A366247 -id:A366247 - cp's OEIS frontend

A366246 The number of infinitary divisors of n that are "Fermi-Dirac primes" (A050376) and terms of A366242.

0, 1, 1, 0, 1, 2, 1, 1, 0, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 0, 2, 1, 1, 1, 3, 1, 2, 2, 2, 2, 0, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 2, 2, 2

Offset: 1

Amiram Eldar, Oct 05 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A050376, A064547, A077761, A139351, A366242, A366243, A366247.

s[0] = 0; s[n_] := s[n] = s[Floor[n/4]] + If[OddQ[Mod[n, 4]], 1, 0]; f[p_, e_] := s[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = if(e>3, s(e\4)) + e%2 \\ after Charles R Greathouse IV at A139351
a(n) = vecsum(apply(s, factor(n)[, 2]));

Additive with a(p^e) = A139351(e).
a(n) = A064547(n) - A366247(n).
a(n) = A064547(A366244(n)).
a(n) >= 0, with equality if and only if n is in A366243.
a(n) <= A064547(n), with equality if and only if n is in A366242.
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = -0.25705126777012995187..., where f(x) = - x + Sum_{k>=0} (x^(4^k)/(1+x^(4^k))).

A366309 The number of infinitary divisors of n that are terms of A366243.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Oct 06 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A037445, A139352, A366242, A366243, A366245, A366247, A366246, A366308.

s[0] = 0; s[n_] := s[n] = s[Floor[n/4]] + If[Mod[n, 4] > 1, 1, 0]; f[p_, e_] := 2^s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = if(e > 3, s(e\4)) + e%4\2 \\ after Charles R Greathouse IV at A139352
a(n) = vecprod(apply(x -> 2^s(x), factor(n)[, 2]));

Multiplicative with a(p^e) = 2^A139352(e).
a(n) = 2^A366247(n).
a(n) = A037445(n)/A366308(n).
a(n) = A037445(A366245(n)).
a(n) >= 1, with equality if and only if n is in A366242.
a(n) <= A037445(n), with equality if and only if n is in A366243.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 - 1/p)*(1 + Sum_{k>=1} 2^A139352(k)/p^k) = 1.44736831993091923328... .

A370650 Numbers whose number of infinitary divisors that are terms of A366242 is equal to the number of infinitary divisors that are terms of A366243.

Original entry on oeis.org

1, 8, 12, 18, 20, 27, 28, 44, 45, 50, 52, 63, 64, 68, 75, 76, 92, 98, 99, 116, 117, 124, 125, 144, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 216, 236, 242, 244, 245, 261, 268, 275, 279, 284, 292, 316, 324, 325, 332, 333, 338, 343, 356, 360, 363, 369, 387, 388, 400

Offset: 1

Amiram Eldar, Feb 24 2024

nonn

Numbers k such that A366308(k) = A366309(k).
Numbers k such that A366246(k) = A366247(k) = A064547(k)/2.
If k is a term, then all the numbers with the same prime signature as k are terms. The least terms with each prime signature are listed in A370651.
p^A039004(k) is a term for all primes p and all k >= 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A037445, A039004, A064547, A366242, A366243, A366246, A366247, A366308, A366309.

A370651 is a subsequence.

s1[0] = 0; s1[n_] := s1[n] = s1[Floor[n/4]] + If[OddQ[Mod[n, 4]], 1, 0]; f1[p_, e_] := s1[e]; a1[1] = 0; a1[n_] := Plus  @@ f1 @@@ FactorInteger[n];
s2[0] = 0; s2[n_] := s2[n] = s2[Floor[n/4]] + If[Mod[n, 4] > 1, 1, 0]; f2[p_, e_] := s2[e]; a2[1] = 0; a2[n_] := Plus @@ f2 @@@ FactorInteger[n];
q[n_] := a1[n] == a2[n]; Select[Range[400], q]

cp's OEIS Frontend

A366246 The number of infinitary divisors of n that are "Fermi-Dirac primes" (A050376) and terms of A366242.

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A366309 The number of infinitary divisors of n that are terms of A366243.

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A370650 Numbers whose number of infinitary divisors that are terms of A366242 is equal to the number of infinitary divisors that are terms of A366243.

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