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%I #49 Sep 08 2022 08:45:01
%S 2,3,4,5,8,17,27,41,49,59,64,71,89,101,125,131,167,169,173,256,289,
%T 293,383,512,529,677,701,729,743,761,773,827,839,841,857,911,1091,
%U 1097,1163,1181,1193,1217,1373,1427,1487,1559,1583,1709,1811,1847,1849,1931
%N Numbers n for which sigma(n^2) is prime.
%C sigma(n) is the sum of the divisors of n (A000203).
%C 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
%C This is a subsequence of A000961, see A248963 for its complement therein. - _M. F. Hasler_, Oct 19 2014
%C a(n) nearly always has digitsum of the form 1 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
%C Number of terms < 10^k: 5, 13, 36, 137, 735, 4730, 33732, 253393, ..., . _Robert G. Wilson v_, Mar 09 2017
%C Primes in the sequence are A053182. - _Thomas Ordowski_, Nov 18 2017
%H Robert G. Wilson v, <a href="/A055638/b055638.txt">Table of n, a(n) for n = 1..10000</a> (first 4730 terms from T. D. Noe)
%F a(n) = sqrt(A023194(n+1)).
%F Equal to A000961 \ A248963. - _M. F. Hasler_, Oct 19 2014
%t Select[Range[2000], PrimeQ[DivisorSigma[1, #^2]] &]
%o (PARI) for(n=1,9999,isprime(sigma(n^2))&&print1(n",")) \\ _M. F. Hasler_, Oct 18 2014
%o (Magma) [n: n in [1..2000] | IsPrime(SumOfDivisors(n^2))]; // _Vincenzo Librandi_, Oct 18 2014
%Y Cf. A023194 (sigma(n) is prime).
%Y Cf. A023195 (primes of the form sigma(n)), A062700 (in order of appearance).
%Y Cf. A000961, A248963.
%K nonn
%O 1,1
%A _Robert G. Wilson v_, Jun 07 2000
%E Minor edits by _M. F. Hasler_, Oct 18 2014
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