A065985 - OEIS

%I #17 Sep 10 2024 08:44:35

%S 6,8,10,12,14,15,18,20,21,22,26,27,28,32,33,34,35,38,39,44,45,46,48,

%T 50,51,52,55,57,58,62,63,65,68,69,74,75,76,77,80,82,85,86,87,91,92,93,

%U 94,95,98,99,106,111,112,115,116,117,118,119,122,123,124,125,129,133,134

%N Numbers k such that d(k) / 2 is prime, where d(k) = number of divisors of k.

%C Numbers whose sorted prime signature (A118914) is either of the form {2*p-1} or {1, p-1}, where p is a prime. Equivalently, disjoint union of numbers of the form q^(2*p-1) where p and q are primes, and numbers of the form r * q^(p-1), where p, q and r are primes and r != q. - _Amiram Eldar_, Sep 09 2024

%H Harry J. Smith, <a href="/A065985/b065985.txt">Table of n, a(n) for n = 1..1000</a>

%t Select[Range[1, 1000], PrimeQ[DivisorSigma[0, # ] / 2] == True &]

%o (PARI) n=0; for (m=1, 10^9, f=numdiv(m)/2; if (frac(f)==0 && isprime(f), write("b065985.txt", n++, " ", m); if (n==1000, return))) \\ _Harry J. Smith_, Nov 05 2009

%o (PARI) is(n)=n=numdiv(n)/2; denominator(n)==1 && isprime(n) \\ _Charles R Greathouse IV_, Oct 15 2015

%Y Cf. A000005, A118914.

%Y Numbers with exactly 2*p divisors: A030513 (p=2), A030515 (p=3), A030628 \ {1} (p=5), A030632 (p=7), A137485 (p=11), A137489 (p=13), A175744 (p=17), A175747 (p=19).

%Y Subsequences: A006881, A030078, A050997, A054753, A095990, A096156, A138031, A178739, A179665, A189987.

%K nonn

%O 1,1

%A _Joseph L. Pe_, Dec 10 2001