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 1-4 of 4 results.

A365296 The smallest coreful infinitary divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 6, 25, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 36, 37, 38, 39, 10, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 6, 55, 14, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69
Offset: 1

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Author

Amiram Eldar, Aug 31 2023

Keywords

Comments

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n.
The number of coreful infinitary divisors of n is A363329(n).
All the terms are in A138302.

Links

Crossrefs

Cf. A006519, A037445, A077609, A007947, A034444, A138302, A268335, A302792, A339597, A363329.

Programs

  • Mathematica
    f[p_, e_] := p^(2^IntegerExponent[e, 2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(2^valuation(f[i,2], 2)));}
  • Python
    from math import prod
from sympy import factorint
def A365296(n): return prod(p**(e&-e) for p, e in factorint(n).items()) # Chai Wah Wu, Sep 01 2023

Formula

Multiplicative with a(p^e) = p^A006519(e).
a(n) = n if and only if n is in A138302.
a(n) >= A007947(n) with equality if and only if n is an exponentially odd number (A268335).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 1/p + Sum_{e>=1} 1/p^f(e)-1/p^(f(e)+1)) = 0.4459084041..., where f(k) = 2*k - A006519(k) = A339597(k-1).
A037445(a(n)) = A034444(n). - Amiram Eldar, Oct 19 2023

A363331 a(n) is the sum of divisors of n that are both coreful and infinitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 14, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 42, 25, 26, 39, 28, 29, 30, 31, 50, 33, 34, 35, 36, 37, 38, 39, 70, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 78, 55, 98, 57, 58, 59, 60, 61, 62, 63, 84, 65, 66, 67, 68
Offset: 1

Views

Author

Amiram Eldar, May 28 2023

Keywords

Comments

First differs from A363334 at n = 16.
The number of these divisors is A363329(n).

Examples

			a(8) = 14 since 8 has 3 divisors that are both infinitary and coreful, 2, 4 and 8, and 2 + 4 + 8 = 14.

Links

Crossrefs

Cf. A049417, A057723, A077609, A138302, A361810, A363329, A363334.

Programs

  • Mathematica
    f[p_, e_] := Times @@ (1 + Flatten[p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], ?(# == 1 &)]))]) - 1; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n), b); prod(i = 1, #f~, b = binary(f[i, 2]); prod(k = 1, #b, if(b[k], f[i, 1]^(2^(#b - k)) + 1, 1)) - 1);}

Formula

Multiplicative with a(p^e) = (Product_{k>=0} (p^(2^k*(b(k)+1)) - 1)/(p^(2^k) - 1)) - 1, where e = Sum_{k >= 0} b(k) * 2^k is the binary representation of e.
a(n) >= n, with equality if and only if n is in A138302.
a(n) >= A361810(n), with equality if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1 - 1/p) * Sum_{k>=1} a(p^k)/p^(2*k)) = 0.53906337497505398777... .

A370079 The product of the odd exponents of the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Feb 09 2024

Keywords

Comments

First differs from A363329 at n = 32.

Links

Crossrefs

Cf. A005361, A013661, A268335, A335275, A350387, A350389, A363329, A368472, A370080.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> if(x%2, x, 1), factor(n)[, 2]));

Formula

a(n) = A005361(A350389(n)).
Multiplicative with a(p^e) = e if e is odd, and 1 if e is even.
a(n) = A005361(n)/A370080(n).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= A005361(n), with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) + 1/p^(3*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4) = 1.32800597172596287374... .
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 2/((p^s - 1)*(p^s + 1)^2)). - Vaclav Kotesovec, Feb 11 2024

A363330 Numbers with a record number of divisors that are both coreful and infinitary.

Original entry on oeis.org

1, 8, 128, 216, 3456, 27000, 279936, 432000, 9261000, 34992000, 148176000, 8957952000, 12002256000, 197222256000, 3072577536000, 7501410000000, 15975002736000, 433297296432000, 1920360960000000, 4089600700416000, 9984376710000000, 35097081010992000, 2128789617370416000
Offset: 1

Views

Author

Amiram Eldar, May 28 2023

Keywords

Comments

Indices of records in A363329.
The corresponding record values are 1, 3, 7, 9, 21, 27, 49, 63, 81, 147, 189, 315, ... (see the link for more values).

Links

Crossrefs

Cf. A363329.
Subsequence of A025487.
Similar sequences: A005934, A037992.

Programs

  • Mathematica
    f[p_, e_] := 2^DigitCount[e, 2, 1] - 1; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n];
v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]];
seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq
Showing 1-4 of 4 results.

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