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 1-6 of 6 results.

A327564 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)^(k_j - 1)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 9, 4, 1, 1, 3, 1, 1, 1, 27, 1, 4, 1, 3, 1, 1, 1, 9, 6, 1, 16, 3, 1, 1, 1, 81, 1, 1, 1, 12, 1, 1, 1, 9, 1, 1, 1, 3, 4, 1, 1, 27, 8, 6, 1, 3, 1, 16, 1, 9, 1, 1, 1, 3, 1, 1, 4, 243, 1, 1, 1, 3, 1, 1, 1, 36, 1, 1, 6, 3, 1, 1, 1, 27, 64, 1, 1, 3, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 03 2020

Keywords

Examples

			a(12) = a(2^2 * 3) = (2 + 1)^(2 - 1) * (3 + 1)^(1 - 1) = 3.

Links

Crossrefs

Cf. A001221, A003557, A003958, A003959, A003968, A005117 (positions of 1's), A007947, A048250, A064478, A064549, A173557, A323363, A325126, A326297, A340368, A348038, A348039, A347960.

Programs

  • Mathematica
    a[n_] := Times @@ ((#[[1]] + 1)^(#[[2]] - 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 85}]
  • PARI
    a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]++; f[k,2]--); factorback(f); \\ Michel Marcus, Mar 03 2020

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA001221(n/d) * A003557(n/d) * a(d).
a(n) = A003959(n) / A048250(n) = A003968(n) / A007947(n).
a(n) = A348038(n) * A348039(n) = A340368(n) / A173557(n). - Antti Karttunen, Oct 29 2021
Dirichlet g.f.: 1/(zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^s)). - Amiram Eldar, Dec 07 2023

A325126 a(1) = 1; a(n) = -Sum_{d|n, dA007947.

Original entry on oeis.org

1, -2, -3, 2, -5, 6, -7, -2, 6, 10, -11, -6, -13, 14, 15, 2, -17, -12, -19, -10, 21, 22, -23, 6, 20, 26, -12, -14, -29, -30, -31, -2, 33, 34, 35, 12, -37, 38, 39, 10, -41, -42, -43, -22, -30, 46, -47, -6, 42, -40, 51, -26, -53, 24, 55, 14, 57, 58, -59, 30
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 04 2019

Keywords

Comments

Dirichlet inverse of A007947.
Moebius transform of A125131.

Links

Crossrefs

Cf. A007947, A008836, A125131, A326297.
Cf. also A327564, A332732, A349340, A349612, A349618.

Programs

  • Mathematica
    a[n_] := If[n == 1, n, -Sum[If[d < n, Last[Select[Divisors[n/d], SquareFreeQ]] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 60}]
f[p_, e_] := -p*(1 - p)^(e - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)
  • PARI
    rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = {my(va=vector(nn)); va[1] = 1; for (n=2, nn, va[n] = -sumdiv(n, d, if (dMichel Marcus, Jun 01 2020

Formula

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} rad(k) * A(x^k).
From Isaac Saffold, May 30 2020: (Start)
a(n) = A008836(n)*A326297(n)*A007947(n).
Proof:
Define lambda(n) := A008836(n); h(n) := A326297(n); rad(n) := A007947(n).
As lambda(n), h(n), and rad(n) are multiplicative, the identity needs only to be proved for prime power n.
It is clear that the identity holds for n = 1 = p^0. For a given nonnegative integer k, assume the identity holds for all v such that 0 <= v <= k. Then, by the recursive formula for Dirichlet inverses,
a(p^(k+1)) = -Sum_{v=0..k} lambda(p^v)*h(p^v)*rad(p^v)*rad(p^(k+1-v))
= -p * (1 + p*Sum_{v=1..k}((-1)^v * (p-1)^(v-1)))
= -p * (1 - p*Sum_{v=0..(k-1)}((1 - p)^v))
= -p * (1 - p*(((1-p)^k - 1) / -p))
= -p * (1-p)^k
= (-1)^(k+1) * (p-1)^k * p
= lambda(p^(k+1)) * h(p^(k+1)) * rad(p^(k+1))
Thus the identity holds for p^(k+1), k >= 0.
As k is arbitrary and the identity holds for p^0, it holds for the prime powers, and thus for all positive integers. Q.E.D. (End)

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

A349340 Dirichlet inverse of A003557, where A003557 is multiplicative with a(p^e) = p^(e-1).

Original entry on oeis.org

1, -1, -1, -1, -1, 1, -1, -1, -2, 1, -1, 1, -1, 1, 1, -1, -1, 2, -1, 1, 1, 1, -1, 1, -4, 1, -4, 1, -1, -1, -1, -1, 1, 1, 1, 2, -1, 1, 1, 1, -1, -1, -1, 1, 2, 1, -1, 1, -6, 4, 1, 1, -1, 4, 1, 1, 1, 1, -1, -1, -1, 1, 2, -1, 1, -1, -1, 1, 1, -1, -1, 2, -1, 1, 4, 1, 1, -1, -1, 1, -8, 1, -1, -1, 1, 1, 1, 1, -1, -2, 1
Offset: 1

Views

Author

Antti Karttunen, Nov 18 2021

Keywords

Links

Crossrefs

Cf. A003557, A076479, A326297 (absolute values).
Cf. also A325126, A349350, A349619.

Programs

  • Mathematica
    f[p_, e_] := -(p - 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)
  • PARI
    A003557(n) = (n/factorback(factorint(n)[, 1]));
memoA349340 = Map();
A349340(n) = if(1==n,1,my(v); if(mapisdefined(memoA349340,n,&v), v, v = -sumdiv(n,d,if(dA003557(n/d)*A349340(d),0)); mapput(memoA349340,n,v); (v)));
  • PARI
    A349340(n) = { my(f=factor(n)); prod(i=1, #f~, -((f[i,1]-1)^(f[i,2]-1))); };

Formula

Multiplicative with a(p^e) = -((p-1)^(e-1)).
a(n) = A076479(n) * A326297(n).
a(1) = 1; a(n) = -Sum_{d|n, d < n} A003557(n/d) * a(d).

A351434 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)^(k_j + 1)).

Original entry on oeis.org

1, 1, 4, 1, 16, 4, 36, 1, 8, 16, 100, 4, 144, 36, 64, 1, 256, 8, 324, 16, 144, 100, 484, 4, 64, 144, 16, 36, 784, 64, 900, 1, 400, 256, 576, 8, 1296, 324, 576, 16, 1600, 144, 1764, 100, 128, 484, 2116, 4, 216, 64, 1024, 144, 2704, 16, 1600, 36, 1296, 784, 3364, 64, 3600, 900, 288, 1, 2304
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 11 2022

Keywords

Links

Crossrefs

Cf. A003557, A003958, A003959, A023900, A064549, A326297, A327564, A351419, A351435.

Programs

  • Maple
    a:= n-> mul((i[1]-1)^(i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..65);  # Alois P. Heinz, Feb 11 2022
  • Mathematica
    f[p_, e_] := (p - 1)^(e + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Table[a[n], {n, 1, 65}]
  • PARI
    a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]--; f[k,2]++); factorback(f); \\ Michel Marcus, Feb 11 2022

Formula

a(n) = A003958(n) * |A023900(n)|.
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (1 - (3*p^2 - 4*p + 2)/(p*(p^3 - p + 1))) = 0.1161464566... . - Amiram Eldar, Nov 19 2022

A351435 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)^(k_j + 1)).

Original entry on oeis.org

1, 9, 16, 27, 36, 144, 64, 81, 64, 324, 144, 432, 196, 576, 576, 243, 324, 576, 400, 972, 1024, 1296, 576, 1296, 216, 1764, 256, 1728, 900, 5184, 1024, 729, 2304, 2916, 2304, 1728, 1444, 3600, 3136, 2916, 1764, 9216, 1936, 3888, 2304, 5184, 2304, 3888, 512, 1944, 5184, 5292, 2916
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 11 2022

Keywords

Links

Crossrefs

Cf. A003557, A003958, A003959, A048250, A064549, A065478, A306633, A326297, A327564, A351434.

Programs

  • Maple
    a:= n-> mul((i[1]+1)^(i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..53);  # Alois P. Heinz, Feb 11 2022
  • Mathematica
    f[p_, e_] := (p + 1)^(e + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Table[a[n], {n, 1, 53}]
  • PARI
    a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]++; f[k,2]++); factorback(f); \\ Michel Marcus, Feb 11 2022

Formula

a(n) = A003959(n) * A048250(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = 1/(3 * Product_{p prime} (1 - p/(p^3-1))) = 1 /(3 * A065478) = 0.5787439255... . - Amiram Eldar, Nov 19 2022
Sum_{n>=1} 1/a(n) = zeta(2)/zeta(3) (A306633). - Amiram Eldar, Dec 15 2023
Showing 1-6 of 6 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.