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 1-6 of 6 results.

A185873 Values of k corresponding to the terms of A185584.

Original entry on oeis.org

1, 4, 11, 19, 31, 74, 65, 80, 107, 107, 107, 107
Offset: 1

Views

Author

N. J. A. Sloane, Feb 05 2011

Keywords

Comments

Based on postings to the Sequence Fans Mailing List by Claudio Meller and Andrew Weimholt, circa Feb 03 2011.

Crossrefs

Cf. A185584.

Programs

  • Mathematica
    First generate the sequence A185584 using the program supplied there.  Then apply this program to that sequence:
    Position[Accumulate[Divisors[#]^2],#][[1,1]]&/@ {A185584}
    (* Harvey P. Dale, Feb 05 2011 *)

Extensions

More terms from Harvey P. Dale, Feb 05 2011
a(8)-a(11) from Donovan Johnson, Feb 07 2011
a(12) from Donovan Johnson, Jan 20 2014

A064510 Numbers m such that the sum of the first k divisors of m is equal to m for some k.

Original entry on oeis.org

1, 6, 24, 28, 496, 2016, 8128, 8190, 42336, 45864, 392448, 714240, 1571328, 33550336, 61900800, 91963648, 211891200, 1931236608, 2013143040, 4428914688, 8589869056, 10200236032, 137438691328, 214204956672
Offset: 1

Views

Author

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 06 2001

Keywords

Comments

Obviously all perfect numbers are included in this sequence.
a(25) > 5*10^11. Other than perfect numbers, 104828758917120, 916858574438400, 967609154764800, 93076753068441600, 215131015678525440 and 1371332329173024768 are also terms. - Donovan Johnson, Dec 26 2012
a(25) > 10^12. - Giovanni Resta, Apr 15 2017

Examples

			Divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. 1+2+3+4+6+8 = 24.
		

Crossrefs

Programs

  • Mathematica
    subtract = If[ #1 < #2, Throw[ #1], #1 - #2]&; f[n_] := Catch @ Fold[subtract, n, Divisors @ n]; lst = {}; Do[ If[ f[n] == 0, AppendTo[lst, n]], {n, 10^8}]; lst (* Bobby R. Treat and Robert G. Wilson v, Jul 14 2005 *)
    Select[Range[2000000],MemberQ[Accumulate[Divisors[#]],#]&] (* Harvey P. Dale, Mar 22 2012 *)
  • PARI
    isok(n) = {my(d = divisors(n)); my(k = 1); while ((k <= #d) && ((sd = sum(j=1, k, d[j])) != n), k++;); (sd == n);} \\ Michel Marcus, Jan 16 2014

Extensions

More terms from Don Reble, Dec 17 2001
a(19)-a(23) from Donovan Johnson, Aug 31 2008
a(24) from Donovan Johnson, Aug 11 2011

A185960 Positive numbers equal to the sum of the cubes of their first k divisors for some k.

Original entry on oeis.org

1, 36, 126144, 236736, 934902, 3447632, 11877300, 67885860, 163704304, 1317560094, 1483434161, 5178258504, 8025266717, 12097787616, 16631037162, 25248156920, 110341933344, 145437730020, 718690577744
Offset: 1

Views

Author

N. J. A. Sloane, Feb 07 2011

Keywords

Comments

2665757000703, 18382539143800, 21465141915672, 34176701510281, 44733530333520 and 82841336110520 are also terms. - Donovan Johnson, Jan 21 2014

References

  • Arie Groeneveld, Posting to Sequence Fans Mailing List, Feb 06, 2011

Crossrefs

Programs

  • PARI
    for(n=1, 67885860, nd=numdiv(n); if(nd<>2, d=divisors(n); s=0; for(j=1, nd, s=s+d[j]^3; if(s>=n, if(s==n, print1(n ", ")); next(2))))) \\ Donovan Johnson, Jan 21 2014

Extensions

a(7)-a(13) from Donovan Johnson, Feb 07 2011
a(14)-a(17) from Donovan Johnson, Feb 08 2011
a(18)-a(19) from Donovan Johnson, Jan 21 2014

A318528 a(n) = least number > 1 that equals the sum of the n-th powers of its first k divisors for some k.

Original entry on oeis.org

6, 130, 36, 41860, 276, 1015690, 2316, 921951940, 20196, 10009766650, 179196, 2387003305930334914, 1602516, 100006103532010, 14381676, 1880100018939820249188604888836, 129271236, 1000003814697527770, 1162785756, 19105043663614041367780, 10462450356, 10000002384185795209930, 94151567436, 226500219158007133816826003223992308820431641700
Offset: 1

Views

Author

Amiram Eldar, Aug 28 2018

Keywords

Comments

a(48) > 10^90. - Max Alekseyev, Jan 17 2025

Examples

			a(2) = 130 since 130 has the divisors 1, 2, 5, 10, ... and 1^2 + 2^2 + 5^2 + 10^2 = 130.
		

Crossrefs

Programs

  • Mathematica
    a[k_] := Module[{n = 2}, While[! MemberQ[Accumulate[Divisors[n]^k], n], n++]; n]; Do[Print[a[n]], {n, 1, 10}]
  • PARI
    a(n) = for(x=2, oo, my(div=divisors(x), s=0); for(k=1, #div, s=sum(i=1, k, div[i]^n); if(s==x, return(x)))) \\ Felix Fröhlich, Aug 28 2018

Formula

a(n) = 1 + 2^n + 3^n for n = p^k with prime p > 2. - Giovanni Resta, Aug 28 2018
From Charlie Neder, Jan 24 2019: (Start)
a(n) = 1 + 2^n + 3^n for n odd,
a(n) = 1 + 2^n + 5^n + 10^n for n congruent to 2 modulo 4,
a(n) = 1 + 2^n + 4^n + 5^n + 7^n + 10^n + 13^n for n congruent to 4 or 8 modulo 12 and not 16 modulo 20.
All other a(n) contain a term at least 24^n. (End)

Extensions

a(12)-a(24) from Giovanni Resta confirmed by Max Alekseyev, Jan 04 2025

A185961 Let d_1=1 < d_2 < d_3 < ... be the divisors of n; sequence lists positive numbers n such that for some k, n = 2(d_1 + ... + d_k).

Original entry on oeis.org

2, 6, 12, 28, 40, 48, 224, 234, 496, 960, 8128, 47616, 174592, 10371840, 15037440, 28090368, 33550336, 134209536, 207516672, 492101632, 1150402560, 8589869056, 59205411720, 137438691328
Offset: 1

Views

Author

N. J. A. Sloane, Feb 07 2011

Keywords

References

  • Arie Groeneveld, Posting to Sequence Fans Mailing List, Feb 06, 2011

Crossrefs

Supersequence of A000396.

Programs

  • PARI
    forstep(n=2, 33550336, 2, d=divisors(n); s=0; for(j=1, numdiv(n), s=s+2*d[j]; if(s>=n, if(s==n, print1(n ", ")); next(2)))) \\ Donovan Johnson, Jan 24 2014

Extensions

a(14)-a(22) from Donovan Johnson, Feb 10 2011
a(23)-a(24) from Donovan Johnson, Jan 24 2014

A194578 Numbers n such that the sum of the first k divisors of n^2+1 is equal to n for some k.

Original entry on oeis.org

1, 3, 43, 68, 93, 228, 612, 1903, 32292, 44868, 252072, 36564198, 55862807, 134735264
Offset: 1

Views

Author

Michel Lagneau, Aug 29 2011

Keywords

Comments

No further at least up to 10^7.
No other terms up to 10^8. - Michel Marcus, Jan 17 2014
a(15) > 2.5*10^8. - Giovanni Resta, Apr 14 2017

Examples

			The divisors of 43^2 + 1 are {1, 2, 5, 10, 25, 37, 50, 74, 185, 370, 925, 1850} and 1+2+5+10 + 25 = 43, hence 43 is in the list.
		

Crossrefs

Cf. A185584.

Programs

  • Mathematica
    Select[Range[10000000], MemberQ[Accumulate[Divisors[#^2+1]], #]&]

Extensions

a(12)-a(13) from Michel Marcus, Jan 17 2014
a(14) from Giovanni Resta, Apr 14 2017
Showing 1-6 of 6 results.