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A361385
a(n) is the number of "Fermi-Dirac prime" factors (or I-components) of the n-th infinitary harmonic number.
1
0, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 4, 3, 5, 5, 5, 4, 6, 5, 5, 6, 6, 5, 6, 5, 6, 6, 6, 5, 7, 4, 5, 5, 6, 7, 6, 6, 6, 7, 6, 6, 7, 6, 6, 6, 7, 6, 8, 7, 7, 7, 6, 7, 7, 7, 6, 8, 6, 5, 6, 7, 6, 7, 7, 6, 8, 7, 7, 8, 7, 6, 7, 8, 7, 6, 8, 7, 7, 7, 7, 9, 6, 8, 6, 8, 8, 7
OFFSET
1,2
COMMENTS
Each term appears a finite number of times in the sequence (Hagis and Cohen, 1990).
LINKS
Peter Hagis, Jr. and Graeme L. Cohen, Infinitary harmonic numbers, Bull. Australian math. Soc., Vol. 41 (1990), pp. 151-158.
FORMULA
a(n) = A064547(A063947(n)).
MATHEMATICA
f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 2/(1 + p^(2^(m - j))), 1], {j, 1, m}]]; ih[1] = 1; ih[n_] := n*Times @@ f @@@ FactorInteger[n]; ic[n_] := Plus @@ (DigitCount[Last /@ FactorInteger[n], 2, 1]); ic[1] = 0; ic /@ Select[Range[10^5], IntegerQ[ih[#]] &]
PROG
(PARI) A064547(n) = {my(f = factor(n)[, 2]); sum(k=1, #f, hammingweight(f[k])); } \\ Michel Marcus at A064547
ihmean(n) = {my(f = factor(n), b); n * prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 2/(f[i, 1]^(2^(#b-k))+1), 1))); };
lista(kmax) = {my(ih); for(k = 1, kmax, ih = ihmean(k); if(denominator(ih) == 1, print1(A064547(k), ", "))); }
CROSSREFS
Cf. A006086, A006087, A361384 (analogous unitary sequence).
Sequence in context: A278959 A377108 A090501 * A126848 A232753 A067085
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 10 2023
STATUS
approved