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.

A134382 a(n) is the smallest number k larger than a(n-1) such that n*d(k)*sopf(k)=sigma(k), where d is the number of divisors (A000005) and sopf the sum of prime factors without repetition (A008472).

Original entry on oeis.org

20, 140, 464, 660, 1276, 1365, 2204, 2508, 2805, 2907, 5590, 5698, 5742, 6006, 7395, 8680, 14645, 15052, 18875, 19170, 19740, 23871, 34579, 34804, 35164, 35244, 35934, 38121, 106805, 114953, 261536, 503082
Offset: 1

Views

Author

Enoch Haga, Oct 23 2007

Keywords

Comments

Sequence suggested by Puzzle 419 in Carlos Rivera's The Prime Puzzles & Problems Connection.
For n=33, the search for terms k that satisfy 33*d(k)*sopf(k)=sigma(k), without being greater than a(32), gives 21070, 25585, 30702, 36120, 41710, 49256, 52269, 68906, 74692, 92785, 95702, 111342, 117626, 383086 with no other terms up to 10^9. So this sequence might well be complete. - Michel Marcus, Oct 02 2019
I confirm that the solutions for n=33 listed above are complete, thus the sequence stops at n=32. - Max Alekseyev, Sep 18 2024

Links

Crossrefs

Subsequence of A070222. - R. J. Mathar, Feb 05 2010
Cf. A134383, A134384, A134385, A134386.

Programs

  • Maple
    A008472 := proc(n) local divs,i ; if n = 1 then 0; else divs := ifactors(n)[2] ; add( op(1,i),i=divs) ; fi ; end: A134382 := proc(n) option remember ; local k,kmin ; if n = 1 then kmin := 1 ; else kmin := procname(n-1)+1 ; fi ; for k from kmin do if numtheory[sigma](k) = n* numtheory[tau](k)*A008472(k) then RETURN(k) ; fi ; od: end: for n from 1 to 30 do print( A134382(n)) ; od: # R. J. Mathar, Nov 16 2007, Jun 24 2009
  • Mathematica
    sopf[1] = 0; sopf[n_] := Total[FactorInteger[n][[All, 1]]]; a[n_] := a[n] = For[k = If[n == 1, 1, a[n-1] + 1], True, k++, If[DivisorSigma[1, k] == n*DivisorSigma[0, k]*sopf[k], Return[k]]]; Table[Print[a[n]]; a[n], {n, 1, 32}] (* Jean-François Alcover, Sep 12 2013 *)
  • PARI
    lista(nn) = {lasta = 2; for (n=1, nn, k = lasta; while ((f = factor(k)) && (n*numdiv(k)*sum(j=1,#f~,f[j,1]) != sigma(k)), k++); print1(k, ", "); lasta = k;);} \\ Michel Marcus, Feb 25 2016

Formula

a(n) > a(n-1): n*A000005(a(n))*A008472(a(n)) = A000203(a(n)). - R. J. Mathar, Nov 16 2007, Jun 24 2009

Extensions

Edited by R. J. Mathar, Nov 16 2007
A-number in formula and Maple program corrected by R. J. Mathar, Jun 24 2009
a(32) from R. J. Mathar, Feb 05 2010
full,fini keywords added by Max Alekseyev, Sep 18 2024

A187771 Numbers whose sum of divisors is the cube of the sum of its prime divisors.

Original entry on oeis.org

245180, 612408, 639198, 1698862, 1721182, 5154168, 7824284, 15817596, 20441848, 25969788, 27688078, 28404862, 35860609, 67149432, 77378782, 91397838, 96462862, 179302264, 191550135, 289772221, 306901244, 311657084, 392802179, 441839706, 572673855, 652117774, 988918364
Offset: 1

Views

Author

Manuel Valdivia, Jan 04 2013

Keywords

Comments

This sequence and A187824 and A187761 are winners in the contest held at the 2013 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 14 2013
The identity sigma(k) = (sopf(k))^m only occurs for m = 3 (this sequence) in the given range, however it is likely that it also occurs for other powers m in higher numbers.
The smallest k such that sigma(k) = sopf(k)^m, for m=4,5,6 are 1056331752 (A221262), 213556659624 (A221263) and 45770980141656, respectively. - Giovanni Resta, Jan 07 2013
Prime divisors are taken without multiplicity. - Harvey P. Dale, Dec 17 2016

Examples

			a(13) = 35860609 = 41 * 71 * 97 * 127, then sigma(35860609) = 37933056 = (41 + 71 + 97 + 127)^3.

References

  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

Links

Crossrefs

Cf. A000203, A008472, A020477, A070222.
Cf. A221262 (sigma(k)=sopf(k)^4), A221263 (sigma(k)=sopf(k)^5).

Programs

  • Mathematica
    d[n_]:= If[Plus@@Divisors[n]-Power[Plus@@Select[Divisors[n], PrimeQ], 3]==0, n]; Select[Range[2,10^9], #==d[#]&]
Select[Range[2, 10^9],DivisorSigma[1,#]==Total[FactorInteger[#][[All, 1]]]^3&] (* Harvey P. Dale, Dec 17 2016 *)
  • PARI
    is(n)=my(f=factor(n));sum(i=1,#f~,f[i,1])^3==sigma(n) \\ Charles R Greathouse IV, Jun 29 2013

Formula

a(n) = k if sigma(k) = (sopf(k))^3, where sigma(k) = A000203(k) and sopf(k) = A008472(k).

A329731 Numbers k such that the sum of distinct prime divisors of k divides the sum of unitary divisors of k.

Original entry on oeis.org

12, 15, 18, 35, 36, 42, 56, 60, 63, 66, 72, 75, 90, 95, 108, 110, 114, 119, 126, 132, 135, 143, 144, 147, 153, 168, 180, 192, 195, 209, 220, 245, 250, 258, 260, 264, 287, 288, 290, 294, 297, 300, 308, 312, 315, 319, 322, 323, 324, 338, 342, 360, 375, 377, 378
Offset: 1

Views

Author

Amiram Eldar, Nov 19 2019

Keywords

Comments

The unitary version of A070222.

Examples

			12 is in the sequence since its sum of unitary divisors, 1 + 3 + 4 + 12 = 20, is divisible by its sum of distinct prime divisors, 2 + 3 = 5.

Links

Crossrefs

Cf. A008472, A034448, A070222.

Programs

  • Mathematica
    aQ[n_] := Divisible[Times @@ (1 + Power @@@ (f = FactorInteger[n])), Total[First /@ f]]; Select[Range[2, 400], aQ]
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.