A286095 - OEIS

%I #30 Jun 05 2017 03:48:43

%S 49,81,121,169,361,529,625,841,961,1024,1369,1849,2209,2809,3721,4489,

%T 5329,6241,6889,9409,10609,11449,11881,12769,14641,16129,18769,19321,

%U 22201,22801,24649,26569,32041,32761,36481,37249,38809,39601,44521,49729,51529,52441,54289

%N 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.

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

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

%C 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.

%H Robert Israel, <a href="/A286095/b286095.txt">Table of n, a(n) for n = 1..10000</a>

%H Peter A. Lindstrom and Andy Liu, <a href="https://cms.math.ca/crux/backfile/Crux_v6n06_Jun.pdf">Problem 465</a>, Crux Mathematicorum, page 188, Vol.6 , Jun. 80.

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

%p for n from 2 to 550000 do p(n):=tau(n);

%p if not isprime(n) and is prime(p(n)) and not isprime(sigma(n)) then print (n,p(n),sigma(n)) else fi; od:

%p # alternative

%p N:= 10^5: # to get all terms <= N

%p P:= select(isprime, [2,seq(i,i=3..isqrt(N),2)]):

%p S:= {}:

%p for p in P do

%p   k:= 1:

%p   do

%p     k:= nextprime(k+1)-1;

%p     if p^k > N then break fi;

%p     if not isprime((p^(k+1)-1)/(p-1)) then S:= S union {p^k} fi

%p   od

%p od:

%p sort(convert(S,list)); # _Robert Israel_, Jun 05 2017

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

%o (PARI) lista(nn) = {forcomposite(n=1, nn, if (isprime(numdiv(n)) && !isprime(sigma(n)), print1(n, ", ")););} \\ _Michel Marcus_, May 24 2017

%Y Cf. A000005, A000203, A023194, A275938.

%K nonn

%O 1,1

%A _Bernard Schott_, May 22 2017