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 1-8 of 8 results.

A374456 The Euler phi function values of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 4, 4, 10, 12, 6, 8, 16, 18, 12, 10, 22, 8, 12, 18, 28, 8, 30, 16, 20, 16, 24, 36, 18, 24, 16, 40, 12, 42, 22, 46, 32, 52, 18, 40, 24, 36, 28, 58, 60, 30, 48, 20, 66, 44, 24, 70, 72, 36, 60, 24, 78, 40, 82, 64, 42, 56, 40, 88, 72, 60, 46, 72, 32, 96
Offset: 1

Views

Author

Amiram Eldar, Jul 09 2024

Keywords

Links

Crossrefs

Cf. A000010, A065463, A268335, A307868.
Similar sequences related to phi: A002618, A049200, A323333, A358039.
Similar sequences related to exponentially odd numbers: A366438, A366439, A366534, A366535, A367417, A368711, A374457.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], (p-1) * p^(e-1), 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100], # > 0 &]
  • PARI
    s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, (f[i, 1]-1) * f[i, 1]^(f[i, 2] - 1), 0));}
lista(kmax) = {my(s1); for(k = 1, kmax, s1 = s(k); if(s1 > 0, print1(s1, ", ")));}

Formula

a(n) = A000010(A268335(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A307868 / A065463^2 = 0.95051132596733153581... .

A367406 The exponentially odd numbers (A268335) multiplied by their squarefree kernels (A007947).

Original entry on oeis.org

1, 4, 9, 25, 36, 49, 16, 100, 121, 169, 196, 225, 289, 361, 441, 484, 529, 144, 676, 81, 841, 900, 961, 64, 1089, 1156, 1225, 1369, 1444, 1521, 400, 1681, 1764, 1849, 2116, 2209, 2601, 2809, 324, 3025, 784, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761
Offset: 1

Views

Author

Amiram Eldar, Nov 17 2023

Keywords

Comments

Analogous to A355038, with the exponentially odd numbers instead of the square numbers (A000290).
This sequence is a permutation of the square numbers.

Links

Crossrefs

Cf. A000290, A007947, A053143, A064549, A065463, A268335, A355038, A367407, A367417, A367418.

Programs

  • Mathematica
    s[n_] := n * Times @@ FactorInteger[n][[;;, 1]]; s /@ Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]
  • PARI
    b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, f[i,1]^(f[i,2]+1), 0));}
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(b1, ", ")));}

Formula

a(n) = A064549(A268335(n)).
a(n) = A268335(n)*A367417(n).
a(n) = A367407(n)^2.
a(n) = A268335(n)^2/A367418(n).
Sum_{k=1..n} a(k) = c * n^3 / 3, where c = (Pi^2/(15*d^3)) * Product_{p prime} (1 - 1/(p^3*(p+1))) = 1.78385074227198915372..., and d = A065463 is the asymptotic density of the exponentially odd numbers.
a(n) = A053143(A268335(n)). - Amiram Eldar, Nov 30 2023

A367407 a(n) = sqrt(A367406(n)).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 4, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 12, 26, 9, 29, 30, 31, 8, 33, 34, 35, 37, 38, 39, 20, 41, 42, 43, 46, 47, 51, 53, 18, 55, 28, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 44, 89, 91, 93, 94, 95, 24, 97
Offset: 1

Views

Author

Amiram Eldar, Nov 17 2023

Keywords

Comments

A permutation of the positive integers.

Links

Crossrefs

Cf. A064549, A268335, A367406, A367417, A367419.
Cf. A065463, A065465, A253905.

Programs

  • Mathematica
    s[n_] := Sqrt[n * Times @@ FactorInteger[n][[;;, 1]]]; s /@ Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]
  • PARI
    b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, f[i,1]^(f[i,2]+1), 0));}
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(sqrtint(b1), ", ")));}

Formula

a(n) = sqrt(A064549(A268335(n))).
a(n) = sqrt(A268335(n)*A367417(n)).
a(n) = A268335(n)/A367419(n).
Sum_{k=1..n} a(k) = c * n^2 / 2, where c = (zeta(3)/(zeta(2)*d^2)) * Product_{p prime} (1 - 1/(p^2*(p+1))) = A253905 * A065465 / d^3 = 1.29812028442810841122..., and d = A065463 is the asymptotic density of the exponentially odd numbers (A268335).

A367418 The exponentially odd numbers (A268335) divided by their squarefree kernels (A007947).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 9, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1
Offset: 1

Views

Author

Amiram Eldar, Nov 17 2023

Keywords

Comments

Analogous to A102631, with the exponentially odd numbers instead of the square numbers (A000290).
All the terms are square numbers.

Links

Crossrefs

Cf. A000290, A008833, A102631, A268335, A367406, A367417, A367419.

Programs

  • Mathematica
    s[n_] := n / Times @@ FactorInteger[n][[;; , 1]]; s /@ Select[Range[200], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]
  • PARI
    b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, f[i, 1]^(f[i, 2]-1), 0)); }
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(b1, ", "))); }

Formula

a(n) = A003557(A268335(n)).
a(n) = A268335(n)/A367417(n).
a(n) = A367419(n)^2.
a(n) = A268335(n)^2/A367406(n).
a(n) = A008833(A268335(n)). - Amiram Eldar, Nov 30 2023

A367419 a(n) = sqrt(A367418(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1
Offset: 1

Views

Author

Amiram Eldar, Nov 17 2023

Keywords

Links

  • Amiram Eldar, Table of n, a(n) for n = 1..10000
  • Michael De Vlieger, Plot f(a(n)) at (x,y) = (n mod m, floor(n/m)) for m = 857 and n = 734449, where f is a color function such that 1 = gray, red indicates primes, gold composite prime powers, green squarefree composites, blue and purple numbers neither squarefree nor prime powers, but purple additionally represents squareful numbers that are not prime powers.

Crossrefs

Cf. A003557, A065463, A268335, A367407, A367417, A367418.

Programs

  • Mathematica
    s[n_] := Sqrt[n / Times @@ FactorInteger[n][[;; , 1]]]; s /@ Select[Range[200], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]
  • PARI
    b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, f[i, 1]^(f[i, 2]-1), 0)); }
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(sqrtint(b1), ", "))); }

Formula

a(n) = sqrt(A003557(A268335(n))) = sqrt(A268335(n)/A367417(n)).
a(n) = A268335(n)/A367407(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/A065463 = 1.41956288050548591931... . - Amiram Eldar, Nov 17 2023

A374457 The Dedekind psi function values of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 4, 6, 12, 8, 12, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 48, 42, 36, 30, 72, 32, 48, 48, 54, 48, 38, 60, 56, 72, 42, 96, 44, 72, 48, 72, 54, 108, 72, 96, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 144, 90
Offset: 1

Views

Author

Amiram Eldar, Jul 09 2024

Keywords

Links

Crossrefs

Cf. A001615, A065463, A268335.
Similar sequences related to psi: A000082, A033196, A323332, A371413, A371415.
Similar sequences related to exponentially odd numbers: A366438, A366439, A366534, A366535, A367417, A368711, A374456.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], (p+1) * p^(e-1), 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100], # > 0 &]
  • PARI
    s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, (f[i, 1]+1) * f[i, 1]^(f[i, 2] - 1), 0));}
lista(kmax) = {my(s1); for(k = 1, kmax, s1 = s(k); if(s1 > 0, print1(s1, ", ")));}

Formula

a(n) = A001615(A268335(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 / A065463^2 = 2.01515877170903249510... .

A382660 The unitary totient function applied to the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 7, 4, 10, 12, 6, 8, 16, 18, 12, 10, 22, 14, 12, 26, 28, 8, 30, 31, 20, 16, 24, 36, 18, 24, 28, 40, 12, 42, 22, 46, 32, 52, 26, 40, 42, 36, 28, 58, 60, 30, 48, 20, 66, 44, 24, 70, 72, 36, 60, 24, 78, 40, 82, 64, 42, 56, 70, 88, 72, 60, 46, 72
Offset: 1

Views

Author

Amiram Eldar, Apr 02 2025

Keywords

Links

Crossrefs

Cf. A013662, A047994, A065463, A268335.
Similar sequences: A358346, A363825, A366438, A366439, A366534, A366535, A367417, A368711, A368979, A374456, A382661.

Programs

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

Formula

a(n) = A047994(A268335(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(4)/(2*d^2)) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^5) = 0.504949539649594981601..., and d = A065463 is the asymptotic density of the exponentially odd numbers.

A382661 The unitary Jordan totient function applied to the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 8, 24, 24, 48, 63, 72, 120, 168, 144, 192, 288, 360, 384, 360, 528, 504, 504, 728, 840, 576, 960, 1023, 960, 864, 1152, 1368, 1080, 1344, 1512, 1680, 1152, 1848, 1584, 2208, 2304, 2808, 2184, 2880, 3024, 2880, 2520, 3480, 3720, 2880, 4032, 2880, 4488, 4224
Offset: 1

Views

Author

Amiram Eldar, Apr 02 2025

Keywords

Links

Crossrefs

Cf. A013664, A065463, A191414, A268335.
Similar sequences: A358346, A363825, A366438, A366439, A366534, A366535, A367417, A368711, A368979, A374456, A382660.

Programs

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

Formula

a(n) = A191414(A268335(n)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(6)/(3*d^3)) * Product_{p prime} (1 - 1/p^2 + 1/p^5 - 2/p^6 + 1/p^7) = 0.59726984314764530141..., and d = A065463 is the asymptotic density of the exponentially odd numbers.
Showing 1-8 of 8 results.

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