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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A382660 The unitary totient function applied to the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 7, 4, 10, 12, 6, 8, 16, 18, 12, 10, 22, 14, 12, 26, 28, 8, 30, 31, 20, 16, 24, 36, 18, 24, 28, 40, 12, 42, 22, 46, 32, 52, 26, 40, 42, 36, 28, 58, 60, 30, 48, 20, 66, 44, 24, 70, 72, 36, 60, 24, 78, 40, 82, 64, 42, 56, 70, 88, 72, 60, 46, 72
Offset: 1

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Author

Amiram Eldar, Apr 02 2025

Keywords

Crossrefs

Programs

  • Mathematica
    f[p_, e_] := p^e-1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; expOddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; uphi /@ Select[Range[100], expOddQ]
  • PARI
    uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2]-1);}
    isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
    list(lim) = apply(uphi, select(isexpodd, vector(lim, i, i)));

Formula

a(n) = A047994(A268335(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(4)/(2*d^2)) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^5) = 0.504949539649594981601..., and d = A065463 is the asymptotic density of the exponentially odd numbers.
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