cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A372502 The number of "Fermi-Dirac primes" (A050376) that divide n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 1, 3, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 3, 3, 2, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 3, 2, 3, 1, 4, 3, 2, 1, 4, 2, 2, 2
Offset: 1

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Author

Amiram Eldar, May 04 2024

Keywords

Comments

Differs from A345222 at n = 64, 128, 192, 320, 384, ... .

Crossrefs

Programs

  • Mathematica
    f[p_, e_] := BitLength[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecsum(apply(x -> exponent(x) + 1, factor(n)[, 2]));
    
  • Python
    from sympy import factorint
    def A372502(n): return sum(e.bit_length() for e in factorint(n).values()) # Chai Wah Wu, Feb 18 2025

Formula

Additive with a(p^e) = A070939(e).
a(n) = A064547(n) + A372332(n).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{k>=1} P(2^k) = 0.53331724743088069672..., and P(s) is the prime zeta function.
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