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.

A076496 Numbers k such that sigma(k) == 12 (mod k).

Original entry on oeis.org

1, 6, 11, 24, 30, 42, 54, 66, 78, 102, 114, 121, 138, 174, 186, 222, 246, 258, 282, 304, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 780, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338
Offset: 1

Views

Author

Labos Elemer, Oct 21 2002

Keywords

Examples

			6*p is a solution if p > 3 is prime, since sigma(6*p) = 1 + 2 + 3 + 6 + p + 2*p + 3*p + 6*p = 12*(p+1) = 2*6*p + 12 = 2*k + 12. These are "regular" solutions. Also k = 121, 304 are "singular" solutions. See other remainders in cross-references.

Links

Crossrefs

Cf. A054024, A045768, A045769, A045770, A076495, A088834.
Cf. A141545 (a subsequence).

Programs

  • Mathematica
    Select[Range[2000], Mod[DivisorSigma[1, #] - 12, #] == 0 &] (* Vincenzo Librandi, Mar 11 2014, corrected by Amiram Eldar, Jan 04 2023 *)
  • PARI
    isok(k) = Mod(sigma(k), k) == 12; \\ Michel Marcus, Jan 04 2023

Extensions

Initial term 1 added by Vincenzo Librandi, Mar 11 2014
Terms 6 and 11 inserted by Michel Marcus, Jan 04 2023

A084306 Numbers x such that sigma(x) mod x = 12 and x is not divisible by 6. Singular solutions mentioned in A076496.

Original entry on oeis.org

121, 304, 127744, 33501184, 8589082624
Offset: 1

Views

Author

Labos Elemer, Jun 11 2003

Keywords

Comments

If n = P*q, where P is a multiple perfect number and q is prime so that gcd(P,q) = 1, then sigma(n) = kn(q+1). Consequently sigma(n) = knq + kn sigma(n) mod n = kn. Such values of n are regular solutions to this and analogous cases. Here, not these but the additional eccentric solutions are collected. Cf. A076496.
a(6) > 10^11. - Donovan Johnson, Sep 20 2012
If p = 2^k - 13 > 3 is a prime number, then 2^(k-1)*p is a term. This happens for k = 5, 9, 13, 17, 57, 105, 137, 3217, ... (A096818). - Giovanni Resta, Apr 01 2014

Examples

			n = 33501184 = 4096*8179; sigma(n) = 2n + 12 = 67002380.

Crossrefs

Cf. A076496, A076495, A000396, A045768-A045770, A007691, A096818.

Programs

  • Mathematica
    Do[s=Mod[DivisorSigma[1, n], n]; If[IntegerQ[n/100000], Print[{n}]]; If[Equal[s, 12]&&!Equal[Mod[n, 6], 0], Print[n]], {n, 1, 100000000}]

Extensions

a(5) from Donovan Johnson, Sep 20 2012

A234642 Smallest x such that x mod phi(x) = n, or 0 if no such x exists.

Original entry on oeis.org

1, 3, 10, 9, 20, 25, 30, 15, 40, 21, 50, 35, 60, 33, 98, 39, 80, 65, 90, 51, 100, 45, 70, 95, 120, 69, 338, 63, 196, 161, 110, 87, 160, 93, 130, 75, 180, 217, 182, 99, 200, 185, 170, 123, 140, 117, 190, 215, 240, 141, 250, 235, 676, 329, 230, 159, 392, 153, 322
Offset: 0

Views

Author

Charles R Greathouse IV, Dec 28 2013

Keywords

Comments

Conjecture: a(n) > 0 for all n. This would follow from a form of Goldbach's (binary) conjecture. Checked up to 10^7; largest term in that range is a(9972987) = 4178506411.
Pomerance proves that x = n (mod phi(x)) has at least two solutions for each n, but this allows x < n and so does not prove the conjecture above.
a(n) > 0 for all n <= 10^9. The largest term in that range is a(990429171) = 1050844225771. - Donovan Johnson, Feb 18 2014

Links

Crossrefs

Cf. A068494, A076495.

Programs

  • Mathematica
    A234642[n_]:=NestWhile[# + 1 &, 1, Not[Mod[#, EulerPhi[#]] == n] &] (* JungHwan Min, Dec 23 2015 *)
A234642[n_]:=Catch[Do[If[Mod[k, EulerPhi[k]] == n, Throw[k]], {k, Infinity}]] (* JungHwan Min, Dec 23 2015 *)
xmp[n_]:=Module[{x=1},While[Mod[x,EulerPhi[x]]!=n,x++];x]; Array[xmp,60,0] (* Harvey P. Dale, Jan 04 2016 *)
  • PARI
    a(n)=my(k=n);while(k++%eulerphi(k)!=n,);k
Showing 1-3 of 3 results.

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