|
| |
|
|
A023194
|
|
Numbers n such that sigma(n) (sum of divisors of n) is prime.
|
|
25
|
|
|
|
2, 4, 9, 16, 25, 64, 289, 729, 1681, 2401, 3481, 4096, 5041, 7921, 10201, 15625, 17161, 27889, 28561, 29929, 65536, 83521, 85849, 146689, 262144, 279841, 458329, 491401, 531441, 552049, 579121, 597529, 683929, 703921, 707281, 734449, 829921, 1190281
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
All numbers except the first are squares. Why? - Zak Seidov, Jun 10 2005
Answer from Gabe Cunningham (gcasey(AT)MIT.EDU): "From the fact that the sigma (the sum-of-divisors function) is multiplicative, we can derive that the sigma(n) is even except when n is a square or twice a square.
"If n = 2(2k+1)^2, that is, n is twice an odd square, then sigma(n) = 3*sigma((2k+1)^2). If n = 2(2k)^2, that is, n is twice an even square, then sigma(n) is only prime if n is a power of 2; otherwise we have sigma(n) = sigma(8*2^m) * sigma(k/2^m) for some positive integer m.
"So the only possible candidates for values of n other than squares such that sigma(n) is prime are odd powers of 2. But sigma(2^(2m+1)) = 2^(2m+2)-1 = (2^(m+1)+1) * (2^(m+1) - 1), which is only prime when m=0, that is, when n=2. So 2 is the only nonsquare n such that sigma(n) is prime."
All numbers on this list also have a prime number of divisors. [From Howard Berman (howard_berman(AT)hotmail.com), Oct 29 2008]
From Gabe Cunningham's comment it follows that the alternate Mathematica program provided below is substantially more efficient as it only tests squares. [From Harvey P. Dale, Dec 12 2010]
Each number of this sequence is a prime power. This follows from the facts that sigma is multiplicative and sigma(n) > 1 for n > 1. Thus for n>1 a(n) is of the form a(n) = k^2 where k =p^m, with p prime,then the divisors of a(n) are {1, p, p^2, p^3,..., (p^m)^2}, and this set is a multiplicative group (modulo q); if q prime, q=sigma(k^2). Reciprocally, if q is a prime of the form 1 + p + p^2 + ... + p^(2*m), then q = sigma(p^(2*m)) (definition of sigma). [from Michel Lagneau, Aug 17 2011], edited by Franklin T. Adams-Watters, Aug 17 2011.
|
|
|
LINKS
|
T. D. Noe and David W. Wilson, Table of n, a(n) for n = 1..10000
|
|
|
MATHEMATICA
|
Select[ Range[ 100000 ], PrimeQ[ DivisorSigma[ 1, # ] ]& ] (* Wilson *)
Prepend[Select[Range[1100]^2, PrimeQ[DivisorSigma[1, #]]&], 2] (* Harvey P. Dale, Dec 12 2010 *)
|
|
|
PROG
|
(PARI) for(x=1, 1000, if(isprime(sigma(x)), print(x))) (Jorge Coveiro (jorgecoveiro(AT)yahoo.com), Dec 23 2004)
(PARI) upTo(lim)=my(v=List([2])); forstep(e=2, log(lim)\log(2), 2, forprime(p=2, lim^(1/e), if(isprime((p^(e+1)-1)/(p-1)), listput(v, p^e)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Aug 17 2011
|
|
|
CROSSREFS
|
Cf. A055638 (the square roots of the squares in this sequence).
Cf. A023195 (the primes produced by these n).
Sequence in context: A006474 A110878 A077137 * A114080 A090676 A000291
Adjacent sequences: A023191 A023192 A023193 * A023195 A023196 A023197
|
|
|
KEYWORD
|
nonn,easy,nice,changed
|
|
|
AUTHOR
|
David W. Wilson
|
|
|
STATUS
|
approved
|
| |
|
|
