A302778 Number of "Fermi-Dirac primes" (A050376) <= n.

0, 1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 8, 9, 10, 10, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 22, 22, 23, 23, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 26, 26, 26, 26, 26, 26, 27, 27, 28, 28, 29, 29, 29, 29, 29, 29, 30

Offset

1,3

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(1) = 0; for n > 1, a(n) = A302777(n) + a(n-1).

For all n >= 1, a(A050376(n)) = n.

From Ridouane Oudra, Jan 18 2026: (Start)

a(n) = Sum_{p prime <=n} (floor(log_2(log_p(n))) + 1).

a(n) = Sum_{k=0..floor(log_2(log_2(n)))} pi(n^(1/2^k)).

a(n) = - Sum_{k=1..floor(log_2(n))} mu(2*k)*A025528(floor(n^(1/k))). (End)

Maple

seq(add(floor(log[2](log[p](n)))+1, p in select(isprime, [$1..n])), n=1..80); # Ridouane Oudra, Jan 18 2026

Mathematica

s[n_] := Boole[n > 1 && Length[(f = FactorInteger[n])] == 1 && (e = f[[;; , 2]]) == 2^IntegerExponent[e, 2]]; Accumulate @ Array[s, 100] (* Amiram Eldar, Nov 27 2020 *)

Prog

(PARI)
A209229(n) = (n && !bitand(n, n-1));
A302777(n) = A209229(isprimepower(n));
s=0; for(n=1, 105, s+=A302777(n); print1(s, ", "));
(Python)
from sympy import primepi, integer_nthroot
def A302778(n): return sum(primepi(integer_nthroot(n, 1<<i)[0]) for i in range(n.bit_length().bit_length())) # Chai Wah Wu, Feb 18-19 2025

Crossrefs

Partial sums of A302777. A left inverse of A050376.

Cf. A302785, A302786.

Differs from A203967 for the first time at n=64, where a(64) = 23, while A203967(64) = 24.

Cf. also A000720, A025528, A025528, A008683.

Sequence in context: A047784 A047742 A383236 * A203967 A365398 A365399

Adjacent sequences: A302775 A302776 A302777 * A302779 A302780 A302781

Keyword

nonn

Author

Antti Karttunen, Apr 16 2018

Status

approved