A289521 -id:A289521 - cp's OEIS frontend

A325973 Arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}: a(n) = (1/2) * (A034448(n) + A048250(n)).

1, 3, 4, 4, 6, 12, 8, 6, 7, 18, 12, 16, 14, 24, 24, 10, 18, 21, 20, 24, 32, 36, 24, 24, 16, 42, 16, 32, 30, 72, 32, 18, 48, 54, 48, 31, 38, 60, 56, 36, 42, 96, 44, 48, 42, 72, 48, 40, 29, 48, 72, 56, 54, 48, 72, 48, 80, 90, 60, 96, 62, 96, 56, 34, 84, 144, 68, 72, 96, 144, 72, 51, 74, 114, 64, 80, 96, 168, 80, 60, 43, 126

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

This is not multiplicative: a(4) = 4, a(9) = 7, but a(36) = 31, not 28. However, the function acts multiplicatively on certain subsequences of natural numbers, like for example when restricted to A048107, where this sequence coincides with A326043.

			For n = 36, its divisors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Of these, unitary divisors are 1, 4, 9 and 36, so A034448(36) = 1+4+9+36 = 50, while the squarefree divisors are 1, 2, 3 and 6, so A048250(36) = 1+2+3+6 = 12, thus a(36) = (50+12)/2 = 31.
For n = 495, its divisors are 1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495. Of these, unitary are 1, 5, 9, 11, 45, 55, 99, 495, whose sum is A034448(495) = 720, while the squarefree divisors are 1, 3, 5, 11, 15, 33, 55, 165, and their sum is A048250(495) = 288. Thus a(495) = (720+288)/2 = 504. Also for 495, whose prime factorization is 3^2 * 5^1 * 11^1 this can be computed faster as the average of ((3^2)+1)*(5+1)*(11+1) and (3+1)*(5+1)*(11+1), thus (1/2)*(3+(3^2)+2)*(5+1)*(11+1) = 504.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A034448, A048107, A048250, A052548, A289521, A306633, A325974, A325975, A325977, A325978, A325981, A326043.

Array[(1/2) If[# == 1, 2, Times @@ (1 + Power @@@ #) + Times @@ (1 + #[[;; , 1]]) &@ FactorInteger[#]] &, 90] (* Michael De Vlieger, Jun 06 2019, after Giovanni Resta at A034448 and Amiram Eldar at A048250. *)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325973(n) = ((A034448(n)+A048250(n))/2);

A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));

a(n) = (1/2) * (A034448(n) + A048250(n)).
a(n) = A000203(n) - A325974(n).
a(n) = n + A325977(n).
a(A048107(n)) = A326043(A048107(n)).
For n >= 1, a(2^n) = A052548(n-1) = 2^(n-1) + 2.
For n >= 1, a(3^n) = A289521(n) = (3^n + 5)/2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) + 1)/4 = 0.5921081944... . - Amiram Eldar, Feb 22 2024

A291773 Domination number of the n-Apollonian network.

Original entry on oeis.org

1, 1, 3, 4, 7, 16, 43, 124, 367, 1096, 3283, 9844, 29527, 88576, 265723, 797164, 2391487, 7174456, 21523363, 64570084, 193710247, 581130736, 1743392203, 5230176604, 15690529807, 47071589416, 141214768243, 423644304724, 1270932914167, 3812798742496

Offset: 1

Eric W. Weisstein, Aug 31 2017

Also, the connected domination number of the n-Apollonian network. - Andrew Howroyd, Jan 16 2018

Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Apollonian Network
Eric Weisstein's World of Mathematics, Connected Domination Number
Eric Weisstein's World of Mathematics, Domination Number
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A298105.

(* Start from Eric W. Weisstein, Jan 17 2018 *)
Join[{1, 1}, Table[(3^(n - 3) + 5)/2, {n, 3, 20}]]
Join[{1, 1}, Table[(3^n + 135)/54, {n, 3, 20}]]
Join[{1, 1}, (3^Range[3, 20] + 135)/54]
Join[{1, 1}, LinearRecurrence[{4, -3}, {3, 4}, 20]]
CoefficientList[Series[(1 - 3 x + 2 x^2 - 5 x^3)/(1 - 4 x + 3 x^2), {x, 0, 20}], x]
(* End *)

\\ here d0..d3 are for 0..3 outside vertices included in dominating set.
D(d0,d1,d2,d3) = {[min(3*d0,1+3*d1), min(d0+2*d1,1+d1+2*d2), min(2*d1+d2,1+2*d2+d3), min(3*d2,1+3*d3)]}
a(n)={my(v=[1,0,0,0]); for(i=2,n,v=D(v[1],v[2],v[3],v[4])); min(min(v[1],1+v[2]),min(2+v[3],3+v[4]))} \\ Andrew Howroyd, Sep 01 2017

Vec(x*(1 - 3*x + 2*x^2 - 5*x^3) / ((1 - x)*(1 - 3*x)) + O(x^40)) \\ Colin Barker, Oct 03 2017

a(n) = (3^(n-3) + 5) / 2 for n > 2. - Andrew Howroyd, Sep 01 2017
From Colin Barker, Oct 03 2017: (Start)
G.f.: x*(1 - 3*x + 2*x^2 - 5*x^3) / ((1 - x)*(1 - 3*x)).
a(n) = 4*a(n-1) - 3*a(n-2) for n>4.
(End)
a(n) = A289521(n-3) for n > 3. - Andrew Howroyd, Jan 16 2018

a(7)-a(30) from Andrew Howroyd, Sep 01 2017

A289022 Wiener index of the n-Apollonian network.

Original entry on oeis.org

6, 27, 204, 1941, 19572, 198567, 1999056, 19931337, 196939572, 1930784091, 18802964760, 182062831005, 1754100012108, 16826739416271, 160799296563312, 1531421717572401, 14540848734272388, 137690120683444995, 1300613432805623496, 12258142039717884549

Offset: 1

Andrew Howroyd, Sep 02 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Apollonian Network
Eric Weisstein's World of Mathematics, Wiener Index
Index entries for linear recurrences with constant coefficients, signature (23, -174, 448, -29, -1221, 2088, -4050, 2916).

Cf. A067771 (edges), A192792, A289521, A289722.

(* Start from Eric W. Weisstein, Sep 07 2017 *)
Table[(6655 + 31 (-1)^n 2^(n + 2) + 5 3^(1 + 2 n) (24 + 11 n) + 3^(n + 1) (1197 + 55 n) + 5 2^(5 + n/2) Cos[n Pi/2] - 155 2^((3 + n)/2) Sin[n Pi/2])/3630, {n, 20}]
LinearRecurrence[{23, -174, 448, -29, -1221, 2088, -4050, 2916}, {6, 27, 204, 1941, 19572, 198567, 1999056, 19931337}, 20]
CoefficientList[Series[(6 - 111 x + 627 x^2 - 741 x^3 - 1497 x^4 + 2862 x^5 - 5670 x^6 + 8748 x^7)/((1 - x) (1 - 3 x)^2 (1 - 9 x)^2 (1 + 2 x) (1 + 2 x^2)), {x, 0, 20}], x]
(* End *)

R(dp, peq, p1, p2, x) = {[3*(dp - x + peq^2 + (2+7*x)*p1^2 + (7+2*x)*p2^2 + (4+2*x)*peq*p1 + 6*peq*p2 + 2*(4+5*x)*p1*p2 + x*(peq+3*p1+3*p2)), x*(1+3*p1), 2*(p1+p2), peq+p2]}
A(n,x) = {my(v=[6*x,x,0,0,x]); for(i=2, n, v=R(v[1],v[2],v[3],v[4],x)); v[1]}
Wiener(dp)=sum(i=1,poldegree(dp),i*polcoeff(dp,i));
a(n) = Wiener(A(n,x));

a(n) = Sum_{k=1..1+floor(2*n/3)} k*A289722(n,k).
a(n) = 23*a(n-1) - 174*a(n-2) + 448*a(n-3) - 29*a(n-4) - 1221*a(n-5) + 2088*a(n-6) - 4050*a(n-7) + 2916*a(n-8).
G.f.: x*(6 - 111*x + 627*x^2 - 741*x^3 - 1497*x^4 + 2862*x^5 - 5670*x^6 + 8748*x^7)/((1 - x)*(1 - 3*x)^2*(1 - 9*x)^2*(1 + 2*x)*(1 + 2*x^2)).

A297558 Number of maximum matchings in the n-Apollonian network.

Original entry on oeis.org

3, 32, 738, 615514464, 5030805301520123200352256, 72175948705150863615789780847197889452411340074927861143342108815392768

Offset: 1

Eric W. Weisstein, Dec 31 2017

nonn

From Andrew Howroyd, May 30 2025: (Start)
For n >= 3, the maximum size of a matching is given by A289521(n-1). For n < 3 the maximum size is n + 1.
Term a(7) has 208 decimal digits and a(8) has 618 decimal digits. (End)

Andrew Howroyd, Table of n, a(n) for n = 1..8
Eric Weisstein's World of Mathematics, Apollonian Network.
Eric Weisstein's World of Mathematics, Matching.
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set.

Cf. A289521, A292429.

\\ Needs a(n,x) defined in A292429.
vector(6,n,pollead(a(n,x))) \\ Andrew Howroyd, May 30 2025

a(5) onwards from Andrew Howroyd, May 30 2025

A205248 Number of (n+1) X 2 0..1 arrays with the number of clockwise edge increases in every 2 X 2 subblock the same.

Original entry on oeis.org

16, 40, 112, 328, 976, 2920, 8752, 26248, 78736, 236200, 708592, 2125768, 6377296, 19131880, 57395632, 172186888, 516560656, 1549681960, 4649045872, 13947137608, 41841412816, 125524238440, 376572715312, 1129718145928, 3389154437776

Offset: 1

R. H. Hardin, Jan 24 2012

Also, the number of cliques in the n-Apollonian network. Cliques in this graph have a maximum size of 4. - Andrew Howroyd, Sep 02 2017

			Some solutions for n=4:
  1  0    0  1    1  1    0  1    1  1    1  1    1  0    1  0    1  1    1  1
  0  1    0  0    1  1    0  1    0  1    0  1    0  1    0  0    1  1    1  1
  1  0    1  1    1  1    0  1    0  1    0  0    1  0    0  1    1  1    1  1
  0  1    1  0    1  1    0  0    0  1    1  0    0  1    1  1    1  1    1  1
  1  0    0  0    1  1    0  1    1  1    1  1    1  0    0  1    1  1    1  1

R. H. Hardin, Table of n, a(n) for n = 1..210
Eric Weisstein's World of Mathematics, Apollonian Network
Eric Weisstein's World of Mathematics, Clique
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Column 1 of A205255.

Cf. A003462, A007051, A067771, A289521.

Table[4*(3^n + 1), {n, 1, 25}] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)
4 (3^Range[30] + 1) (* Eric W. Weisstein, Nov 29 2017 *)
LinearRecurrence[{4, -3}, {16, 40}, 30] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[-8 (-2 + 3 x)/(1 - 4 x + 3 x^2), {x, 0, 30}], x] (* Eric W. Weisstein, Nov 29 2017 *)

Vec(8*(2 - 3*x)/((1 - x)*(1 - 3*x)) + O(x^40)) \\ Andrew Howroyd, Sep 02 2017

a(n) = 4*a(n-1) - 3*a(n-2).
From Andrew Howroyd, Sep 02 2017: (Start)
a(n) = 4*(3^n + 1).
G.f.: 8*x*(2 - 3*x)/((1 - x)*(1 - 3*x)).
a(n) = 8*A007051(n).
a(n) = 1 + A289521(n) + A067771(n) + A003462(n+1) + A003462(n).
(End)

cp's OEIS Frontend

A325973 Arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}: a(n) = (1/2) * (A034448(n) + A048250(n)).

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A291773 Domination number of the n-Apollonian network.

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A289022 Wiener index of the n-Apollonian network.

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A297558 Number of maximum matchings in the n-Apollonian network.

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A205248 Number of (n+1) X 2 0..1 arrays with the number of clockwise edge increases in every 2 X 2 subblock the same.

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