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

1,2

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

Mathematica

Select[Range[3000], !PrimeQ[#] && SquareFreeQ[#] && SquareFreeQ[DivisorSigma[1, #]] &] (* Amiram Eldar, Jun 05 2025 *)

Prog

(PARI) isok(m) = !isprime(m) && moebius(m) && moebius(sigma(m)); \\ Harry J. Smith, Oct 16 2009

Crossrefs

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

Cf. A128607, A128608.

Sequence in context: A043200 A043980 A351538 * A353635 A044164 A044545

Adjacent sequences: A065299 A065300 A065301 * A065303 A065304 A065305

Keyword

nonn

Author

Labos Elemer, Oct 29 2001

Extensions

Name corrected by Amiram Eldar, Jun 05 2025

Status

approved