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.

A318473 Additive with a(p^e) = A000045(e+1).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 5, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 8, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 13, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 6, 5, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 9, 1, 3, 3, 4, 1, 3, 1, 4, 3
Offset: 1

Views

Author

Antti Karttunen, Aug 29 2018

Keywords

Links

Crossrefs

Cf. A000045, A077761, A318471, A318474.
Differs from A008481 for the first time at n=32, where a(32)=8, while A008481(32)=7.

Programs

  • Mathematica
    a[n_] := Total@ Fibonacci[FactorInteger[n][[;; , 2]] + 1]; a[1] = 0; Array[a, 100] (* Amiram Eldar, May 15 2023 *)
  • PARI
    A318473(n) = vecsum(apply(e -> fibonacci(1+e),factor(n)[,2]));

Formula

a(n) = A007814(A318474(n)).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{k>=2} Fibonacci(k-1) * P(k) = 1.30985781707683753402..., where P(s) is the prime zeta function. - Amiram Eldar, Oct 09 2023

A127669 Number of numbers mapped to A127668(n) with the map described there.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 5, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 3, 2, 7, 2, 2, 2, 5, 1, 2, 2, 5, 1, 3, 1, 3, 3, 2, 1, 7, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 5, 1, 3, 3, 11, 2, 3, 1, 3, 2, 3, 1, 7, 2, 2, 3, 3, 2, 3, 2, 7, 5, 2, 1, 5, 2, 2, 2, 5, 1, 5, 2, 3, 2, 2, 2, 11, 1, 3, 3, 5
Offset: 2

Views

Author

Wolfdieter Lang Jan 23 2007

Keywords

Comments

This is not A008481(n), n>=2, which starts similarly, but differs, beginning with n=24.

Examples

			a(4)=2 because two numbers are mapped to 11= A127668(4), namely n=p(1)*p(1)=4 and n=p(11)=31. p(n)=A000041(n) (partition numbers).
a(24)=5 but A008481(24)=4.
The five numbers mapped to A127668(24)= 2111 are: 18433, 2594, 2263, 292, 24.

Links

Crossrefs

Cf. A008481, A127668.

Programs

  • Maple
    f:= proc(n) local S;
   nops(g(sprintf("%d",n)))
end proc:
g:= proc(s) option remember;
    local S,m,k1;
    if s[1] = "0" then return {} fi;
    S:= {[parse(s)]};
    for m from 1 to length(s)-1 do
      k1:= parse(s[1..m]);
      S:= S union map(t -> [k1,op(t)], select(r -> r[1] <= k1, procname(s[m+1..-1])));
    od;
    S;
end proc:
h:= proc(n) local F;
  F:= map(t -> numtheory:-pi(t[1])$t[2], sort(ifactors(n)[2],(a,b) -> a[1] > b[1]));
  parse(cat(op(F)))
end proc:
seq(f(h(i)),i=2..100); # Robert Israel, Dec 08 2024

Formula

a(n) <= pa(Length( A127668(n))), n>=2. Length gives the number of digits and pa(k):=A000041(k) (partition numbers). (It was originally claimed that this is equality, but that is not correct. - Franklin T. Adams-Watters, May 21 2014)

Extensions

Edited by Franklin T. Adams-Watters, May 21 2014
Corrected by Robert Israel, Dec 08 2024

A318312 Multiplicative with a(p^e) = 2^A000041(e).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 4, 32, 2, 8, 2, 8, 4, 4, 2, 16, 4, 4, 8, 8, 2, 8, 2, 128, 4, 4, 4, 16, 2, 4, 4, 16, 2, 8, 2, 8, 8, 4, 2, 64, 4, 8, 4, 8, 2, 16, 4, 16, 4, 4, 2, 16, 2, 4, 8, 2048, 4, 8, 2, 8, 4, 8, 2, 32, 2, 4, 8, 8, 4, 8, 2, 64, 32, 4, 2, 16, 4, 4, 4, 16, 2, 16, 4, 8, 4, 4, 4, 256, 2, 8, 8, 16, 2, 8, 2, 16, 8, 4, 2, 32, 2, 8, 4, 64
Offset: 1

Views

Author

Antti Karttunen, Aug 30 2018

Keywords

Links

Crossrefs

Cf. A000041, A008481.
Differs from A318474 for the first time at n=32, where a(32) = 128, while A318474(32) = 256.

Programs

  • PARI
    A318312(n) = factorback(apply(e -> 2^numbpart(e),factor(n)[,2]));

Formula

a(n) = 2^A008481(n).

A328892 If n = Product (p_j^k_j) then a(n) = Sum (2^(k_j - 1)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 3, 1, 2, 2, 8, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 4, 3, 1, 3, 1, 16, 2, 2, 2, 4, 1, 2, 2, 5, 1, 3, 1, 3, 3, 2, 1, 9, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 4, 1, 2, 3, 32, 2, 3, 1, 3, 2, 3, 1, 6, 1, 2, 3, 3, 2, 3, 1, 9, 8, 2, 1, 4, 2, 2, 2, 5, 1, 4
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 29 2019

Keywords

Examples

			a(72) = 6 because 72 = 2^3 * 3^2 and 2^(3 - 1) + 2^(2 - 1) = 6.

Links

Crossrefs

Cf. A000040 (positions of 1's), A008481, A011782, A162510, A324910.

Programs

  • Maple
    a:= n-> add(2^(i[2]-1), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 29 2019
  • Mathematica
    a[1] = 0; a[n_] := Plus @@ (2^(#[[2]] - 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 90}]
  • PARI
    a(n)={vecsum([2^(k-1) | k<-factor(n)[,2]])} \\ Andrew Howroyd, Oct 29 2019

Formula

If n = Product (p_j^k_j) then a(n) = Sum ordered partition(k_j).
Additive with a(p^e) = 2^(e-1).

A381014 If n = Product (p_j^k_j) then a(n) = Sum partition(p_j^k_j).

Original entry on oeis.org

0, 2, 3, 5, 7, 5, 15, 22, 30, 9, 56, 8, 101, 17, 10, 231, 297, 32, 490, 12, 18, 58, 1255, 25, 1958, 103, 3010, 20, 4565, 12, 6842, 8349, 59, 299, 22, 35, 21637, 492, 104, 29, 44583, 20, 63261, 61, 37, 1257, 124754, 234, 173525, 1960, 300, 106, 329931, 3012, 63, 37, 493, 4567, 831820, 15
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 10 2025

Keywords

Crossrefs

Cf. A000041, A008481.

Programs

  • Mathematica
    Join[{0}, Table[Plus @@ (PartitionsP[#[[1]]^#[[2]]] & /@ FactorInteger[n]), {n, 2, 60}]]
  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, numbpart(f[k,1]^f[k,2])); \\ Michel Marcus, Apr 17 2025

Formula

If n = Product (p_j^k_j) then a(n) = Sum A000041(p_j^k_j).
Showing 1-5 of 5 results.

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