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 1-3 of 3 results.

A347385 Dedekind psi function applied to the odd part of n: a(n) = A001615(A000265(n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 31 2021

Keywords

Comments

Coincides with A000593 on A122132.

Links

Crossrefs

Cf. A000265, A001615, A122132, A185199, A206787, A336651, A347386, A347387, A351035.
Cf. also A000593, A053575.

Programs

  • Mathematica
    f[p_, e_] := If[p == 2, 1, (p + 1)*p^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 31 2021 *)
  • PARI
    A347385(n) = if(1==n,n, my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));

Formula

Multiplicative with a(2^e) = 1, a(p^e) = (p+1)*p^(e-1) for all odd primes p.
a(n) = A001615(A000265(n)).
a(n) = A206787(n) * A336651(n). - Antti Karttunen, Feb 11 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = 4/Pi^2 = 0.405284... (A185199). - Amiram Eldar, Nov 19 2022
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s))*(4^s-2^(s+1))/(4^s-1). - Amiram Eldar, Jan 04 2023

A347386 Number of iterations of A347385 (Dedekind psi function applied to the odd part of n) needed to reach a power of 2.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 0, 2, 2, 2, 1, 2, 1, 2, 0, 3, 2, 3, 2, 1, 2, 2, 1, 3, 2, 3, 1, 3, 2, 1, 0, 2, 3, 2, 2, 4, 3, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 2, 3, 3, 2, 4, 3, 3, 1, 3, 3, 3, 2, 2, 1, 2, 0, 2, 2, 4, 3, 2, 2, 3, 2, 5, 4, 3, 3, 2, 2, 3, 2, 4, 2, 2, 1, 4, 3, 3, 2, 4, 3, 2, 2, 1, 2, 3, 1, 3, 2, 3, 3, 4, 3, 3, 2, 2
Offset: 1

Views

Author

Antti Karttunen, Aug 31 2021

Keywords

Comments

Also, for n > 1, one less than the number of iterations of A347385 to reach 1.

Links

Crossrefs

Cf. A000265, A001615, A209229, A347385, A347387 (the exponent of the eventual power of 2 reached).
Cf. also A003434, A019269, A227944, A256757, A331410, A336361 for similar sequences.

Programs

  • Mathematica
    f[p_, e_] := If[p == 2, 1, (p + 1)*p^(e - 1)]; psiOdd[1] = 1; psiOdd[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := -1 + Length @ NestWhileList[psiOdd, n, # != 2^IntegerExponent[#, 2] &]; Array[a, 100] (* Amiram Eldar, Aug 31 2021 *)
  • PARI
    A347385(n) = if(1==n,n, my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
A347386(n) = if(!bitand(n, n-1), 0, 1+A347386(A347385(n)));

Formula

If A209229(n) = 1, then a(n) = 0, otherwise a(n) = 1 + a(A001615(A000265(n))).
For all n >= 1, a(n) <= A331410(n).

A347388 The first 3-smooth number eventually reached when iterating Dedekind psi function from n, with a(n) = n if n is already a 3-smooth number.

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 8, 8, 9, 18, 12, 12, 24, 24, 24, 16, 18, 18, 36, 36, 32, 36, 24, 24, 72, 96, 27, 48, 72, 72, 32, 32, 48, 54, 48, 36, 144, 144, 96, 72, 96, 96, 72, 72, 72, 72, 48, 48, 96, 216, 72, 192, 54, 54, 72, 96, 144, 216, 144, 144, 96, 96, 96, 64, 192, 144, 108, 108, 96, 144, 72, 72, 576, 576, 288, 288, 96, 384
Offset: 1

Views

Author

Antti Karttunen, Aug 31 2021

Keywords

Comments

See comments and references in A019269, which gives the number of iterations needed to reach a(n).

Crossrefs

Cf. A001615, A003586, A006530, A019269, A065333.
Cf. also A347387.

Programs

  • Mathematica
    psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); a[n_] := NestWhile[psi, n, FactorInteger[#][[-1, 1]] > 3 &]; Array[a, 100] (* Amiram Eldar, Aug 31 2021 *)
  • PARI
    A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A347388(n) = if(A006530(n)<=3,n,A347388(A001615(n)));

Formula

If A006530(n) <= 3, then a(n) = n, otherwise a(n) = a(A001615(n)).
Showing 1-3 of 3 results.

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