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.

A067193 Numbers k such that sigma(k) == 4 (mod phi(k)).

Original entry on oeis.org

24, 27, 44, 66, 75, 170, 944, 1200, 16064, 260864, 4189184, 17179541504, 274876596224
Offset: 1

Views

Author

Benoit Cloitre, Feb 19 2002

Keywords

Comments

a(14) > 10^12. 1125899822956544 and 4611686013058678784 are also terms. - Donovan Johnson, Feb 29 2012
a(14) > 10^13. If 2^j-5 is prime (A059608) and j > 3, then 2^(j-2)*(2^j-5) is a term. - Giovanni Resta, Mar 29 2020

Crossrefs

Cf. A063514, A059608, A067192.

Extensions

a(11) from Donovan Johnson, Dec 14 2009
a(12)-a(13) from Donovan Johnson, Feb 29 2012

A072808 Smallest m such that sigma(m) mod phi(m) = n or 0 if no solution exists.

Original entry on oeis.org

4, 5, 8, 24, 0, 22, 16, 21, 450, 40, 25, 48, 50, 136, 32, 110, 100, 90, 144, 88, 0, 656, 121, 102, 0, 80, 169, 96, 0, 68, 64, 55, 676, 464, 289, 65, 0, 117, 162, 91, 0, 116, 225, 85, 0, 272, 529, 95, 0, 148, 288, 133, 0, 164, 0, 115, 0, 160, 841, 147, 0, 333, 128, 247
Offset: 1

Views

Author

Labos Elemer, Jul 12 2002

Keywords

Comments

Warning: It is only conjectured that there are no solutions for n such that a(n) = 0. The search for solutions tested all m <= 10^10 for these n.
For odd remainders a(n) is a square or twice a square. See A028982, except terms 1 and 2.
All zeros corresponding to odd terms a(n) with n < 64 confirmed up to m <= 10^24. - Giovanni Resta, Apr 02 2020

Examples

			For n=4: a(4)=24 since sigma(24)=60, phi(24)=8 and Mod(60, 8)=4.

Crossrefs

Cf. A000203, A000010, A028982, A063514, A067192, A067193, A071390.

Programs

  • Maple
    V:= Vector(100):
for m from 2 to 10^7 do
 v:= numtheory:-sigma(m) mod numtheory:-phi(m);
 if v > 0 and v <= 100 and V[v] = 0 then V[v]:= m fi
od:
convert(V,list); # Robert Israel, Nov 30 2024
  • Mathematica
    f[x_] := Mod[DivisorSigma[1, x], EulerPhi[x]]; t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10000000000}]; t

Formula

a(n) = Min{x; Mod(A000203(x), A000010(x))=n} or 0 if no solutions.

Extensions

Name corrected by Sean A. Irvine, Oct 30 2024
Name corrected by Robert Israel, Nov 30 2024

A333633 Smallest m such that sigma(m) == 2*n (mod phi(m)) or 0 if no solution exists.

Original entry on oeis.org

1, 5, 24, 22, 21, 40, 48, 136, 110, 90, 88, 656, 102, 80, 96, 68, 55, 464, 65, 117, 91, 116, 85, 272, 95, 148, 133, 164, 115, 160, 147, 333, 247, 212, 145, 243968, 155, 244, 217, 405, 230, 11072, 185, 292, 259, 1184, 205, 237824, 215, 657, 301, 356, 189, 343, 329, 388, 559, 404
Offset: 0

Views

Author

Michel Marcus, Mar 30 2020

Keywords

Comments

a(221) <= 288230257234804736 = 2^(k-2)*(2^k-443) for k=30. - Michel Marcus, Apr 02 2020
a(221) > 10^13. - Giovanni Resta, Apr 12 2020

Links

Crossrefs

Bisection of A072808.
Cf. A000203, A000010, A063514, A067192, A067193.

Programs

  • PARI
    g(n) = my(f=factor(n)); sigma(f) % eulerphi(f);
a(n) = {n *= 2; my(k=1); while (g(k) != n, k++); k;} \\ Michel Marcus, Mar 30 2020

Extensions

a(0) prepended by Jinyuan Wang, Mar 30 2020
Showing 1-3 of 3 results.

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

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