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 1-4 of 4 results.

A087250 Duplicate of A065301.

Original entry on oeis.org

1, 2, 5, 13, 26, 29, 37, 41, 61, 73, 74, 101, 109, 113, 122, 137, 146, 157, 173, 181, 193
Offset: 1

Views

Author

Keywords

A087249 Squarefree numbers k such that sigma(k) is not squarefree.

Original entry on oeis.org

3, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 35, 38, 39, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 62, 65, 66, 67, 69, 70, 71, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 102, 103, 105, 106, 107, 110, 111, 114, 115, 118, 119, 123, 127, 129, 130
Offset: 1

Views

Author

Labos Elemer, Sep 05 2003

Keywords

Examples

			For k=7: sigma(7) = 2*2*2 = 8.
For k=10: sigma(10) = 1 + 2 + 5 + 10 = 18 = 2*3*3.

Links

Crossrefs

Complement of A065301 within A005117.
Cf. A065303, A087248.

Programs

  • Mathematica
    Select[Range[150],SquareFreeQ[#]&&!SquareFreeQ[DivisorSigma[1,#]]&]  (* Harvey P. Dale, Feb 06 2011 *)
  • PARI
    is(k) = issquarefree(k) && !issquarefree(sigma(k)); \\ Amiram Eldar, Jun 15 2024

A065302 Squarefree nonprime numbers whose sum of divisors is also squarefree.

Original entry on oeis.org

1, 26, 74, 122, 146, 218, 314, 362, 386, 458, 554, 626, 746, 794, 818, 842, 866, 914, 1082, 1202, 1226, 1322, 1346, 1418, 1466, 1514, 1538, 1658, 1706, 1754, 1874, 1994, 2018, 2042, 2066, 2138, 2186, 2234, 2258, 2306, 2402, 2426, 2474, 2594, 2642, 2762
Offset: 1

Views

Author

Labos Elemer, Oct 29 2001

Keywords

Comments

All elements except the first, a(1)=1, are of the form 2*p, where p is a prime and p == 1 (mod 12). Also, sigma(2*p) = (1+2)*(1+p) = 6m where m = (1+p)/2 and m == 1 (mod 6). A squarefree composite number not of the form 2*p cannot be in the sequence since sigma is multiplicative. For example, sigma(p*q) = (1+p)*(1+q) is divisible by 4 for p,q > 2. - Walter Kehowski, Mar 21 2007

Links

Crossrefs

Cf. A000203, A008683, A065299, A065300, A065301, A065303.
Cf. A128607, A128608.

Programs

  • Mathematica
    Select[Range[3000], !PrimeQ[#] && SquareFreeQ[#] && SquareFreeQ[DivisorSigma[1, #]] &] (* Amiram Eldar, Jun 05 2025 *)
  • PARI
    isok(m) = !isprime(m) && moebius(m) && moebius(sigma(m)); \\ Harry J. Smith, Oct 16 2009

Extensions

Name corrected by Amiram Eldar, Jun 05 2025

A086368 Nonsquarefree numbers k such that sigma(k) is squarefree.

Original entry on oeis.org

4, 8, 9, 16, 18, 20, 25, 36, 45, 49, 50, 64, 72, 80, 100, 104, 116, 117, 121, 128, 144, 148, 169, 180, 196, 200, 208, 225, 234, 242, 244, 256, 261, 289, 292, 296, 320, 325, 333, 361, 369, 404, 436, 441, 450, 452, 464, 488, 512, 529, 548, 549, 576, 578, 584, 592
Offset: 1

Views

Author

Labos Elemer, Sep 05 2003

Keywords

Examples

			k = 45 = 3*3*5 and sigma(45) = 78 = 2*3*13.

Links

Crossrefs

Intersection of A013929 and A065300.
Cf. A000203 (sigma), A005117, A065303, A087249, A065301.

Programs

  • Maple
    filter:= n -> not numtheory:-issqrfree(n) and numtheory:-issqrfree(numtheory:-sigma(n)):
select(filter, [$1..1000]); # Robert Israel, May 11 2018
  • Mathematica
    q[k_] := ! SquareFreeQ[k] && SquareFreeQ[DivisorSigma[1, k]]; Select[Range[600], q] (* Amiram Eldar, Feb 24 2025 *)
  • PARI
    isok(k) = !issquarefree(k) && issquarefree(sigma(k)); \\ Amiram Eldar, Feb 24 2025
Showing 1-4 of 4 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.