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

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

Author

Labos Elemer, Oct 29 2001

Keywords

• nonn

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

• Harry J. Smith, Table of n, a(n) for n = 1..1000

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