A286095 Composite numbers n such that tau(n) (number of divisors of n) is prime and sigma(n) (sum of divisors of n) is not prime.

49, 81, 121, 169, 361, 529, 625, 841, 961, 1024, 1369, 1849, 2209, 2809, 3721, 4489, 5329, 6241, 6889, 9409, 10609, 11449, 11881, 12769, 14641, 16129, 18769, 19321, 22201, 22801, 24649, 26569, 32041, 32761, 36481, 37249, 38809, 39601, 44521, 49729, 51529, 52441, 54289

Offset: 1

Author

Bernard Schott, May 22 2017

Keywords

• nonn

Comments

If sigma(n) is prime (A023194) then tau(n) is prime too. (See Crux Mathematicorum link.)

But the reverse is false; the numbers which verify tau(n) prime and sigma(n) not prime are in the sequence A275938.

All odd primes belong to the sequence A275938, but there are also in this sequence composite numbers which are all prime powers, these prime powers are here.

Examples

			tau(49) = 3 and sigma(49) = 57 = 3 * 19.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Peter A. Lindstrom and Andy Liu, Problem 465, Crux Mathematicorum, page 188, Vol.6 , Jun. 80.

Crossrefs

Cf. A000005, A000203, A023194, A275938.

Programs

Maple

for n from 2 to 550000 do p(n):=tau(n);
if not isprime(n) and is prime(p(n)) and not isprime(sigma(n)) then print (n,p(n),sigma(n)) else fi; od:
# alternative
N:= 10^5: # to get all terms <= N
P:= select(isprime, [2,seq(i,i=3..isqrt(N),2)]):
S:= {}:
for p in P do
  k:= 1:
  do
    k:= nextprime(k+1)-1;
    if p^k > N then break fi;
    if not isprime((p^(k+1)-1)/(p-1)) then S:= S union {p^k} fi
  od
od:
sort(convert(S,list)); # Robert Israel, Jun 05 2017

Mathematica

Select[Range[10^5], Function[n, And[CompositeQ@ n, Map[PrimeQ@ DivisorSigma[#, n] &, {0, 1}] == {True, False}]]] (* Michael De Vlieger, May 24 2017 *)

PARI

lista(nn) = {forcomposite(n=1, nn, if (isprime(numdiv(n)) && !isprime(sigma(n)), print1(n, ", ")););} \\ Michel Marcus, May 24 2017