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.

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... .

A379832 The second Jordan totient function applied to the exponentially odd numbers.

Original entry on oeis.org

1, 3, 8, 24, 24, 48, 48, 72, 120, 168, 144, 192, 288, 360, 384, 360, 528, 384, 504, 648, 840, 576, 960, 768, 960, 864, 1152, 1368, 1080, 1344, 1152, 1680, 1152, 1848, 1584, 2208, 2304, 2808, 1944, 2880, 2304, 2880, 2520, 3480, 3720, 2880, 4032, 2880, 4488, 4224
Offset: 1

Views

Author

Amiram Eldar, Jan 03 2025

Keywords

Links

Crossrefs

Cf. A007434, A065463, A185197, A268335, A374456 (analogous with J_1 = phi), A379715, A379716, A379717, A379718, A379833.

Programs

  • Mathematica
    f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; expoddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; j2 /@ Select[Range[100], expoddQ]
  • PARI
    j2(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));}
isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
list(lim) = apply(j2, select(isexpodd, vector(lim, i, i)));

Formula

a(n) = A007434(A268335(n)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/(Pi^2 * Product_{p prime} (1 - 1/(p*(p+1)))^3) = A185197 / A065463^3 = 0.57968779180803379088... .
Sum_{n>=1} 1/a(n) = (Pi^6/540) * Product_{p prime} (1 - 1/p^4 + 1/p^6) = 1.67479534964539923068...
In general, Sum_{m exponentially odd} 1/J_k(m) = zeta(k) * zeta(2*k) * Product_{p prime} (1 - 1/p^(2*k) + 1/p^(3*k)), for k >= 2, where J_k is the k-th Jordan totient function.

A380165 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is an exponentially odd number.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jan 14 2025

Keywords

Links

Crossrefs

Cf. A000010, A000290, A013662, A077591, A268335, A350389, A351569, A365402, A374456, A380164.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], (p-1)*p^(e-1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(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), 1));}

Formula

a(n) = A000010(A350389(n)).
a(n) >= 1, with equality if and only if n is either a square (A000290) or twice and odd square (A077591 \ {1}).
a(n) <= A000010(n), with equality if and only if n is an exponentially odd number (A268335).
Multiplicative with a(p^e) = (p-1)*p^(e-1) if e is odd, and 1 otherwise.
Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 - 1/p^s + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-1) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^5) = 0.50115112192510092436... .

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 1-5 of 5 results.

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