A065985 Numbers k such that d(k) / 2 is prime, where d(k) = number of divisors of k.

6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, 38, 39, 44, 45, 46, 48, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 125, 129, 133, 134

Offset: 1

Author

Joseph L. Pe, Dec 10 2001

Keywords

• nonn

Comments

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

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A118914.

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).

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

Programs

Mathematica

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

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

PARI

is(n)=n=numdiv(n)/2; denominator(n)==1 && isprime(n) \\ Charles R Greathouse IV, Oct 15 2015