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A372331
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The number of infinitary divisors of the smallest number k such that k*n is a number whose number of divisors is a power of 2 (A036537).
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3
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1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1
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OFFSET
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1,4
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COMMENTS
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The number of divisors d of n that are infinitarily relatively prime to n (see A064379), i.e., d have no common infinitary divisors with n.
Equivalently, the number of divisors d of n such that for each prime divisor p of d, bitand(v_p(n), v_p(d)) = 0, where v_p(k) is the highest power of p that divides k. Note that for infinitary divisors d of n (A077609), bitand(v_p(n), v_p(d)) = v_p(d).
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LINKS
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FORMULA
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a(n) = 1 if and only if n is in A036537.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} A080100(k)/p^k) = 1.51209151045338102469... .
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MATHEMATICA
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f[p_, e_] := 2^DigitCount[e, 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) a(n) = vecprod(apply(x -> 2^(logint(x, 2) + 1 - hammingweight(x)), factor(n)[, 2]));
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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