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.

A131908 Smallest positive integer k with the same sum of cubes of divisors as the n-th integer for which such a k exists.

Original entry on oeis.org

184926, 291741, 583482, 1166964, 1458705, 1880574, 2042187, 2333928, 2404038, 2917410, 3209151, 3513594, 3792633, 4084374, 4253298, 4667856, 4959597, 5543079, 5362854, 5834820, 5732706, 6418302, 6710043, 7293525, 7585266
Offset: 1

Views

Author

Peter Pein (petsie(AT)dordos.net), Jul 26 2007, Jul 28 2007

Keywords

Examples

			a(1)=184926 because A131907(1)=194315 with the sum of cubes of its divisors being 7401260364550416. This is the same as the sum of cubes of divisors of 184926 and there is no positive integer less than 184926 with this cubesum of its divisors

Links

Crossrefs

Cf. A069822, A131902-A131907.

Programs

  • Mathematica
    Last@Transpose[Reap[For[n = 1, n <= 5*10^6, n++, (If[Head[ #1] === tmp, #1 = n, Sow[{n, #1}]] & )[ tmp[DivisorSigma[3, n]]]]][[2, 1]]]

Formula

a(n)=min(k>0, k has the same sum of cubes of divisors as A131907(n))

Extensions

More terms from Max Alekseyev, Jul 28 2007

A131902 Smallest positive integer k with the same sum of divisors as the n-th integer for which such a k exists.

Original entry on oeis.org

6, 14, 10, 14, 16, 20, 21, 33, 24, 28, 20, 30, 33, 30, 34, 30, 54, 40, 24, 42, 44, 42, 66, 30, 48, 42, 60, 57, 68, 44, 54, 40, 60, 66, 54, 52, 63, 85, 102, 74, 66, 104, 88, 66, 80, 60, 84, 99, 93, 96, 86, 114, 76, 132, 105, 102, 60, 88, 111, 90, 138, 105, 114, 102, 105, 138, 96
Offset: 1

Views

Author

Peter Pein (petsie(AT)dordos.net), Jul 26 2007

Keywords

Examples

			a(3)=10 because 17 is the third integer for which a smaller integer with same sum of divisors exists and sigma(17)=1+17=18 and sigma(10)=1+2+5+10=18 and there is no k>0 less than 10 with sigma(k)=18.

Links

Crossrefs

Cf. A069822, A131903, A131904, A131905, A131906, A131907, A131908.

Programs

  • Maple
    N:= 1000: # to use values of sigma <= N
V:= Vector(N): A:= Vector(N):
for n from 1 to N do
  v:= numtheory:-sigma(n);
  if v <= N then
    if V[v] = 0 then V[v]:= n
    else A[n]:= V[v]
    fi
  fi
od:
subs(0=NULL, convert(A,list)); # Robert Israel, Mar 30 2018
  • Mathematica
    Clear[tmp]; Function[n,If[Head[ #1]===tmp,#1=n;Unevaluated[Sequence[]],#1]& [tmp[DivisorSigma[1,n]]]]/@Range[200]
  • PARI
    list(lim) = my(m); for(k = 1, lim, m = invsigmaMin(sigma(k)); if(m < k, print1(m, ", "))); \\ Amiram Eldar, Dec 20 2024, using Max Alekseyev's invphi.gp

Formula

Let S = {n>0 : there exists a k>0 and k0: sigma(k) = sigma(n-th element of S)).

A110928 Pairs of distinct numbers m and n, m

Original entry on oeis.org

6, 7, 24, 26, 30, 35, 40, 47, 66, 77, 78, 91, 102, 119, 114, 133, 120, 130, 120, 141, 130, 141, 136, 157, 138, 161, 150, 175, 168, 182, 174, 203, 186, 215, 186, 217, 215, 217, 222, 259, 230, 249, 246, 287, 258, 301, 264, 286, 280, 282, 280, 329, 282, 329, 318
Offset: 1

Views

Author

Walter Kehowski, Sep 23 2005

Keywords

Comments

There do not appear to be any pairs (m,n) such that sigma_k(m)=sigma_k(n) for k>2.
For sigma_3, the first pair is (184926, 194315). Other terms may be found in A131907 and A131908. See A158915.

Examples

			sigma_2(30)=1^1+2^2+3^2+5^2+6^2+10^2+15^2+30^2=1300 and sigma_2(35)=1^2+5^2+7^2+35^2=1300.

Links

Crossrefs

Cf. A001157, A002025, A002046, A063990.
Cf. A110926, A110927, A110929.

Programs

  • Maple
    with(numtheory); sigmap := proc(p,n) convert(map(proc(z) z^p end, divisors(n)),`+`) end; SA2:=[]: for z from 1 to 1 do for m to 1500 do M:=sigmap(2,m); for n from m+1 to 1500 do N:=sigmap(2,n); if N=M then SA2:=[op(SA2),[m,n,N]] fi od od od; SA2; select(proc(z) z[1]<=1000 end, SA2); #just to shorten it a bit
  • Mathematica
    a[n_] := Module[{s = DivisorSigma[2, n], ans = {}}, kmax = Ceiling[Sqrt[s]]; Do[If[DivisorSigma[2, k] == s, AppendTo[ans, k]], {k, n + 1, kmax}]; ans];  s = {}; Do[v = a[n]; Do[s = Join[s, {n, v[[k]]}], {k, 1, Length[v]}], {n, 1, 400}]; s (* Amiram Eldar, Sep 08 2019 *)

Formula

sigma_2(m)=sigma_2(n), m
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.