A323363 -id:A323363 - cp's OEIS frontend

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

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));

A323372 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A003557(i) = A003557(j) and A323363(i) = A323363(j).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 44, 49, 50, 51, 44, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 58, 62, 29, 65, 66, 67, 68, 69, 58, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 79

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

Restricted growth sequence transform of ordered pair [A003557(n), A323363(n)].
For all i, j:
  a(i) = a(j) => A291751(i) = A291751(j),
  a(i) = a(j) => A323364(i) = A323364(j).

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

Cf. A001615, A003557, A291750, A291751, A323363, A323364.

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); };
v323372 = rgs_transform(vector(up_to, n, [A003557(n), A323363(n)]));
A323372(n) = v323372[n];

A324335 a(n) = A323363(A005940(1+n)), where A005940 is the Doudna sequence and A323363 is the Dirichlet inverse of Dedekind's psi.

Original entry on oeis.org

1, -3, -4, 3, -6, 12, 4, -3, -8, 18, 24, -12, 6, -12, -4, 3, -12, 24, 32, -18, 48, -72, -24, 12, 8, -18, -24, 12, -6, 12, 4, -3, -14, 36, 48, -24, 72, -96, -32, 18, 96, -144, -192, 72, -48, 72, 24, -12, 12, -24, -32, 18, -48, 72, 24, -12, -8, 18, 24, -12, 6, -12, -4, 3, -18, 42, 56, -36, 84, -144, -48, 24, 112, -216, -288, 96, -72, 96, 32, -18

Offset: 0

Antti Karttunen, Feb 23 2019

sign

Antti Karttunen, Table of n, a(n) for n = 0..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A001615, A005940, A323363, A323915, A324336.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
memoA323363 = Map();
A323363(n) = if(1==n,1,my(v); if(mapisdefined(memoA323363,n,&v), v, v = -sumdiv(n,d,if(dA001615(n/d)*A323363(d),0)); mapput(memoA323363,n,v); (v)));
A324335(n) = A323363(A005940(1+n));

a(n) = A323363(A005940(1+n)).

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];

A344695 a(n) = gcd(sigma(n), psi(n)), where sigma is the sum of divisors function, A000203, and psi is the Dedekind psi function, A001615.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 3, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32, 108

Offset: 1

Antti Karttunen and Peter Munn, May 26 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 8, although a(4) = 1 and a(27) = 4. See A344702.
A more specific property holds: for prime p that does not divide n, a(p*n) = a(p) * a(n). In particular, on squarefree numbers (A005117) this sequence coincides with sigma and psi, which are multiplicative.
If prime p divides the squarefree part of n then p+1 divides a(n). (For example, 20 has square part 4 and squarefree part 5, so 5+1 divides a(20) = 6.) So a(n) = 1 only if n is square. The first square n with a(n) > 1 is a(196) = 21. See A344703.
Conjecture: the set of primes that appear in the sequence is A065091 (the odd primes). 5 does not appear as a term until a(366025) = 5, where 366025 = 5^2 * 11^4. At this point, the missing numbers less than 22 are 2, 10 and 17. 17 appears at the latest by a(17^2 * 103^16) = 17.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A001615, A005117, A244963, A344696, A344697, A344702, A344703 (numbers k for which a(k^2) > 1).
Cf. also A048250, A291750, A291751, A340070, A323363.
Subsets of range: A008864, A065091 (conjectured).

Table[GCD[DivisorSigma[1,n],DivisorSum[n,MoebiusMu[n/#]^2*#&]],{n,100}] (* Giorgos Kalogeropoulos, Jun 03 2021 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344695(n) = gcd(sigma(n), A001615(n));
(Python 3.8+)
from math import prod, gcd
from sympy import primefactors, divisor_sigma
def A001615(n):
    plist = primefactors(n)
    return n*prod(p+1 for p in plist)//prod(plist)
def A344695(n): return gcd(A001615(n),divisor_sigma(n)) # Chai Wah Wu, Jun 03 2021

a(n) = gcd(A000203(n), A001615(n)).
For prime p, a(p^e) = (p+1)^(e mod 2).
For prime p with gcd(p, n) = 1, a(p*n) = a(p) * a(n).
a(A007913(n)) | a(n).
a(n) = gcd(A000203(n), A244963(n)) = gcd(A001615(n), A244963(n)).
a(n) = A000203(n) / A344696(n).
a(n) = A001615(n) / A344697(n).

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

Ilya Gutkovskiy, Mar 03 2020

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

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

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

a[n_] := Times @@ ((#[[1]] + 1)^(#[[2]] - 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 85}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]++; f[k,2]--); factorback(f); \\ Michel Marcus, Mar 03 2020

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

A328639 Dirichlet g.f.: zeta(2s) / (zeta(s) zeta(s-2)).

Original entry on oeis.org

1, -5, -10, 5, -26, 50, -50, -5, 10, 130, -122, -50, -170, 250, 260, 5, -290, -50, -362, -130, 500, 610, -530, 50, 26, 850, -10, -250, -842, -1300, -962, -5, 1220, 1450, 1300, 50, -1370, 1810, 1700, 130, -1682, -2500, -1850, -610, -260, 2650, -2210, -50, 50, -130

Offset: 1

Ilya Gutkovskiy, Oct 23 2019

Dirichlet inverse of A065958.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008683, A008836, A026424 (positions of negative terms), A046970, A065958, A323363, A328640.

a[1] = 1; a[n_] := -Sum[DirichletConvolve[j^2, MoebiusMu[j]^2, j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 50}]
Table[DivisorSum[n, LiouvilleLambda[n/#] MoebiusMu[#] #^2 &], {n, 1, 50}]
f[p_, e_] := (-1)^e*(p^2 + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

a(n)={sumdiv(n, d, (-1)^bigomega(n/d)*moebius(d)*d^2)} \\ Andrew Howroyd, Oct 25 2019

a(1) = 1; a(n) = -Sum_{d|n, dA065958(n/d) * a(d).
a(n) = Sum_{d|n} lambda(n/d) * mu(d) * d^2, where lambda = A008836 and mu = A008683.
Multiplicative with a(p^e) = (-1)^e*(p^2 + 1). - Amiram Eldar, Nov 30 2020

A328640 Dirichlet g.f.: zeta(2s) / (zeta(s) zeta(s-3)).

Original entry on oeis.org

1, -9, -28, 9, -126, 252, -344, -9, 28, 1134, -1332, -252, -2198, 3096, 3528, 9, -4914, -252, -6860, -1134, 9632, 11988, -12168, 252, 126, 19782, -28, -3096, -24390, -31752, -29792, -9, 37296, 44226, 43344, 252, -50654, 61740, 61544, 1134, -68922, -86688, -79508, -11988, -3528

Offset: 1

Ilya Gutkovskiy, Oct 23 2019

Dirichlet inverse of A065959.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008683, A008836, A026424 (positions of negative terms), A063453, A065959, A323363, A328639.

a[1] = 1; a[n_] := -Sum[DirichletConvolve[j^3, MoebiusMu[j]^2, j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 45}]
Table[DivisorSum[n, LiouvilleLambda[n/#] MoebiusMu[#] #^3 &], {n, 1, 45}]
f[p_, e_] := (-1)^e*(p^3+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 03 2022 *)

a(n)={sumdiv(n, d, (-1)^bigomega(n/d)*moebius(d)*d^3)} \\ Andrew Howroyd, Oct 25 2019

a(1) = 1; a(n) = -Sum_{d|n, dA065959(n/d) * a(d).
a(n) = Sum_{d|n} lambda(n/d) * mu(d) * d^3, where lambda = A008836 and mu = A008683.
Multiplicative with a(p^) = (-1)^e*(p^3+1). - Amiram Eldar, Dec 03 2022

A347092 Dirichlet inverse of A322577, which is the convolution of Dedekind psi with Euler phi.

Original entry on oeis.org

1, -4, -6, 5, -10, 24, -14, -4, 10, 40, -22, -30, -26, 56, 60, 5, -34, -40, -38, -50, 84, 88, -46, 24, 26, 104, -6, -70, -58, -240, -62, -4, 132, 136, 140, 50, -74, 152, 156, 40, -82, -336, -86, -110, -100, 184, -94, -30, 50, -104, 204, -130, -106, 24, 220, 56, 228, 232, -118, 300, -122, 248, -140, 5, 260, -528, -134

Offset: 1

Antti Karttunen, Aug 18 2021

Multiplicative because A322577 is.

Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Cf. A000010, A001615, A322577, A347093.

Cf. A023900, A323363.

import Math.NumberTheory.Primes
a n = product . map (\(p, e) -> if even e then 1 + unPrime p^2 else -2*unPrime p) . factorise $ n -- Sebastian Karlsson, Oct 29 2021

f[p_, e_] := If[EvenQ[e], p^2 + 1, -2*p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 23 2023 *)

up_to = 16384;
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA322577(n)));
A347092(n) = v347092[n];

A347092(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]%2, -2*f[i, 1], 1+(f[i, 1]^2))); }; \\ (after Sebastian Karlsson's multiplicative formula) - Antti Karttunen, Nov 11 2021

from sympy import factorint, prod
def f(p, e): return 1 + p**2 if e%2 == 0 else -2*p
def a(n):
    factors = factorint(n)
    return prod(f(p, factors[p]) for p in factors) # Sebastian Karlsson, Oct 29 2021

a(1) = 1; a(n) = -Sum_{d|n, d < n} A322577(n/d) * a(d).
a(n) = A347093(n) - A322577(n).
From Sebastian Karlsson, Oct 29 2021: (Start)
Dirichlet g.f.: zeta(2*s)/zeta(s-1)^2.
a(n) = Sum_{d|n} A323363(n/d)*A023900(d).
Multiplicative with a(p^e) = 1 + p^2 if e is even, -2*p if e is odd. (End)

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A323372 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A003557(i) = A003557(j) and A323363(i) = A323363(j).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A324335 a(n) = A323363(A005940(1+n)), where A005940 is the Doudna sequence and A323363 is the Dirichlet inverse of Dedekind's psi.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A344695 a(n) = gcd(sigma(n), psi(n)), where sigma is the sum of divisors function, A000203, and psi is the Dedekind psi function, A001615.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A328639 Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(s-2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A328640 Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(s-3)).

Views

A328639 Dirichlet g.f.: zeta(2s) / (zeta(s) zeta(s-2)).

A328640 Dirichlet g.f.: zeta(2s) / (zeta(s) zeta(s-3)).