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 1-5 of 5 results.

A066109 Numbers k such that sigma_4(k)/sigma_2(k) is prime.

Original entry on oeis.org

4, 9, 20, 25, 169, 289, 961, 1849, 3721, 6889, 11881, 14641, 15625, 17161, 52441, 57121, 66049, 69169, 72361, 96721, 97969, 117649, 130321, 196249, 214369, 253009, 326041, 351649, 358801, 383161, 410881, 418609, 426409, 434281, 491401
Offset: 1

Views

Author

Labos Elemer, Dec 05 2001

Keywords

Comments

Numbers k such that A001159(k)/A001157(k) is prime.
Except for the 3rd term 20, below 10000000 all the other terms are even powers of a prime. These primes are listed in A066111. It is not known whether other numbers similar to 20 exist or not.
20 is the only exception within the first 2000 terms. - Amiram Eldar, Feb 25 2025

Examples

			For k = 20: divisors(20) = {20, 10, 5, 4, 2, 1}, sigma_4 = 160000 + 10000 + 625 + 256 + 16 + 1 = 170898, sigma_2 = 400 + 100 + 25 + 16 + 4 + 1 = 546; p = 170898/546 = 73 is prime.

Links

Crossrefs

Cf. A000040, A001157, A001159, A046871, A066111, A066112.

Programs

  • Mathematica
    Do[s = DivisorSigma[4, n]; z = DivisorSigma[2, n]; If[PrimeQ[s/z], Print[{n, s, z, s/z}]], {n, 1, 10000000}]
Select[Range[500000],PrimeQ[DivisorSigma[4,#]/DivisorSigma[2,#]]&] (* Harvey P. Dale, May 02 2011 *)
  • PARI
    isok(k) = { my(f=sigma(k, 4)/sigma(k, 2)); !frac(f) && isprime(f) } \\ Harry J. Smith, Nov 16 2009

A066112 Numbers k such that sigma_4(k)/sigma_2(k) is an integer but not a prime.

Original entry on oeis.org

1, 16, 36, 48, 49, 64, 81, 100, 121, 144, 162, 180, 196, 225, 245, 256, 324, 361, 400, 432, 441, 484, 500, 529, 576, 605, 625, 648, 676, 729, 784, 841, 900, 931, 980, 1024, 1089, 1156, 1200, 1225, 1280, 1296, 1369, 1444, 1521, 1600, 1620, 1681, 1764, 1805
Offset: 1

Views

Author

Labos Elemer, Dec 06 2001

Keywords

Examples

			The sequence includes squares, twice squares (such as 162 and 648), and other numbers (such as 48 and 180). The sigma_4/sigma_2 quotients usually have more than one distinct prime factor. Exception: sigma_4(48)/sigma_2(48) = 5732210/3410 = 1681 = 41^2.

Links

Crossrefs

Cf. A001157, A001159, A046871, A066109, A066110, A066111.

Programs

  • Mathematica
    Do[s=DivisorSigma[4, n]; z=DivisorSigma[2, n]; If[IntegerQ[s/z]&&!PrimeQ[s/z], Print[n]], {n, 1, 10000}]
  • PARI
    isok(k) = { my(f=sigma(k, 4)/sigma(k, 2)); !frac(f) && !isprime(f) } \\ Harry J. Smith, Feb 01 2010

Extensions

Edited by Jon E. Schoenfield, Dec 24 2016

A066134 Numbers from A066112 that are neither square nor twice a square, i.e., are not in A028982 but are in A028983.

Original entry on oeis.org

48, 180, 245, 432, 500, 605, 931, 980, 1200, 1280, 1620, 1805, 2205, 2352, 2420, 3380, 3724, 3888, 3920, 4500, 4655, 5445, 5780, 5808, 6125, 6845, 7203, 7220, 7936, 8112, 8379, 8405, 8820, 9072, 9251, 9680, 10580, 10800, 11520, 11760, 12500
Offset: 1

Views

Author

Labos Elemer, Dec 06 2001

Keywords

Examples

			180 is neither square nor twice a square, but sigma_4(180)/sigma_2(180) = 1135275414/49686 = 22849 = 73*313.

Links

Crossrefs

Cf. A001157, A001159, A028982, A028983, A046871, A066109, A066110, A066111, A066112.

Programs

  • Mathematica
    q[k_] := !IntegerQ[Sqrt[k]] && !IntegerQ[Sqrt[k/2]] && IntegerQ[r = DivisorSigma[4, k]/DivisorSigma[2, k]] && !PrimeQ[r]; Select[Range[12500], q] (* Amiram Eldar, Feb 23 2025 *)
  • PARI
    { n=0; for (m=1, 10^9, if (issquare(m) || issquare(m/2), next); if (frac(f=sigma(m, 4)/sigma(m, 2)), next); if (!isprime(f), write("b066134.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 02 2010

A074632 Numbers k such that the sum of 2nd, 3rd, 4th and 5th powers of divisors of k are divisible by sum of divisors of k.

Original entry on oeis.org

1, 20, 64, 500, 729, 1024, 1280, 4096, 4352, 14580, 15625, 32000, 39168, 46656, 47360, 59049, 65536, 117649, 144640, 161024, 262144, 312500, 364500, 509184, 531441, 746496, 796797, 933120, 1000000, 1180980, 1184000, 1449216, 1771561
Offset: 1

Views

Author

Labos Elemer, Aug 27 2002

Keywords

Examples

			For k = 20: sigma(k) = 42 ,sigma_2(k) = 546 = 13 * 42, sigma_3(k) = 9198 = 219 * 42, sigma_4(k) = 170898 = 4069 * 42, sigma_5(k) = 3304182 = 78671 * 42.

Links

Crossrefs

See also A020487, A046841, A046870, A046871.
Cf. A000203, A001157, A001158, A001159, A001160.

Programs

  • Mathematica
    Select[Range[2000000],And@@Divisible[DivisorSigma[Range[2,5],#], DivisorSigma[ 1,#]]&] (* Harvey P. Dale, Jan 01 2012 *)
  • PARI
    is(k) = {my(f = factor(k), s = sigma(f)); for(k = 2, 5, if(sigma(f, k) % s, return(0))); 1; }  \\ Amiram Eldar, Jun 15 2024

A374170 a(n) is the least nonsquare k such that sigma_n(k) divides sigma_2n(k).

Original entry on oeis.org

20, 20, 6050, 7203
Offset: 1

Views

Author

Mohammed Yaseen, Jun 30 2024

Keywords

Comments

a(1) = A227771(1); a(2) = A046871(5).
a(5) > 10^9 if it exists.
a(6) = 17328, a(7) = 50, a(13) = 761378.

Crossrefs

Cf. A020487, A227771, A046871.

Programs

  • PARI
    a(n) = my(k=1, f=factor(k)); while (issquare(k) || (sigma(f, 2*n) % sigma(f, n)), f=factor(k++)); k; \\ Michel Marcus, Jun 30 2024
Showing 1-5 of 5 results.

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