A055638 Numbers k for which sigma(k^2) is prime.

2, 3, 4, 5, 8, 17, 27, 41, 49, 59, 64, 71, 89, 101, 125, 131, 167, 169, 173, 256, 289, 293, 383, 512, 529, 677, 701, 729, 743, 761, 773, 827, 839, 841, 857, 911, 1091, 1097, 1163, 1181, 1193, 1217, 1373, 1427, 1487, 1559, 1583, 1709, 1811, 1847, 1849, 1931

Offset: 1

Author

Robert G. Wilson v, Jun 07 2000

Keywords

• nonn

Comments

sigma(n) is the sum of the divisors of n (A000203).

If sigma(x) is prime, then x=2 or x=p^(2m), an even power of a prime, cf. A023194. This sequence lists the values n = p^m such that sigma(n^2) is prime, i.e., sqrt( A023194 \ {2} ). The corresponding primes sigma(n^2)=A062700(n) are 1+p+...+p^(2m) = (p^(2m+1)-1)/(p-1), and any prime of that form (cf. A023195) corresponds to a term p^m is in this sequence. - M. F. Hasler, Oct 14 2014

This is a subsequence of A000961, see A248963 for its complement therein. - M. F. Hasler, Oct 19 2014

a(n) nearly always has digitsum of the form 2 mod 3. Specifically, 99.8% of the first 33733 entries examined conformed. The first exceptions are 3, 4, 27, 49, 64, 169, 256, 289, 529, 729. The exceptions (examined) appear to be integer powers themselves excepting the initial 3. Similarly, except for the initial 3, all entries of A023195 appear to have digitsum = 1 mod 3. - Bill McEachen, Mar 05 2017, Mar 20 2025

Number of terms < 10^k: 5, 13, 36, 137, 735, 4730, 33732, 253393, ..., . Robert G. Wilson v, Mar 09 2017

Primes in the sequence are A053182. - Thomas Ordowski, Nov 18 2017

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 4730 terms from T. D. Noe)

Crossrefs

Cf. A023194 (sigma(n) is prime).

Cf. A023195 (primes of the form sigma(n)), A062700 (in order of appearance).

Cf. A000961, A248963.

Programs

Magma

[n: n in [1..2000] | IsPrime(SumOfDivisors(n^2))]; // Vincenzo Librandi, Oct 18 2014

Mathematica

Select[Range[2000], PrimeQ[DivisorSigma[1, #^2]] &]

PARI

for(n=1,9999,isprime(sigma(n^2))&&print1(n",")) \\ M. F. Hasler, Oct 18 2014

Formula

a(n) = sqrt(A023194(n+1)).

Equal to A000961 \ A248963. - M. F. Hasler, Oct 19 2014

Extensions

Minor edits by M. F. Hasler, Oct 18 2014