A322328 -id:A322328 - cp's OEIS frontend

A370297 Inverse Moebius transform of A322328.

1, 5, 5, 13, 5, 25, 5, 25, 13, 25, 5, 65, 5, 25, 25, 41, 5, 65, 5, 65, 25, 25, 5, 125, 13, 25, 25, 65, 5, 125, 5, 61, 25, 25, 25, 169, 5, 25, 25, 125, 5, 125, 5, 65, 65, 25, 5, 205, 13, 65, 25, 65, 5, 125, 25, 125, 25, 25, 5, 325, 5, 25, 65, 85, 25, 125, 5, 65, 25, 125, 5, 325

Offset: 1

Werner Schulte, Feb 14 2024

Cf. A034444, A048691, A322328.

f[p_, e_] := 2*e^2 + 2*e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 14 2024 *)

a(n) = factorback(apply(e->1+2*e*(e+1), factor(n)[,2]))

Multiplicative with a(p^e) = 1 + 2*e*(e+1) for prime p and e >= 0.
Dirichlet g.f.: (zeta(s))^5 / (zeta(2*s))^2.
Dirichlet convolution of A034444 and A048691.
Dirichlet inverse sequence b(n) for n > 0 is multiplicative with b(p) = -5 and b(p^e) = (-1)^e * (8*e-4) for prime p and e > 1.

A113413 A Riordan array of coordination sequences.

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 2, 8, 6, 1, 2, 12, 18, 8, 1, 2, 16, 38, 32, 10, 1, 2, 20, 66, 88, 50, 12, 1, 2, 24, 102, 192, 170, 72, 14, 1, 2, 28, 146, 360, 450, 292, 98, 16, 1, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1, 2, 36, 258, 952, 1970, 2364, 1666, 688, 162, 20, 1, 2, 40, 326

Offset: 0

Paul Barry, Oct 29 2005

Columns include A040000, A008574, A005899, A008412, A008413, A008414. Row sums are A078057(n)=A001333(n+1). Diagonal sums are A001590(n+3). Reverse of A035607. Signed version is A080246. Inverse is A080245.
For another version see A122542. - Philippe Deléham, Oct 15 2006
T(n,k) is the number of length n words on alphabet {0,1,2} with no two consecutive 1's and no two consecutive 2's and having exactly k 0's. - Geoffrey Critzer, Jun 11 2015
From Eric W. Weisstein, Feb 17 2016: (Start)
Triangle of coefficients (from low to high degree) of x^-n * vertex cover polynomial of the n-ladder graph P_2 \square p_n:
  Psi_{L_1}: x*(2 + x) -> {2, 1}
  Psi_{L_2}: x^2*(2 + 4 x + x^2) -> {2, 4, 1}
  Psi_{L_3}: x^3*(2 + 8 x + 6 x^2 + x^3) -> {2, 8, 6, 1}
(End)
Let c(n, k), n > 0, be multiplicative sequences for some fixed integer k >= 0 with c(p^e, k) = T(e+k, k) for prime p and e >= 0. Then we have Dirichlet g.f.: Sum_{n>0} c(n, k) / n^s = zeta(s)^(2*k+2) / zeta(2*s)^(k+1). Examples: For k = 0 see A034444 and for k = 1 see A322328. Dirichlet convolution of c(n, k) and lambda(n) is Dirichlet inverse of c(n, k). - Werner Schulte, Oct 31 2022

			Triangle begins
  1;
  2,  1;
  2,  4,  1;
  2,  8,  6,  1;
  2, 12, 18,  8,  1;
  2, 16, 38, 32, 10,  1;
  2, 20, 66, 88, 50, 12,  1;

G. C. Greubel, Table of n, a(n) for the first 50 rows
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
P. Holub, M. Miller, H. Perez-Roses, and J. Ryan, Degree diameter problem on honeycomb networks, Disc. Appl. Math. 179 (2014) 139-151.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 2.
Mirka Miller, Hebert Perez-Roses, and Joe Ryan, The Maximum Degree-and-Diameter-Bounded Subgraph in the Mesh, arXiv preprint arXiv:1203.4069 [math.CO], 2012.

Other versions: A035607, A119800, A122542, A266213.

nn = 10; Map[Select[#, # > 0 &] &, CoefficientList[Series[1/(1 - 2 x/(1 + x) - y x), {x, 0, nn}], {x, y}]] // Grid (* Geoffrey Critzer, Jun 11 2015 *)
CoefficientList[CoefficientList[Series[1/(1 - 2 x/(1 + x) - y x), {x, 0, 10}, {y, 0, 10}], x], y] (* Eric W. Weisstein, Feb 17 2016 *)

T = lambda n,k : binomial(n, k)*hypergeometric([-k-1, k-n], [-n], -1).simplify_hypergeometric()
A113413 = lambda n,k : 1 if n==0 and k==0 else T(n, k)
for n in (0..12): print([A113413(n,k) for k in (0..n)]) # Peter Luschny, Sep 17 2014 and Mar 16 2016

# Alternatively:
def A113413_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)+2*sum(prec(n-i,k-1) for i in (2..n-k+1))
    return [prec(n, k) for k in (1..n)]
for n in (1..10): print(A113413_row(n)) # Peter Luschny, Mar 16 2016

From Paul Barry, Nov 13 2005: (Start)
Riordan array ((1+x)/(1-x), x(1+x)/(1-x)).
T(n, k) = Sum_{i=0..n-k} C(k+1, i)*C(n-i, k).
T(n, k) = Sum_{j=0..n-k} C(k+j, j)*C(k+1, n-k-j).
T(n, k) = D(n, k) + D(n-1, k) where D(n, k) = Sum_{j=0..n-k} C(n-k, j)*C(k, j)*2^j = A008288(n, k).
T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k-1).
T(n, k) = Sum_{j=0..n} C(floor((n+j)/2), k)*C(k, floor((n-j)/2)). (End)
T(n, k) = C(n, k)*hypergeometric([-k-1, k-n], [-n], -1). - Peter Luschny, Sep 17 2014
T(n, k) = (Sum_{i=2..k+2} A137513(k+2, i) * (n-k)^(i-2)) / (k!) for 0 <= k < n (conjectured). - Werner Schulte, Oct 31 2022

A322327 a(n) = A005361(n) * A034444(n).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 8, 2, 4, 4, 8, 2, 8, 2, 8, 4, 4, 2, 12, 4, 4, 6, 8, 2, 8, 2, 10, 4, 4, 4, 16, 2, 4, 4, 12, 2, 8, 2, 8, 8, 4, 2, 16, 4, 8, 4, 8, 2, 12, 4, 12, 4, 4, 2, 16, 2, 4, 8, 12, 4, 8, 2, 8, 4, 8, 2, 24, 2, 4, 8, 8, 4, 8, 2, 16, 8, 4, 2, 16, 4, 4, 4, 12, 2, 16, 4, 8, 4, 4, 4, 20, 2, 8, 8, 16

Offset: 1

Werner Schulte, Dec 03 2018

Conjecture: Let k be some fixed integer and a_k(n) = A005361(n) * k^A001221(n) for n > 0 with 0^0 = 1. Then a_k(n) is multiplicative with a_k(p^e) = k*e for prime p and e > 0. For k = 0 see A000007 (offset 1), for k = 1 see A005361, for k = 2 see this sequence, for k = 3 see A226602 (offset 1), and for k = 4 see A322328.

Dirichlet inverse b(n) [= A355837(n)] is multiplicative with b(p^e) = 2 * (e mod 2) * (-1)^((e+1)/2) for prime p and e > 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A000007, A001221, A001620, A005361, A034444, A226602, A227291, A286324, A322328, A355837 (Dirichlet inverse).

a[n_] := If[n==1, 1, Module[{f = FactorInteger[n]}, 2^Length[f] * Times@@f[[;;,2]]]]; Array[a, 100] (* Amiram Eldar, Dec 03 2018 *)

a(n) = my(f=factor(n)); vecprod(f[,2])*2^omega(n); \\ Michel Marcus, Dec 04 2018

A322327(n) = factorback(apply(e -> e+e, factor(n)[, 2])); \\ Antti Karttunen, Jul 18 2022

from math import prod
from sympy import factorint
def A322327(n): return prod(e<<1 for e in factorint(n).values()) # Chai Wah Wu, Dec 26 2022

Multiplicative with a(p^e) = 2*e for prime p and e > 0.
Dirichlet g. f.: (zeta(s))^2 * zeta(2*s) / zeta(4*s).
Equals Dirichlet convolution of A000005 and A227291.
Sum_{k=1..n} a(k) ~ 15*(log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 - 360*zeta'(4)/Pi^4) * n / Pi^2 + 6*zeta(1/2)^2 * sqrt(n) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 20 2020
a(n) = A005361(n^2) = A286324(n^2). - Amiram Eldar, Dec 09 2023

Data section extended up to a(100) by Antti Karttunen, Jul 18 2022

A361132 Multiplicative with a(p^e) = e^4, p prime and e > 0.

Original entry on oeis.org

1, 1, 1, 16, 1, 1, 1, 81, 16, 1, 1, 16, 1, 1, 1, 256, 1, 16, 1, 16, 1, 1, 1, 81, 16, 1, 81, 16, 1, 1, 1, 625, 1, 1, 1, 256, 1, 1, 1, 81, 1, 1, 1, 16, 16, 1, 1, 256, 16, 16, 1, 16, 1, 81, 1, 81, 1, 1, 1, 16, 1, 1, 16, 1296, 1, 1, 1, 16, 1, 1, 1, 1296, 1, 1, 16, 16

Offset: 1

Vaclav Kotesovec, Mar 02 2023, following a suggestion from Amiram Eldar

In general, if the function is multiplicative with a(p^e) = e^k, where k>=1, then Sum_{m=1..n} a(m) ~ c(k) * n, where c(k) = Product_{primes p} (1 + PolyLog(-k, 1/p)) * (1 - 1/p).
Equivalently, c(k) = Product_{primes p} (1 + Sum_{j>=2} (j^k - (j-1)^k) / p^j).
Sum_{m=1..n} A005361(m)^k ~ c(k) * n.
Table of logarithms of the first twenty constants c(k):
log(c1)  =    0.6645400902595784780106197346845697376257107319484837534113838...
log(c2)  =    2.1027190979191945200514651557327047986978773488049101019457040...
log(c3)  =    4.6968549904993458045898305766669061238379561861949323835425304...
log(c4)  =    8.6865032221694100694964858752580123427478996289429265630701524...
log(c5)  =   14.2913129298819954890384122051888143114132125173972994127345117...
log(c6)  =   21.8135511355940060754244319875442802379763506456537810297977335...
log(c7)  =   31.6936244245134941047326145621097555406387768809071583785926496...
log(c8)  =   44.5357450879229051636129496942971942282070021854681649075237793...
log(c9)  =   61.1279313139359633940353674601273793850149492879803908371116076...
log(c10) =   82.5520903493060704390063479960346732401820956158379186266389560...
log(c11) =  110.2954981238150788264027780431082219466660734768697563026966486...
log(c12) =  146.3390378386537094475359791093275236623437203145309460650602987...
log(c13) =  193.3102629498150337396691694808577709247583271151043344733643302...
log(c14) =  254.7562108044458078036208253682699240853829328072028848109791635...
log(c15) =  335.5155584889434205169760027607421364026263435517505529418223175...
log(c16) =  442.1708823748701851244490135727342670822854621013078138839028927...
log(c17) =  583.6971600757633563987486782501478518757572163549653222049269791...
log(c18) =  772.3363960260522276224001927946529683262139600086441840227950538...
log(c19) = 1024.7789861796186438478485897805332932014500908873437888887485298...
log(c20) = 1363.8429394936892771815120584792965902670785987496833459129791344...
Conjecture: log(log(c(k)))/k converges to a constant (around 0.315).

Vaclav Kotesovec, Plot of log(log(c(k))) / k, for k = 1..40

Cf. A005361, A360969, A360970, A322328, A361148, A361179.

Cf. A082695 (c1).

g[p_, e_] := e^4; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, (1 - 4*X + 21*X^2 + X^3 + 6*X^4 - X^5)/(1-X)^5)[n], ", "))

a(n) = A005361(n)^4.
Dirichlet g.f.: Product_{primes p} (1 + p^s*(p^(3*s) + 11*p^(2*s) + 11*p^s + 1) / (p^s - 1)^5).
Sum_{k=1..n} a(k) ~ c * n, where c = Product_{primes p} (1 + (15*p^3 + 5*p^2 + 5*p - 1) / (p*(p-1)^4)) = 5922.43654748315227690838901234893132297258444672...

A360909 Multiplicative with a(p^e) = 3*e + 2.

Original entry on oeis.org

1, 5, 5, 8, 5, 25, 5, 11, 8, 25, 5, 40, 5, 25, 25, 14, 5, 40, 5, 40, 25, 25, 5, 55, 8, 25, 11, 40, 5, 125, 5, 17, 25, 25, 25, 64, 5, 25, 25, 55, 5, 125, 5, 40, 40, 25, 5, 70, 8, 40, 25, 40, 5, 55, 25, 55, 25, 25, 5, 200, 5, 25, 40, 20, 25, 125, 5, 40, 25, 125

Offset: 1

Vaclav Kotesovec, Feb 25 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), A360997 (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), this sequence (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).

a[n_] := Times @@ ((3*Last[#] + 2) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)

for(n=1, 100, print1(direuler(p=2, n, (1+3*X-X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 3/p^s - 1/p^(2*s)).
Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)), (with a product that converges for s=1).
Sum_{k=1..n} a(k) ~ c * n * log(n)^4 / 24, where c = Product_{primes p} (1 - 7/p^2 + 11/p^3 - 6/p^4 + 1/p^5) = 0.091414252314317101861531055690354339957600046..., more precise (but very complicated) asymptotics can be obtained (in Mathematica notation) as Residue[Zeta[s]^5 * f[s] * n^s / s, {s, 1}], where f[s] = Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)).

A360997 Multiplicative with a(p^e) = e + 3.

Original entry on oeis.org

1, 4, 4, 5, 4, 16, 4, 6, 5, 16, 4, 20, 4, 16, 16, 7, 4, 20, 4, 20, 16, 16, 4, 24, 5, 16, 6, 20, 4, 64, 4, 8, 16, 16, 16, 25, 4, 16, 16, 24, 4, 64, 4, 20, 20, 16, 4, 28, 5, 20, 16, 20, 4, 24, 16, 24, 16, 16, 4, 80, 4, 16, 20, 9, 16, 64, 4, 20, 16, 64, 4, 30, 4, 16

Offset: 1

Vaclav Kotesovec, Feb 28 2023

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

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), this sequence (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), A360909 (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).

Cf. A000005, A007425, A007426, A074816.

g[p_, e_] := e+3; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, (1+2*X-2*X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: Product_{primes p} (1 + (4*p^s - 3)/(p^s - 1)^2).
Dirichlet g.f.: zeta(s)^4 * Product_{primes p} (1 - 5/p^(2*s) + 6/p^(3*s) - 2/p^(4*s)).
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = A000005(A361264(n)).
a(n) = A074816(n)*A007426(n)/A007425(n). (End)

A360996 Multiplicative with a(p^e) = 5*e, p prime and e > 0.

Original entry on oeis.org

1, 5, 5, 10, 5, 25, 5, 15, 10, 25, 5, 50, 5, 25, 25, 20, 5, 50, 5, 50, 25, 25, 5, 75, 10, 25, 15, 50, 5, 125, 5, 25, 25, 25, 25, 100, 5, 25, 25, 75, 5, 125, 5, 50, 50, 25, 5, 100, 10, 50, 25, 50, 5, 75, 25, 75, 25, 25, 5, 250, 5, 25, 50, 30, 25, 125, 5, 50, 25, 125, 5, 150

Offset: 1

Vaclav Kotesovec, Feb 28 2023

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

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), A360997 (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), A360909 (3*e+2), A360911 (3*e-2), A322328 (4*e).

Cf. A082476.

g[p_, e_] := 5*e; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, (1+3*X+X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: Product_{primes p} (1 + 5*p^s/(p^s - 1)^2).

a(n) = A005361(n) * A082476(n).

A328854 Dirichlet g.f.: Product_{p prime} 1 / (1 - 2 * p^(-s))^2.

Original entry on oeis.org

1, 4, 4, 12, 4, 16, 4, 32, 12, 16, 4, 48, 4, 16, 16, 80, 4, 48, 4, 48, 16, 16, 4, 128, 12, 16, 32, 48, 4, 64, 4, 192, 16, 16, 16, 144, 4, 16, 16, 128, 4, 64, 4, 48, 48, 16, 4, 320, 12, 48, 16, 48, 4, 128, 16, 128, 16, 16, 4, 192, 4, 16, 48, 448, 16, 64, 4, 48, 16, 64, 4, 384, 4, 16, 48

Offset: 1

Ilya Gutkovskiy, Oct 28 2019

Dirichlet convolution of A061142 with itself.

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

Cf. A000005, A001222, A061142, A123667, A322328.

Table[2^PrimeOmega[n] DivisorSigma[0, n], {n, 1, 75}]
f[p_, e_] := (e + 1)*2^e; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

a(n) = numdiv(n)*2^bigomega(n); \\ Michel Marcus, Dec 02 2020

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - 2*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021

If n = Product (p_j^k_j) then a(n) = Product (2^k_j * (k_j + 1)).

a(n) = 2^bigomega(n) * tau(n), where bigomega = A001222 and tau = A000005.

A349770 a(n) = Sum_{d|n} usigma(d) * usigma(n/d).

Original entry on oeis.org

1, 6, 8, 19, 12, 48, 16, 48, 36, 72, 24, 152, 28, 96, 96, 113, 36, 216, 40, 228, 128, 144, 48, 384, 88, 168, 136, 304, 60, 576, 64, 258, 192, 216, 192, 684, 76, 240, 224, 576, 84, 768, 88, 456, 432, 288, 96, 904, 164, 528, 288, 532, 108, 816, 288, 768, 320, 360, 120, 1824

Offset: 1

Ilya Gutkovskiy, Nov 29 2021

Dirichlet convolution of A034448 with itself.

Cf. A034448, A034761, A322328, A328485, A343569.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; a[n_] := Sum[usigma[d] usigma[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]

Dirichlet g.f.: ( zeta(s) * zeta(s-1) / zeta(2*s-1) )^2.
Multiplicative with a(p^e) = e * (p^e + 1) + (p+1) * (p^e - 1)/(p-1). - Amiram Eldar, Nov 29 2021
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / zeta(3)^2 * (Pi^2 * log(n)/72 + gamma * Pi^2/36 - Pi^2/144 + zeta'(2)/6 - Pi^2 * zeta'(3)/(18*zeta(3))), where zeta(3) = A002117, zeta'(2) = -A073002, zeta'(3) = -A244115 and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 05 2021

cp's OEIS Frontend

A370297 Inverse Moebius transform of A322328.

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A113413 A Riordan array of coordination sequences.

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A322327 a(n) = A005361(n) * A034444(n).

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A361132 Multiplicative with a(p^e) = e^4, p prime and e > 0.

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A360909 Multiplicative with a(p^e) = 3*e + 2.

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A360997 Multiplicative with a(p^e) = e + 3.

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A360996 Multiplicative with a(p^e) = 5*e, p prime and e > 0.

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A328854 Dirichlet g.f.: Product_{p prime} 1 / (1 - 2 * p^(-s))^2.

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