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 1-7 of 7 results.

A368711 The maximal exponent in the prime factorization of the exponentially odd numbers (A268335).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 3, 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, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1
Offset: 1

Views

Author

Amiram Eldar, Jan 04 2024

Keywords

Comments

Differs from A368472 at n = 1, 154, 610, 707, 762, ... .

Links

Crossrefs

Cf. A033150, A051903, A268335, A368472.
Similar sequences: A368710, A368712, A368713.

Programs

  • Mathematica
    f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, OddQ], Max @@ e, Nothing]]; f[1] = 0; Array[f, 150]
  • PARI
    lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = factor(k)[,2]; if(vecprod(e)%2, print1(vecmax(e), ", ")));}

Formula

a(n) = A051903(A268335(n)).
a(n) is odd for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + 2 * Sum_{k>=1} (1 - Product_{p prime} (1 - 1/(p^(2*k-1)*(p^2+p-1)))) = 1.34877064483679975726... .

A368710 The maximal exponent in the prime factorization of the powerful numbers.

Original entry on oeis.org

0, 2, 3, 2, 4, 2, 3, 5, 2, 2, 6, 3, 4, 2, 3, 2, 3, 7, 4, 2, 2, 3, 3, 2, 5, 8, 5, 2, 4, 3, 2, 3, 4, 4, 2, 2, 3, 9, 2, 6, 4, 4, 3, 2, 6, 4, 5, 2, 5, 2, 2, 3, 5, 3, 10, 2, 3, 7, 2, 2, 4, 3, 3, 3, 2, 3, 2, 2, 5, 6, 2, 6, 2, 3, 2, 4, 5, 4, 4, 11, 2, 7, 3, 2, 8, 3, 4
Offset: 1

Views

Author

Amiram Eldar, Jan 04 2024

Keywords

Links

Crossrefs

Cf. A001694, A033150, A048250, A051903, A059956, A090699.
Similar sequences: A368711, A368712, A368713.

Programs

  • Mathematica
    s[n_] := If[n == 1, 0, Max @@ Last /@ FactorInteger[n]]; s /@ Select[Range[3000], # == 1 || Min[FactorInteger[#][[;;, 2]]] > 1 &]
(* or *)
f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[n == 1, 0, If[Min[e] > 1, Max[e], Nothing]]]; Array[f, 3000]
  • PARI
    lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[,2]; if(k == 1, print1(0, ", "), if(vecmin(e) > 1, print1(vecmax(e), ", "))));}

Formula

a(n) = A051903(A001694(n)).
a(n) >= 2 for n >= 2.
Sum_{a(n)<=x} = D_{2,1} * sqrt(x) + O(sqrt(x)), where D_{2,1} = (6/Pi^2) * (2 + Sum_{k>=1} (A051903(k)+2)/(sqrt(k) * A048250(k))) (Jakimczuk, 2018; Theorem 2.1 and Remark 2.3).
Asymptotic mean (consequence of the formula above): Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = D_{2,1} * zeta(3)/zeta(3/2) = D_{2,1} / A090699.
The sum in the formula for D_{2,1} converges slowly: for k up to 10^8, 10^9 and 10^10 the sums are 14.845..., 14.908... and 14.938..., respectively. Thus, a lower bound for the value of this mean, calculated by summing over k=1..10^10, is 4.738... .

A368713 The maximal exponent in the prime factorization of the nonsquarefree numbers.

Original entry on oeis.org

2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 3, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 2, 2, 3, 3, 4, 2, 2, 3, 2, 2, 3, 2, 7, 2, 3, 3, 2, 4, 2, 2, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 2, 2, 4, 2, 3, 2, 3, 6, 2, 2, 3, 2, 2, 4, 2, 3, 2, 5, 2, 2, 3, 2, 2, 4
Offset: 1

Views

Author

Amiram Eldar, Jan 04 2024

Keywords

Comments

The terms of A051903 that are larger than 1.

Links

Crossrefs

Cf. A013661, A013929, A033150, A051903.
Similar sequences: A368710, A368711, A368712.

Programs

  • Mathematica
    s[n_] := Max @@ Last /@ FactorInteger[n]; s /@ Select[Range[250], !SquareFreeQ[#] &]
(* or *)
f[n_] := Module[{e = Max @@ FactorInteger[n][[;; , 2]]}, If[e > 1, e, Nothing]]; Array[f, 250]
  • PARI
    lista(kmax) = {my(e); for(k = 2, kmax, e = vecmax(factor(k)[,2]); if(e > 1, print1(e, ", ")));}

Formula

a(n) = A051903(A013929(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (c * zeta(2) - 1)/(zeta(2) - 1) = 2.798673520766..., where c = 1.705211... is Niven's constant (A033150).

A382419 The product of exponents in the prime factorization of the cubefree numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 4, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Mar 25 2025

Keywords

Comments

Differs from A368712 at n = 1, 31, 85, 151, 164, 189, ... .
All the terms are powers of 2.

Links

Crossrefs

Cf. A002117, A004709, A005361, A330594, A368712, A376366, A382421, A382422.

Programs

  • Mathematica
    s[n_] := Times @@ FactorInteger[n][[;; , 2]]; cubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 3; s /@ Select[Range[120], cubeFreeQ]
  • PARI
    list(kmax) = {my(e); print1(1, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(vecmax(e) < 3, print1(vecprod(e), ", "))); }

Formula

a(n) = A005361(A004709(n)).
a(n) = 2^A376366(n).
a(n) >= A368712(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3) * Product_{p prime} (1 + 1/p^2 - 2/p^3) = A002117 * A330594 = 1.33062904409568262931... .

A372466 The maximal exponent in the prime factorization of the numbers whose number of divisors is a power of 2 (A036537).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 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, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3
Offset: 1

Views

Author

Amiram Eldar, May 01 2024

Keywords

Comments

All the terms are of the form 2^k-1 (A000225).

Links

Crossrefs

Cf. A000225, A036537, A051903, A327839, A372467.
Similar sequences: A368710, A368711, A368712, A368713, A369933, A369935.

Programs

  • Mathematica
    pow2Q[n_] := n == 2^IntegerExponent[n, 2]; f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, pow2Q[# + 1] &], Max @@ e, Nothing]]; f[1] = 0; Array[f, 150]
  • PARI
    ispow2(n) = n >> valuation(n, 2) == 1;
lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(ispow2(vecprod(apply(x -> x + 1, e))), print1(vecmax(e), ", "))); }

Formula

a(n) = A051903(A036537(n)).
a(n) = 2^A372467(n) - 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (d(1) + Sum_{k>=2} (2^k-1) * (d(k) - d(k-1))) / A327839 = 1.25306367526166810834..., where d(k) = Product_{p prime} (1 - 1/p + Sum_{i=1..k} (1/p^(2^i-1)-1/p^(2^i))).

A382662 The unitary totient function applied to the cubefree numbers (A004709).

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 8, 4, 10, 6, 12, 6, 8, 16, 8, 18, 12, 12, 10, 22, 24, 12, 18, 28, 8, 30, 20, 16, 24, 24, 36, 18, 24, 40, 12, 42, 30, 32, 22, 46, 48, 24, 32, 36, 52, 40, 36, 28, 58, 24, 60, 30, 48, 48, 20, 66, 48, 44, 24, 70, 72, 36, 48, 54, 60, 24, 78, 40
Offset: 1

Views

Author

Amiram Eldar, Apr 02 2025

Keywords

Links

Crossrefs

Cf. A002117, A004709, A047994.
Similar sequences: A366440, A366536, A366537, A368712, A368779, A369889, A376365, A376366, A379717, A382419, A382663.

Programs

  • Mathematica
    f[p_, e_] := p^e-1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; uphi /@ Select[Range[100], cubeFreeQ]
  • PARI
    uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2]-1); }
iscubefree(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] > 2, return (0))); 1; }
list(lim) = apply(uphi, select(iscubefree, vector(lim, i, i)));

Formula

a(n) = A047994(A004709(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)^2/2) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.41625329674394407438... .

A382663 The unitary Jordan totient function applied to the cubefree numbers (A004709).

Original entry on oeis.org

1, 3, 8, 15, 24, 24, 48, 80, 72, 120, 120, 168, 144, 192, 288, 240, 360, 360, 384, 360, 528, 624, 504, 720, 840, 576, 960, 960, 864, 1152, 1200, 1368, 1080, 1344, 1680, 1152, 1848, 1800, 1920, 1584, 2208, 2400, 1872, 2304, 2520, 2808, 2880, 2880, 2520, 3480, 2880
Offset: 1

Views

Author

Amiram Eldar, Apr 02 2025

Keywords

Links

Crossrefs

Cf. A002117, A004709, A191414.
Similar sequences: A366440, A366536, A366537, A368712, A368779, A369889, A376365, A376366, A379717, A382419, A382662.

Programs

  • Mathematica
    f[p_, e_] := p^(2*e)-1; uj2[1] = 1; uj2[n_] := Times @@ f @@@ FactorInteger[n]; cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; uj2 /@ Select[Range[100], cubeFreeQ]
  • PARI
    uj2(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2*f[i, 2])-1); }
iscubefree(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] > 2, return (0))); 1; }
list(lim) = apply(uj2, select(iscubefree, vector(lim, i, i)));

Formula

a(n) = A191414(A004709(n)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)^3/3) * Product_{p prime} (1 - 2/p^3 + 1/p^4 - 1/p^6 + 1/p^7) = 0.42656661743049439763... .
Showing 1-7 of 7 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.