zeta(3)) / Product_{p prime} (1 - (q(p)-p)/p^2 - q(p)/p^3) = 5.6488805... , and q(p) = A151800(p). - Amiram Eldar, Dec 24 2023

A347125 Möbius transform of A341529, sigma(n) * A003961(n).

Original entry on oeis.org

1, 8, 19, 54, 41, 152, 87, 342, 305, 328, 155, 1026, 237, 696, 779, 2106, 341, 2440, 459, 2214, 1653, 1240, 695, 6498, 1477, 1896, 4675, 4698, 929, 6232, 1183, 12798, 2945, 2728, 3567, 16470, 1557, 3672, 4503, 14022, 1805, 13224, 2067, 8370, 12505, 5560, 2543, 40014, 6809, 11816, 6479, 12798, 3185, 37400, 6355
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2021

Keywords

Comments

Multiplicative because A341529 is.

Links

Crossrefs

Cf. A002117, A003961, A003973, A008683, A151800, A341529, A341512, A346239, A347124.

Programs

  • Mathematica
    f[p_, e_] := Module[{q = NextPrime[p]}, q^(e-1) * (p^e * (q*p-1)-q+1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2023 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A341529(n) = (sigma(n)*A003961(n));
A347125(n) = sumdiv(n,d,moebius(n/d)*A341529(d))

Formula

a(n) = Sum_{d|n} A008683(n/d) * A341529(d).
a(n) = A346239(n) + A347124(n).
Multiplicative with a(p^e) = q^(e-1)*(p^e*(q*p-1)-q+1)/(p-1), where q = A151800(p). - Sebastian Karlsson, Sep 02 2021
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = (1/zeta(3)) / Product_{p prime} ((p^2-q)*(p^3-q))/(p^4*(p-1)) = 7.6530842... , and q(p) = A151800(p). - Amiram Eldar, Dec 24 2023
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.