cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Oct 05 2023

Keywords

Comments

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.

Links

Crossrefs

Cf. A000695, A050376, A366243, A366244, A366245.
Subsequence of A252895.

Programs

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

Formula

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

Views

Author

Amiram Eldar, Oct 05 2023

Keywords

Comments

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).

Links

Crossrefs

Cf. A062880, A050376, A366242, A366244, A366245.
A062503 is a subsequence.

Programs

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

Formula

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).

A366244 The largest infinitary divisor of n that is a term of A366242.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 2, 1, 10, 11, 3, 13, 14, 15, 16, 17, 2, 19, 5, 21, 22, 23, 6, 1, 26, 3, 7, 29, 30, 31, 32, 33, 34, 35, 1, 37, 38, 39, 10, 41, 42, 43, 11, 5, 46, 47, 48, 1, 2, 51, 13, 53, 6, 55, 14, 57, 58, 59, 15, 61, 62, 7, 16, 65, 66, 67, 17, 69, 70, 71
Offset: 1

Views

Author

Amiram Eldar, Oct 05 2023

Keywords

Links

Crossrefs

Cf. A063694, A366242, A366243, A366245.
See the formula section for the relationships with A007913, A046100, A059895, A059896, A059897, A225546, A247503, A352780.

Programs

  • Mathematica
    f[p_, e_] := p^BitAnd[e, Sum[2^k, {k, 0, Floor@ Log2[e], 2}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    s(e) = sum(k = 0, e, (-2)^k*floor(e/2^k));
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^s(f[i,2]));}

Formula

Multiplicative with a(p^e) = p^A063694(e).
a(n) = n / A366245(n).
a(n) >= 1, with equality if and only if n is a term of A366243.
a(n) <= n, with equality if and only if n is a term of A366242.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1-1/p)*(Sum_{k>=1} p^(A063694(k)-2*k)) = 0.35319488024808595542... .
From Peter Munn, Jan 09 2025: (Start)
a(n) = max({k in A366242 : A059895(k, n) = k}).
a(n) = Product_{k >= 0} A352780(n, 2k).
Also defined by:
- for n in A046100, a(n) = A007913(n);
- a(n^4) = (a(n))^4;
- a(A059896(n,k)) = A059896(a(n), a(k)).
Other identities:
a(n) = sqrt(A366245(n^2)).
a(A059897(n,k)) = A059897(a(n), a(k)).
a(A225546(n)) = A225546(A247503(n)).
(End)

A366247 The number of infinitary divisors of n that are "Fermi-Dirac primes" (A050376) and terms of A366243.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0
Offset: 1

Views

Author

Amiram Eldar, Oct 05 2023

Keywords

Comments

First differs from A101436 at n = 32.

Links

Crossrefs

Cf. A050376, A064547, A101436, A139352, A366242, A366243, A366246.

Programs

  • Mathematica
    s[0] = 0; s[n_] := s[n] = s[Floor[n/4]] + If[Mod[n, 4] > 1, 1, 0]; f[p_, e_] := s[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    s(e) = if(e>3, s(e\4)) + e%4\2 \\ after Charles R Greathouse IV at A139352
a(n) = vecsum(apply(s, factor(n)[, 2]));

Formula

Additive with a(p^e) = A139352(e).
a(n) = A064547(n) - A366246(n).
a(n) = A064547(A366245(n)).
a(n) >= 0, with equality if and only if n is in A366242.
a(n) <= A064547(n), with equality if and only if n is in A366243.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.39310573826635831710..., where f(x) = Sum_{k>=0} (x^(2*4^k)/(1+x^(2*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

Views

Author

Amiram Eldar, Oct 06 2023

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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]
  • PARI
    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]));

Formula

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... .
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.