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A055638 Numbers n for which sigma(n^2) is prime. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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 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

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

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

FORMULA

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

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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

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

Cf. A000961, A248963.

Sequence in context: A281303 A264011 A081711 * A057657 A293591 A116657

Adjacent sequences:  A055635 A055636 A055637 * A055639 A055640 A055641

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Jun 07 2000

EXTENSIONS

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

STATUS

approved

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Last modified September 22 08:23 EDT 2018. Contains 315270 sequences. (Running on oeis4.)