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.

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A327967 Positions of records in A327966.

Original entry on oeis.org

0, 1, 2, 6, 9, 14, 33, 62, 177, 414, 1155, 1719, 2625, 4018, 6849, 9770, 17675, 30206, 90609, 164088, 336006, 757995, 1290874, 2029875, 4059746, 7037655, 17594075, 50850483, 68589598, 186888243, 373659254
Offset: 0

Views

Author

Antti Karttunen, Oct 01 2019

Keywords

Comments

Starting offset is zero to align with the indexing used in A189760, as this is conjecturally also the least k such that A327966(k) = n.
For at least n = 1, 3, 4, 5, 6, 7, 10, 14, 15, 17, 18, 21, 23, 24, 25, 26, 27, 28, 29, a(n) = A327965(a(1+n)). For example, 30206 = A327965(90609) and 90609 = A327965(164088).
Applying A327968 to these terms yields: 0, 0, 1, 5, 5, 5, 5, 5, 5, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, ...
See also comments in A189760.

Crossrefs

Cf. A327965, A327966.
Differs from A189760 for the first time at n=19, as a(19) = 164088, while A189760(19) = 260343.

Programs

  • PARI
    A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A327938(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]%f[k,1])); factorback(f); };
A327965(n) = if(n<=1,0,A327938(A003415(n)));
A327966(n) = if(!n,0,1+A327966(A327965(n)));
k=0; n=0; m=-1; while(k<32, if(!(n%2^27),print1("(",n,"),")); if((t=A327966(n))>m, write("b327967.txt", k, " ", n); m = t; k++); n++);

A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 8, 18, 12, 12, 14, 24, 24, 3, 18, 24, 20, 18, 32, 36, 24, 12, 24, 42, 16, 24, 30, 72, 32, 3, 48, 54, 48, 24, 38, 60, 56, 18, 42, 96, 44, 36, 48, 72, 48, 12, 48, 72, 72, 42, 54, 48, 72, 24, 80, 90, 60, 72, 62, 96, 64, 3, 84, 144, 68, 54
Offset: 1

Views

Author

José María Grau Ribas, Jan 04 2021

Keywords

Comments

Starting with any integer and repeatedly applying the map x -> a(x) reaches the fixed point 12 or the loop {3, 4}.

Examples

			a(2^s) = 3 for all s>0.

Links

Crossrefs

Cf. A003958, A003963, A048250, A068468, A064989, A167344, A327938, A293442, A064988, A108548, A124859, A279513, A324106, A322360, A326297, A340324, A340325, A340368.

Programs

  • Maple
    f:= proc(n) local  t;
  mul((t[1]+1)*(t[1]-1)^(t[2]-1),t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jan 07 2021
  • Mathematica
    fa[n_]:=fa[n]=FactorInteger[n];
phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i,Length[fa[n]]}];
Array[phi, 245]
  • PARI
    A340323(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,(f[i,1]+1)*((f[i,1]-1)^(f[i,2]-1)))); \\ Antti Karttunen, Jan 06 2021

Formula

a(n) = A167344(n) / A340368(n) = A048250(n) * A326297(n). - Antti Karttunen, Jan 06 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(6)/(2*zeta(2)*zeta(3))) * Product_{p prime} (1 + 2/p^2) = 0.56361239505... . - Amiram Eldar, Nov 12 2022

A328617 Multiplicative with a(p^e) = p^e, if e = 0 mod p, otherwise a(p^e) = p^((p*floor(e/p)) + A124223(A000720(p),e mod p)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 125, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 2401, 250, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 375, 76, 77, 78, 79, 80, 81
Offset: 1

Views

Author

Antti Karttunen, Oct 23 2019

Keywords

Links

Crossrefs

Cf. A124223, A327938, A289234.
Cf. also A328618, A328619.

Programs

  • PARI
    A328617(n) = { my(f = factor(n), m, q); for(k=1, #f~, q = (f[k, 2]\f[k, 1]); m = (f[k, 2]%f[k, 1]); if(m, f[k, 2] = q*f[k, 1] + lift(1/Mod(m,f[k, 1])))); factorback(f); };

Formula

For all n >= 0, A276085(a(A276086(n))) = A289234(n).

A328621 Multiplicative with a(p^e) = p^(2e mod p).

Original entry on oeis.org

1, 1, 9, 1, 25, 9, 49, 1, 3, 25, 121, 9, 169, 49, 225, 1, 289, 3, 361, 25, 441, 121, 529, 9, 625, 169, 1, 49, 841, 225, 961, 1, 1089, 289, 1225, 3, 1369, 361, 1521, 25, 1681, 441, 1849, 121, 75, 529, 2209, 9, 2401, 625, 2601, 169, 2809, 1, 3025, 49, 3249, 841, 3481, 225, 3721, 961, 147, 1, 4225, 1089, 4489, 289, 4761, 1225, 5041, 3, 5329
Offset: 1

Views

Author

Antti Karttunen, Oct 23 2019

Keywords

Links

Crossrefs

Cf. A011262, A327938, A328618 (a bijective variant).

Programs

  • PARI
    A328621(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = ((2*f[k,2])%f[k,1])); factorback(f); };

A359593 Multiplicative with a(p^e) = 1 if p divides e, p^e otherwise.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 8, 9, 10, 11, 3, 13, 14, 15, 1, 17, 18, 19, 5, 21, 22, 23, 24, 25, 26, 1, 7, 29, 30, 31, 32, 33, 34, 35, 9, 37, 38, 39, 40, 41, 42, 43, 11, 45, 46, 47, 3, 49, 50, 51, 13, 53, 2, 55, 56, 57, 58, 59, 15, 61, 62, 63, 1, 65, 66, 67, 17, 69, 70, 71, 72, 73, 74, 75, 19, 77, 78, 79, 5, 81, 82, 83, 21
Offset: 1

Views

Author

Antti Karttunen, Jan 09 2023

Keywords

Comments

Each term a(n) is a multiple of both A083346(n) and A327938(n).

Links

Crossrefs

Cf. A072873 (positions of 1's), A359594.
Cf. also A083346, A327938, A342014, A359552.

Programs

  • Mathematica
    f[p_, e_] := If[Divisible[e, p], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 09 2023 *)
  • PARI
    A359593(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^(f[k,2]*!!(f[k, 2]%f[k, 1]))); };
  • Python
    from math import prod
from sympy import factorint
def A359593(n): return prod(p**e for p, e in factorint(n).items() if e%p) # Chai Wah Wu, Jan 10 2023

Formula

a(n) = n / A359594(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - p^(p-1)*(p-1)/(p^(2*p)-1)) = 0.4225104173... . - Amiram Eldar, Jan 11 2023
Previous Showing 11-15 of 15 results.

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