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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.

A174725 a(n) = (A074206(n) + A008683(n))/2.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 2, 2, 4, 0, 4, 0, 4, 2, 2, 0, 10, 1, 2, 2, 4, 0, 6, 0, 8, 2, 2, 2, 13, 0, 2, 2, 10, 0, 6, 0, 4, 4, 2, 0, 24, 1, 4, 2, 4, 0, 10, 2, 10, 2, 2, 0, 22, 0, 2, 4, 16, 2, 6, 0, 4, 2, 6, 0, 38, 0, 2, 4, 4, 2
Offset: 1

Views

Author

Mats Granvik, Mar 28 2010

Keywords

Comments

From Mats Granvik, May 25 2017: (Start)
A074206(n) = A002033(n-1) = a(n) + A174726(n).
A008683(n) = a(n) - A174726(n).
Let m = size of matrix a matrix T, and let T be defined as follows:
T(n,k) = if m = 1 then 1 else if mod(n, k) = 0 then if and(n = k, n = m) then 0 else 1 else if and(n = 1, k = m) then 1 else 0
a(n) is then the number of permutation matrices with a positive contribution in the determinant of matrix T. The determinant of T is equal to the Möbius function A008683, see Mathematica program below for how to compute the determinant.
A174726 is the number of permutation matrices with a negative contribution in the determinant of matrix T.
(End)
From Gus Wiseman, Jan 04 2021: (Start)
Also the number of ordered factorizations of n into an even number of factors > 1. The non-ordered case is A339846. For example, the a(n) factorizations for n = 12, 24, 30, 32, 36 are:
(2*6) (3*8) (5*6) (4*8) (4*9)
(3*4) (4*6) (6*5) (8*4) (6*6)
(4*3) (6*4) (10*3) (16*2) (9*4)
(6*2) (8*3) (15*2) (2*16) (12*3)
(12*2) (2*15) (2*2*2*4) (18*2)
(2*12) (3*10) (2*2*4*2) (2*18)
(2*2*2*3) (2*4*2*2) (3*12)
(2*2*3*2) (4*2*2*2) (2*2*3*3)
(2*3*2*2) (2*3*2*3)
(3*2*2*2) (2*3*3*2)
(3*2*2*3)
(3*2*3*2)
(3*3*2*2)
(End)

Links

Crossrefs

Cf. A008683, A051731.
The odd version is A174726.
The unordered version is A339846.
A001055 counts factorizations, with strict case A045778.
A058696 counts partitions of even numbers, ranked by A300061.
A074206 counts ordered factorizations, with strict case A254578.
A251683 counts ordered factorizations by product and length.
Other cases of even length:
- A024430 counts set partitions of even length.
- A027187 counts partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A067661 counts strict partitions of even length.
- A332305 counts strict compositions of even length
Cf. A002033, A027193, A028260, A050320, A058695, A236913, A316439, A320655, A320656, A339890, A378216 (Dirichlet inverse).

Programs

  • Mathematica
    (* From Mats Granvik, May 25 2017: (Start) *)
Clear[t, nn]; nn = 77; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k == 1, Sum[t[n, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; Monitor[Table[Sum[If[Mod[n, k] == 0, MoebiusMu[k]*t[n/k, 1], 0], {k, 1, 77}], {n, 1, nn}], n]
(* The Möbius function as a determinant *) Table[Det[Table[Table[If[m == 1, 1, If[Mod[n, k] == 0, If[And[n == k, n == m], 0, 1], If[And[n == 1, k == m], 1, 0]]], {k, 1, m}], {n, 1, m}]], {m, 1, 42}]
(* (End) *)
ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n],EvenQ@*Length]],{n,100}] (* Gus Wiseman, Jan 04 2021 *)

Formula

a(n) = (Mobius transform of a(n)) + (Mobius transform of A174726). - Mats Granvik, Apr 04 2010
From Mats Granvik, May 25 2017: (Start)
This sequence is the Moebius transform of A074206.
a(n) = (A074206(n) + A008683(n))/2.
(End)
G.f. A(x) satisfies: A(x) = x + Sum_{i>=2} Sum_{j>=2} A(x^(i*j)). - Ilya Gutkovskiy, May 11 2019

Extensions

References to A002033(n-1) changed to A074206(n) by Antti Karttunen, Nov 23 2024

A344587 Deficiency of prime-shifted n: a(n) = 2*A003961(n) - sigma(A003961(n)).

Original entry on oeis.org

1, 2, 4, 5, 6, 6, 10, 14, 19, 10, 12, 12, 16, 18, 22, 41, 18, 26, 22, 22, 38, 22, 28, 30, 41, 30, 94, 42, 30, 18, 36, 122, 46, 34, 58, 47, 40, 42, 62, 58, 42, 42, 46, 52, 102, 54, 52, 84, 109, 66, 70, 72, 58, 126, 70, 114, 86, 58, 60, 6, 66, 70, 178, 365, 94, 54, 70, 82, 110, 78, 72, 110, 78, 78, 148, 102, 118, 78
Offset: 1

Views

Author

Antti Karttunen, May 28 2021

Keywords

Comments

First negative value occurs as a(120) = -30.
Questions: Which subsets of natural numbers generate the "cut sigmoid" graph(s) that cross the X-axis in the (lowermost) scatter plot?

Links

Crossrefs

Cf. A000203, A003961, A003973, A033879, A153881, A336851, A337386 (positions of terms <= 0), A346246 (Dirichlet inverse), A349387, A378216, A378231 [= a(n^2)].
Inverse Möbius transform of A337544.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A344587(n) = { my(u=A003961(n)); (u+u - sigma(u)); };

Formula

a(n) = A033879(A003961(n)) = 2*A003961(n) - A003973(n).
a(n) = Sum_{d|n} A337544(d).
From Antti Karttunen, Nov 23 2024: (Start)
a(n) = Sum_{d|n} A003961(d)*A153881(n/d) = A003961(n) - A336851(n).
a(n) = Sum_{d|n} A033879(d)*A349387(n/d).
a(n) = Sum_{d|n} A003972(d)*A378216(n/d).
(End)

A378220 Dirichlet inverse of phi(A003961(n)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, -2, -4, -2, -6, 8, -10, -2, -4, 12, -12, 8, -16, 20, 24, -2, -18, 8, -22, 12, 40, 24, -28, 8, -6, 32, -4, 20, -30, -48, -36, -2, 48, 36, 60, 8, -40, 44, 64, 12, -42, -80, -46, 24, 24, 56, -52, 8, -10, 12, 72, 32, -58, 8, 72, 20, 88, 60, -60, -48, -66, 72, 40, -2, 96, -96, -70, 36, 112, -120, -72, 8, -78, 80, 24
Offset: 1

Views

Author

Antti Karttunen, Nov 23 2024

Keywords

Links

Crossrefs

Cf. A023900, A003961, A048673, A151800, A349385, A378216.
Dirichlet inverse of A003972.
Inverse Möbius transform of A346234.
After the initial term, A349385 doubled.
Cf. A346246, A378216.

Programs

  • Mathematica
    f[p_, e_] := 1 - NextPrime[p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 13 2025 *)
  • PARI
    A378220(n) = factorback(apply(p -> 1-nextprime(1+p), factor(n)[, 1]));
  • Python
    from math import prod
from sympy import nextprime, primefactors
def A378220(n): return prod(1-nextprime(p) for p in primefactors(n)) # Chai Wah Wu, Nov 23 2024

Formula

Multiplicative with a(p^e) = (1-q), where q = A151800(p), i.e., the least prime > p.
a(n) = A023900(A003961(n)).
For n > 1, a(n) = 2*A349385(n).
a(n) = Sum_{d|n} A346234(d).
a(n) = Sum_{d|n} A346246(d)*A378216(n/d).
Showing 1-3 of 3 results.

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