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.

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A329859 Numbers k such that k and uphi(k) have the same set of prime divisors, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

1, 12, 36, 168, 240, 504, 702, 720, 1176, 1200, 1344, 1404, 1620, 3528, 3600, 4032, 4050, 6480, 8100, 9408, 14880, 19656, 22680, 23250, 28080, 28224, 32400, 44640, 46500, 53460, 63882, 65280, 69750, 74400, 113400, 127764, 132678, 133650, 137592, 139500
Offset: 1

Views

Author

Amiram Eldar, Nov 22 2019

Keywords

Examples

			12 is in the sequence since 12 = 2^2 * 3 and uphi(12) = 6 = 2 * 3 both have the same set of prime divisors, {2, 3}.

Links

Crossrefs

The unitary version of A055744.
Cf. A007947, A027598, A047994.

Programs

  • Mathematica
    rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); uphi[1] = 1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); Select[Range[10^5], rad[#] == rad[uphi[#]] &]

A329878 Numbers k such that k and psi(k) have the same set of prime divisors, where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 150, 162, 192, 216, 288, 294, 300, 324, 384, 432, 450, 486, 576, 588, 600, 648, 726, 750, 768, 864, 882, 900, 972, 1152, 1176, 1200, 1296, 1350, 1452, 1458, 1500, 1536, 1728, 1734, 1764, 1800, 1944, 2058, 2178
Offset: 1

Views

Author

Amiram Eldar, Nov 23 2019

Keywords

Comments

Numbers k such that rad(psi(k)) = rad(k), where rad(k) is the squarefree kernel of k (A007947).

Examples

			6 is in the sequence since 6 = 2 * 3 and psi(6) = 12 = 2^2 * 3 have the same set of prime divisors, {2, 3}.

Links

Crossrefs

Cf. A001615, A007947, A027598, A055744.

Programs

  • Mathematica
    rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[2000], rad[psi[#]] == rad[#] &]

A286884 Odd numbers k such that the set of distinct prime divisors of k is equal to the set of distinct prime divisors of the sum of proper divisors of k.

Original entry on oeis.org

108927, 448335, 544635, 53781261243, 92526188391, 612887145325
Offset: 1

Views

Author

Altug Alkan, Aug 02 2017

Keywords

Comments

The first four terms that are divisible by 108927 are 108927, 544635, 92526188391, 4094089374375.
a(7) > 10^12. 2981095241355 is also a term. - Giovanni Resta, Aug 03 2017

Examples

			92526188391 is a term because sigma(92526188391) - 92526188391 = 3^2*7*13^3*19*181^2 and 92526188391 = 3^2*7^2*13^2*19^3*181.

Crossrefs

Cf. A001065, A007947, A027598, A027748, A286876.

Programs

  • Mathematica
    fQ[n_] := Transpose[ FactorInteger[ n]][[1]] == Transpose[ FactorInteger[ DivisorSigma[1, n] - n]][[1]];  (* Robert G. Wilson v, Aug 02 2017 *)
  • PARI
    a001065(n) = if(n==0, 0, sigma(n) - n)
a027748(n) = factor(n)[, 1]~
is(n) = n%2==1 && a027748(n)==a027748(a001065(n)) \\ Felix Fröhlich, Aug 02 2017
  • PARI
    list(lim)=my(v=List(),f,t,o); forfactored(n=108927,lim\1, f=n[2]; if(f[1,1]==2, next); t=sigma(f)-n[1]; for(i=1,#f~, o=valuation(t,f[i,1]); if(o==0, next(2)); t/=f[i,1]^o); if(t==1, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Aug 02 2017

Extensions

a(4)-a(6) from Giovanni Resta, Aug 03 2017

A290051 Least k > 1 such that the set of distinct prime divisors of k is equal to the set of distinct prime divisors of sigma_n(k) where sigma_n (k) is result of applying sum-of-divisors function n times to k.

Original entry on oeis.org

6, 2, 294, 2, 126, 112, 310, 14, 150, 840, 3200, 98, 45360, 10500, 57120, 40320, 242250, 9548, 21839790, 3756480, 200425680, 678810, 1359540
Offset: 1

Views

Author

Altug Alkan, Aug 03 2017

Keywords

Comments

a(24) > 10^9.

Examples

			a(3) = 294 because sigma(sigma(sigma(2*3*7^2))) = 2^5*3*7^2 and 2*3*7^2 = 294 is the least number with this property.

Crossrefs

Cf. A000203, A027598.

Programs

  • Mathematica
    f[n_] := Block[{k = 2}, While[ Transpose[ FactorInteger[ Nest[ DivisorSigma[1, #] &, k, n]]][[1]] != Transpose[ FactorInteger[ k]][[1]], k += 2]; k]; (* Robert G. Wilson v, Aug 03 2017 *)

Extensions

a(10)-a(23) from Giovanni Resta, Aug 03 2017
Previous Showing 11-14 of 14 results.

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

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