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.

A048699 Nonprime numbers whose sum of aliquot divisors (A001065) is a perfect square.

Original entry on oeis.org

1, 9, 12, 15, 24, 26, 56, 75, 76, 90, 95, 119, 122, 124, 140, 143, 147, 153, 176, 194, 215, 243, 287, 332, 363, 386, 407, 477, 495, 507, 511, 524, 527, 536, 551, 575, 688, 738, 791, 794, 815, 867, 871, 892, 924, 935, 963, 992, 1075, 1083, 1159, 1196, 1199, 1295, 1304
Offset: 1

Views

Author

Enoch Haga

Keywords

Comments

The sum of aliquot divisors of prime numbers is 1.
If a^2 is an odd square for which a^2-1 = p + q with p,q primes, then p*q is a term. If m = 2^k-1 is a Mersenne prime then m*(2^k) (twice an even perfect number) is a term. If b = 2^j is a square and b-7 = 3s is a semiprime then 4s is a term. - Metin Sariyar, Apr 02 2020

Examples

			a(3)=15; aliquot divisors are 1,3,5; sum of aliquot divisors = 9 and 3^2=9.

Links

Crossrefs

Cf. A001065, A006532, A020477, A048698, A073040 (includes primes).

Programs

  • Maple
    a := []; for n from 1 to 2000 do if sigma(n) <> n+1 and issqr(sigma(n)-n) then a := [op(a), n]; fi; od: a;
  • Mathematica
    nn=1400;Select[Complement[Range[nn],Prime[Range[PrimePi[nn]]]],IntegerQ[ Sqrt[DivisorSigma[1,#]-#]]&] (* Harvey P. Dale, Apr 25 2011 *)
  • PARI
    isok(k) = !ispseudoprime(k) && issquare(sigma(k) - k); \\ Michel Marcus, May 13 2025

A073040 Numbers n such that sum of proper divisors of n is a square.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 12, 13, 15, 17, 19, 23, 24, 26, 29, 31, 37, 41, 43, 47, 53, 56, 59, 61, 67, 71, 73, 75, 76, 79, 83, 89, 90, 95, 97, 101, 103, 107, 109, 113, 119, 122, 124, 127, 131, 137, 139, 140, 143, 147, 149, 151, 153, 157, 163, 167, 173, 176, 179, 181, 191
Offset: 1

Views

Author

N. J. A. Sloane, Aug 24 2002

Keywords

Comments

Old name was: Numbers n such that sum of divisors of n, sigma (n), minus n is a square.
All primes are terms, since for p prime, A001065(p)=1 and 1 is a square. - Michel Marcus, Apr 22 2018

Examples

			a(6) = 9 because the divisors of 9 are 1, 3, 9, and (1+3+9)-9 = 4 = 2^2.
The number 10 is not in the sequence because (1+2+5+10)-10 = 8, which is not a square.
a(7) = 11 because (1+11)-11 = 1, a square.

Links

Crossrefs

Cf. A000290, A001065, A048698 (which excludes primes).

Programs

  • Maple
    with(numtheory); a := []; for n from 1 to 2000 do if issqr(sigma(n)-n) then a := [op(a), n]; fi; od: a;
  • Mathematica
    Select[Range[200], IntegerQ[Sqrt[-# + Plus@@Divisors[#]]] &] (* Alonso del Arte, Dec 08 2010 *)
  • PARI
    isok(n) = issquare(sigma(n) - n); \\ Michel Marcus, Apr 22 2018

Formula

{n: A001065(n) in A000290} - R. J. Mathar, Dec 11 2010

Extensions

Name edited by Altug Alkan, Apr 22 2018

A194948 Numbers k such that sum of aliquot divisors of k, sigma(k) - k, is a cube.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 56, 59, 61, 67, 69, 71, 73, 76, 79, 83, 89, 97, 101, 103, 107, 109, 113, 122, 127, 131, 133, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233
Offset: 1

Views

Author

Martin Renner, Oct 13 2011

Keywords

Comments

For prime numbers, the sum of their aliquot divisors is exactly 1 = 1^3.

Examples

			a(6) = 10, since the sum of aliquot divisors 1 + 2 + 5 = 8 = 2^3.

Links

Crossrefs

Union of A000040 and A048698.
Cf. A020477, A073040.

Programs

  • Maple
    for n do s:=numtheory[sigma](n)-n; if root(s,3)=trunc(root(s,3)) then print(n); fi; od:
  • Mathematica
    Select[Range[250],IntegerQ[Power[DivisorSigma[1,#]-#, (3)^-1]]&] (* Harvey P. Dale, Nov 25 2011 *)

A333258 Numbers k that are not powers of primes such that the sum of proper unitary divisors of k is a cube.

Original entry on oeis.org

10, 12, 69, 122, 133, 153, 236, 363, 504, 752, 844, 992, 1001, 1018, 1243, 1685, 1819, 1940, 1994, 2295, 2323, 2619, 2871, 2900, 3184, 3403, 3483, 3641, 3763, 3981, 3984, 4024, 5482, 6471, 6892, 7128, 7925, 7928, 8186, 8856, 9077, 9352, 9641, 9664, 10113, 10404
Offset: 1

Views

Author

Amiram Eldar, Mar 13 2020

Keywords

Comments

Powers of primes are excluded since they are trivial terms: their sum of proper unitary divisors is 1 (except for 1 whose sum of proper unitary divisors is 0) .

Examples

			10 is a term since A034460(10) = 8 = 2^3.

Links

Crossrefs

The unitary version of A048698.
Cf. A000961, A024619, A034448, A034460, A063936, A329239.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); us[n_] := usigma[n] - n; Select[Range[10000], PrimeNu[#] > 1 && IntegerQ @ Surd[us [#], 3] &]
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.