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.

A319800 Numbers k such that Sum_{d|k} nphi(d) = k where the sum is over nonunitary divisors of k and nphi(k) is the nonunitary totient function (A254503).

Original entry on oeis.org

3960, 5220, 1873080, 6733440, 8447040, 18685336320, 255306083760, 341863562880, 357274165248, 765899971200, 1018887932160, 16733804567040, 19602402019200, 21205959667200, 79205761958400, 166967788700160, 189585719769600, 279604120561920, 623380501094400
Offset: 1

Views

Author

Amiram Eldar, Sep 28 2018

Keywords

Comments

Ligh and Wall found the first 6 terms and also the terms a(8) = 341863562880, a(9) = 357174165248, a(11) = 1018887932160, and 20993596382889043200. They showed that each term has a powerful part with at least 2 distinct prime factors, and conjectured that it is only even.

References

  • Jozsef Sandor and Borislav Crstici, Handbook of Number Theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 287.

Links

Crossrefs

Cf. A057521, A055231, A254503.

Programs

  • Mathematica
    rad[n_] := Times @@ First /@ FactorInteger[n]; powerFree[n_] := Denominator[ n/rad[n]^2 ]; powerPart[n_] := n/powerFree[n]; nuphi[n_] := powerFree[ n ] * EulerPhi[powerPart[n]]; ndiv[n_] := Block[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; a[n_] := Module[{d = ndiv[n]}, Total@Map[nuphi, d]]; s={}; Do[ If[a[n] == n, AppendTo[s, n]], {n, 1, 10^8}]; s
  • PARI
    nphi(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, moebius(d)^2 * eulerphi(n/d)));
isok(n) = sumdiv(n, d, if(gcd(n/d, d) != 1, nphi(d))) == n; \\ Michel Marcus, Sep 28 2018

Extensions

a(6)-a(9) from Giovanni Resta, Sep 29 2018
a(10)-a(11) from Giovanni Resta, Oct 11 2018
a(12)-a(19) from Max Alekseyev, Jun 05 2025

A328258 a(n) = Sum_{d|n, gcd(d,n/d) = 1} (-1)^(d + 1) * d.

Original entry on oeis.org

1, -1, 4, -3, 6, -4, 8, -7, 10, -6, 12, -12, 14, -8, 24, -15, 18, -10, 20, -18, 32, -12, 24, -28, 26, -14, 28, -24, 30, -24, 32, -31, 48, -18, 48, -30, 38, -20, 56, -42, 42, -32, 44, -36, 60, -24, 48, -60, 50, -26, 72, -42, 54, -28, 72, -56, 80, -30, 60, -72, 62, -32, 80, -63, 84
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 09 2019

Keywords

Comments

Excess of sum of odd unitary divisors of n over sum of even unitary divisors of n.
a(n) = n+1 iff n is in A061345 \ {1}. - Bernard Schott, Mar 05 2023

Links

Crossrefs

Cf. A002129, A034448, A077610, A109712, A192066, A254503, A306633, A360156.

Programs

  • Magma
    [&+[(-1)^(d+1)*d:d in Divisors(n)|Gcd(d, n div d) eq 1]:n in [1..70]]; // Marius A. Burtea, Oct 10 2019
  • Maple
    f:= proc(n) local t;
  mul(1 - (-1)^t[1] * t[1]^t[2], t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Oct 10 2019
  • Mathematica
    a[n_] := Sum[Boole[GCD[d, n/d] == 1] (-1)^(d + 1) d, {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]
a[1] = 1; a[n_] := Times @@ (1 - (-1)^First[#] First[#]^Last[#] & /@ FactorInteger[n]); Table[a[n], {n, 1, 65}]
  • PARI
    a(n) = sumdiv(n, d, if (gcd(d,n/d) == 1, (-1)^(d + 1) * d)); \\ Michel Marcus, Oct 10 2019

Formula

If n = Product (p_j^k_j) then a(n) = Product (1 - (-1)^p_j * p_j^k_j).
If n odd, a(n) = usigma(n), where usigma = A034448.
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2)/(14*zeta(3)) = A306633 / 14 = 0.0977451... . - Amiram Eldar, Nov 17 2022
From Amiram Eldar, Jan 28 2023: (Start)
a(n) = 2 * A192066(n) - A034448(n).
a(n) = A192066(n) - A360156(n/2) if n is even, and A192066(n) otherwise.
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-1))*(2^(2*s)-2^(s+2)+2)/(2^(2*s)-2). (End)

A348513 Möbius transform of A048146, the sum of non-unitary divisors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 4, 3, 0, 0, 6, 0, 0, 0, 8, 0, 6, 0, 10, 0, 0, 0, 12, 5, 0, 9, 14, 0, 0, 0, 16, 0, 0, 0, 24, 0, 0, 0, 20, 0, 0, 0, 22, 15, 0, 0, 24, 7, 10, 0, 26, 0, 18, 0, 28, 0, 0, 0, 30, 0, 0, 21, 32, 0, 0, 0, 34, 0, 0, 0, 48, 0, 0, 15, 38, 0, 0, 0, 40, 27, 0, 0, 42, 0, 0, 0, 44, 0, 30, 0, 46, 0, 0, 0, 48, 0, 14
Offset: 1

Views

Author

Antti Karttunen, Oct 29 2021

Keywords

Links

Crossrefs

Cf. A008683, A048146, A254503.

Programs

  • Mathematica
    nusigma[1] = 0; nusigma[n_] := DivisorSigma[1, n] - Times @@ (1 + Power @@@ FactorInteger[n]); a[n_] := DivisorSum[n, MoebiusMu[n/#]*nusigma[#] &]; Array[a, 100] (* Amiram Eldar, Oct 30 2021 *)
  • PARI
    A254503(n) = {my(f = factor(n)); for (i=1, #f~, if ((e=f[i, 2]) > 1, f[i, 1] = eulerphi(f[i, 1]^e); f[i, 2] = 1); ); factorback(f); }; \\ From A254503
A348513(n) = (n - A254503(n));

Formula

a(n) = n - A254503(n).
a(n) = Sum_{d|n} (A008683(n/d) * A048146(d)).
Showing 1-3 of 3 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.