A323372 -id:A323372 - cp's OEIS frontend

A323363 Dirichlet inverse of Dedekind's psi, A001615.

1, -3, -4, 3, -6, 12, -8, -3, 4, 18, -12, -12, -14, 24, 24, 3, -18, -12, -20, -18, 32, 36, -24, 12, 6, 42, -4, -24, -30, -72, -32, -3, 48, 54, 48, 12, -38, 60, 56, 18, -42, -96, -44, -36, -24, 72, -48, -12, 8, -18, 72, -42, -54, 12, 72, 24, 80, 90, -60, 72, -62, 96, -32, 3, 84, -144, -68, -54, 96, -144, -72, -12, -74, 114, -24

Offset: 1

Antti Karttunen, Jan 13 2019

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A048250 (absolute values).

Cf. A001615, A008836, A323364, A323372.

psi[n_] := If[n == 1, 1, n Times @@ (1 + 1/FactorInteger[n][[All, 1]])];
a[n_] := a[n] = If[n == 1, 1, -Sum[psi[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 75] (* Jean-François Alcover, Feb 15 2020 *)
f[p_, e_] := (-1)^e * (p + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
A323363(n) = if(1==n,1,-sumdiv(n,d,if(dA001615(n/d)*A323363(d),0)));

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} psi(k) * A(x^k). - Ilya Gutkovskiy, Sep 04 2019
From Amiram Eldar, Oct 14 2020: (Start)
Multiplicative with a(p^e) = (-1)^e * (p+1).
a(n) = A008836(n) * A048250(n). (End)
Dirichlet g.f.: zeta(2*s)/(zeta(s-1)*zeta(s)). - Amiram Eldar, Dec 05 2022

A323364 Sum of Dedekind's psi, A001615, and its Dirichlet inverse, A323363.

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 9, 16, 36, 0, 12, 0, 48, 48, 27, 0, 24, 0, 18, 64, 72, 0, 60, 36, 84, 32, 24, 0, 0, 0, 45, 96, 108, 96, 84, 0, 120, 112, 90, 0, 0, 0, 36, 48, 144, 0, 84, 64, 72, 144, 42, 0, 120, 144, 120, 160, 180, 0, 216, 0, 192, 64, 99, 168, 0, 0, 54, 192, 0, 0, 132, 0, 228, 96, 60, 192, 0, 0, 126, 112, 252, 0, 288, 216, 264, 240, 180, 0

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001615, A323363.

Cf. also A319340, A322581, A323365, A323372, A323399.

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
A323363(n) = if(1==n,1,-sumdiv(n,d,if(dA001615(n/d)*A323363(d),0)));
A323364(n) = (A001615(n) + A323363(n));

A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 6, 7, 3, 8, 3, 9, 9, 10, 3, 11, 3, 12, 12, 13, 3, 14, 14, 15, 15, 16, 3, 17, 3, 18, 18, 19, 19, 20, 3, 21, 21, 22, 3, 23, 3, 24, 24, 25, 3, 26, 26, 27, 27, 28, 3, 29, 29, 30, 30, 31, 3, 32, 3, 33, 33, 34, 34, 35, 3, 36, 36, 37, 3, 38, 3, 39, 39, 40, 40, 41, 3, 42, 42, 43, 3, 44, 44, 45, 45, 46, 3, 47, 47, 48, 48, 49, 49, 50, 3, 51, 51, 52, 3, 53, 3, 54, 54

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

This sequence is a restricted growth sequence transform of a function f which is defined as f(n) = A004526(n), unless n is an odd prime, in which case f(n) = -1, which is a constant not in range of A004526. See the Crossrefs section for a list of similar sequences.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A039636(i) = A039636(j).
For all i, j: a(i) = a(j) <=> A323161(i+1) = A323161(j+1).
The shifted version of this filter, A323161, has a remarkable ability to find many sequences related to primes and prime chains. - Antti Karttunen, Jan 06 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A004526, A039636, A305801, A323161.
A list of few similarly constructed sequences follows, where each sequence is an rgs-transform of such function f, for which the value of f(n) is the n-th term of the sequence whose A-number follows after a parenthesis, unless n is of the form ..., in which case f(n) is given a constant value outside of the range of that sequence:
  A322809 (A004526, unless an odd prime) [This sequence],
  A322589 (A007913, unless an odd prime),
  A322591 (A007947, unless an odd prime),
  A322805 (A252463, unless an odd prime),
  A323082 (A300840, unless an odd prime),
  A322822 (A300840, unless n > 2 and a Fermi-Dirac prime, A050376),
  A322988 (A322990, unless a prime power > 2),
  A323078 (A097246, unless an odd prime),
  A322808 (A097246, unless a squarefree number > 2),
  A322816 (A048675, unless an odd prime),
  A322807 (A285330, unless an odd prime),
  A322814 (A286621, unless an odd prime),
  A322824 (A242424, unless an odd prime),
  A322973 (A006370, unless an odd prime),
  A322974 (A049820, unless n > 1 and n is in A046642),
  A323079 (A060681, unless an odd prime),
  A322587 (A295887, unless an odd prime),
  A322588 (A291751, unless an odd prime),
  A322592 (A289625, unless an odd prime),
  A323369 (A323368, unless an odd prime),
  A323371 (A295886, unless an odd prime),
  A323374 (A323373, unless an odd prime),
  A323401 (A323372, unless an odd prime),
  A323405 (A323404, unless an odd prime).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A322809aux(n) = if((n>2)&&isprime(n),-1,(n>>1));
v322809 = rgs_transform(vector(up_to,n,A322809aux(n)));
A322809(n) = v322809[n];

a(n) = A323161(n+1) - 1.

A323401 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A323363(n)] for all other numbers, except f(n) = 0 for odd primes.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 10, 11, 3, 12, 3, 13, 14, 15, 3, 16, 17, 18, 19, 20, 3, 21, 3, 22, 23, 24, 23, 25, 3, 26, 27, 28, 3, 29, 3, 30, 31, 32, 3, 33, 34, 35, 32, 36, 3, 37, 32, 38, 39, 40, 3, 41, 3, 42, 43, 44, 45, 46, 3, 47, 42, 46, 3, 48, 3, 49, 50, 51, 42, 52, 3, 53, 54, 55, 3, 56, 57, 58, 59, 60, 3, 61, 62, 63, 64, 65, 59, 66, 3, 67, 68, 69, 3, 70, 3

Offset: 1

Antti Karttunen, Jan 15 2019

nonn

Restricted growth sequence transform of function f, defined as f(n) = A323372(n) for all other numbers n, except f(p) = 0 for odd primes p.
For all i, j:
  A323400(i) = A323400(j) => a(i) = a(j),
  a(i) = a(j) => A322588(i) = A322588(j),
  a(i) = a(j) => A323364(i) = A323364(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001615, A322588, A323363, A323364, A323400,

Cf. also A323371, A323372.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
v323363 = DirInverse(vector(up_to,n,A001615(n)));
A323363(n) = v323363[n];
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
Aux323401(n) = if((n>2)&&isprime(n), 0, [A003557(n), A323363(n)]);
v323401 = rgs_transform(vector(up_to, n, Aux323401(n)));
A323401(n) = v323401[n];

A323400 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000035(n), A003557(n), A323363(n)] for all other numbers, except f(n) = 0 for odd primes.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 11, 12, 3, 13, 3, 14, 15, 16, 3, 17, 18, 19, 20, 21, 3, 22, 3, 23, 24, 25, 24, 26, 3, 27, 28, 29, 3, 30, 3, 31, 32, 33, 3, 34, 35, 36, 37, 38, 3, 39, 37, 40, 41, 42, 3, 43, 3, 44, 45, 46, 47, 48, 3, 49, 50, 48, 3, 51, 3, 52, 53, 54, 50, 55, 3, 56, 57, 58, 3, 59, 60, 61, 62, 63, 3, 64, 65, 66, 67, 68, 62, 69, 3, 70, 71, 72, 3, 73, 3

Offset: 1

Antti Karttunen, Jan 15 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A323369(i) = A323369(j),
  a(i) = a(j) => A323401(i) = A323401(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000035, A001615, A322588, A323363, A323369, A323400.

Cf. also A305801, A323370, A323372.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
v323363 = DirInverse(vector(up_to,n,A001615(n)));
A323363(n) = v323363[n];
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
Aux323400(n) = if((n>2)&&isprime(n), 0, [(n%2), A003557(n), A323363(n)]);
v323400 = rgs_transform(vector(up_to, n, Aux323400(n)));
A323400(n) = v323400[n];

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A323363 Dirichlet inverse of Dedekind's psi, A001615.

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A323364 Sum of Dedekind's psi, A001615, and its Dirichlet inverse, A323363.

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A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

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A323401 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A323363(n)] for all other numbers, except f(n) = 0 for odd primes.

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A323400 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000035(n), A003557(n), A323363(n)] for all other numbers, except f(n) = 0 for odd primes.

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