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.

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

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

A066110 Primes of the form sigma_4(k)/sigma_2(k), arising in A066109.

Original entry on oeis.org

13, 73, 313, 601, 28393, 83233, 922561, 3416953, 13842121, 47451433, 141146281, 212601841, 234750601, 294482761, 2750006041, 3262751521, 4362404353, 4784281393, 5236041961, 9354855121, 9597826993, 13564461457, 16936647121
Offset: 1

Views

Author

Labos Elemer, Dec 05 2001

Keywords

Examples

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

Links

Crossrefs

Cf. A000040, A001157, A001159, A066109, A066111, A066112.

Programs

  • Mathematica
    Do[s=DivisorSigma[4, n]; z=DivisorSigma[2, n]; If[PrimeQ[s/z], Print[s/z]], {n, 1, 10000000}]
Select[Table[DivisorSigma[4,n]/DivisorSigma[2,n],{n,200000}],PrimeQ] (* Harvey P. Dale, Jan 31 2022 *)
  • PARI
    { n=0; for (m=1, 10^9, if (frac(f=sigma(m, 4)/sigma(m, 2)), next); if (isprime(f), write("b066110.txt", n++, " ", f); if (n==250, return)) ) } \\ Harry J. Smith, Feb 01 2010

Formula

Primes of the form A001159(A066109(k))/A001157(A066109(k)).
Showing 1-3 of 3 results.

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