A366244 -id:A366244 - cp's OEIS frontend

A366242 Numbers that are products of "Fermi-Dirac primes" (A050376) that are powers of primes with exponents that are powers of 4.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 96, 97

Offset: 1

Amiram Eldar, Oct 05 2023

A subsequence of A252895, and first differs from it at n = 172. A252895(172) = 256 = 2^(2^3) is not a term of this sequence.
Equivalently, numbers that are products of "Fermi-Dirac primes" that are powers of primes with exponents that are powers of 2 with even exponents.
Products of distinct numbers of the form p^(2^(2*k)), where p is prime and k >= 0.
Numbers whose prime factorization has exponents that are positive terms of the Moser-de Bruijn sequence (A000695).
Every integer k has a unique representation as a product of 2 numbers: one is in this sequence and the other is in A366243: k = A366244(k) * A366245(k).
The asymptotic density of this sequence is 1/c = 0.65531174251481086750..., where c is given in the formula section.

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

Cf. A000695, A050376, A366243, A366244, A366245.

Subsequence of A252895.

mdQ[n_] := AllTrue[IntegerDigits[n, 4], # < 2 &]; Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], mdQ] &]

ismd(n) = {my(d = digits(n, 4)); for(i = 1, #d, if(d[i] > 1, return(0))); 1;}
is(n) = {my(e = factor(n)[ ,2]); for(i = 1, #e, if(!ismd(e[i]), return(0))); 1;}

a(n) ~ c * n, where c = Product_{k>=0} zeta(2^(2*k+1))/zeta(2^(2*k+2)) = 1.52599127273749217982... .

A366243 Numbers that are products of "Fermi-Dirac primes" (A050376) that are powers of primes with exponents that are not powers of 4.

Original entry on oeis.org

1, 4, 9, 25, 36, 49, 100, 121, 169, 196, 225, 256, 289, 361, 441, 484, 529, 676, 841, 900, 961, 1024, 1089, 1156, 1225, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2304, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761, 4900, 5041, 5329

Offset: 1

Amiram Eldar, Oct 05 2023

Equivalently, numbers that are products of "Fermi-Dirac primes" that are powers of primes with exponents that are powers of 2 with odd exponents.
Products of distinct numbers of the form p^(2^(2*k-1)), where p is prime and k >= 1.
Numbers whose prime factorization has exponents that are positive terms of A062880.
Every integer k has a unique representation as a product of 2 numbers: one is in this sequence and the other is in A366242: k = A366245(k) * A366244(k).

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

Cf. A062880, A050376, A366242, A366244, A366245.

A062503 is a subsequence.

mdQ[n_] := AllTrue[IntegerDigits[n, 4], # < 2 &]; q[e_] := EvenQ[e] && mdQ[e/2]; Select[Range[6000], # == 1 || AllTrue[FactorInteger[#][[;; , 2]], q] &]

ismd(n) = {my(d = digits(n, 4)); for(i = 1, #d, if(d[i] > 1, return(0))); 1;}
is(n) = {my(e = factor(n)[ ,2]); for(i = 1, #e, if(e[i]%2 || !ismd(e[i]/2), return(0))); 1;}

Sum_{n>=1} 1/a(n) = Product_{k>=0} zeta(2^(2*k+1))/zeta(2^(2*k+2)) = 1.52599127273749217982... (this is the constant c in A366242).

A366245 The largest infinitary divisor of n that is a term of A366243.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 1, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1

Offset: 1

Amiram Eldar, Oct 05 2023

First differs from A335324 at n = 256.

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

Cf. A063695, A335324, A366242, A366243, A366244.

See the formula section for the relationships with A008833, A046100, A059895, A059896, A059897, A225546, A248101, A352780.

f[p_, e_] := p^BitAnd[e, Sum[2^k, {k, 1, Floor@ Log2[e], 2}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = -sum(k = 1, e, (-2)^k*floor(e/2^k));
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^s(f[i,2]));}

Multiplicative with a(p^e) = p^A063695(e).
a(n) = n / A366244(n).
a(n) >= 1, with equality if and only if n is a term of A366242.
a(n) <= n, with equality if and only if n is a term of A366243.
From Peter Munn, Jan 09 2025: (Start)
a(n) = max({k in A366243 : A059895(k, n) = k}).
a(n) = Product_{k >= 0} A352780(n, 2k+1).
Also defined by:
- for n in A046100, a(n) = A008833(n);
- a(n^4) = (a(n))^4;
- a(A059896(n,k)) = A059896(a(n), a(k)).
Other identities:
a(n) = sqrt(A366244(n^2)).
a(A059897(n,k)) = A059897(a(n), a(k)).
a(A225546(n)) = A225546(A248101(n)).
(End)

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

Original entry on oeis.org

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))).

A366308 The number of infinitary divisors of n that are terms of A366242.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 2, 1, 4, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 4, 4, 2, 4, 1, 4, 2, 2, 2, 8, 2, 4, 4, 4, 4, 1, 2, 4, 4, 4, 2, 8, 2, 2, 2, 4, 2, 4, 1, 2, 4, 2, 2, 4, 4, 4, 4, 4, 2, 4, 2, 4, 2, 2, 4, 8, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 2, 4, 2, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Oct 06 2023

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

Cf. A037445, A139351, A366242, A366243, A366244, A366246, A366309.

s[0] = 0; s[n_] := s[n] = s[Floor[n/4]] + If[OddQ[Mod[n, 4]], 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%2 \\ after Charles R Greathouse IV at A139351
a(n) = vecprod(apply(x -> 2^s(x), factor(n)[, 2]));

Multiplicative with a(p^e) = 2^A139351(e).
a(n) = 2^A366246(n).
a(n) = A037445(n)/A366309(n).
a(n) = A037445(A366244(n)).
a(n) >= 1, with equality if and only if n is in A366243.
a(n) <= A037445(n), with equality if and only if n is in A366242.

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A366242 Numbers that are products of "Fermi-Dirac primes" (A050376) that are powers of primes with exponents that are powers of 4.

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A366243 Numbers that are products of "Fermi-Dirac primes" (A050376) that are powers of primes with exponents that are not powers of 4.

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A366245 The largest infinitary divisor of n that is a term of A366243.

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

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A366308 The number of infinitary divisors of n that are terms of A366242.

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