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A366905
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The largest exponentially odious divisor of n.
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6
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1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 16, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 16, 65, 66, 67, 68
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OFFSET
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1,2
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COMMENTS
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First differs from A353897 at n = 128.
The largest divisor of n that is an exponentially odious number (A270428).
The number of exponentially odious divisors of n is A366901(n) and their sum is A366903(n).
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LINKS
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FORMULA
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Multiplicative with a(p^e) = p^max{k=1..e, k odious}.
a(n) <= n, with equality if and only if n is exponentially odious number (A270428).
Sum_{k=1..n} a(k) ~ c*n^2, where c = (1/2) * Product_{p prime} (1 + Sum_{e>=1} (p^f(e) - p^(f(e-1)+1))/p^(2*e)) = 0.4636829525..., f(e) = max{k=1..e, k odious} for e >= 1, and f(0) = 0.
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MATHEMATICA
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maxOdious[e_] := Module[{k = e}, While[EvenQ[DigitCount[k, 2, 1]], k--]; k]; f[p_, e_] := p^maxOdious[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) s(n) = {my(k = n); while(!(hammingweight(k)%2), k--); k; }
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 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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