A151821 -id:A151821 - cp's OEIS frontend

A258935 Independence number of Keller graphs.

4, 5, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 1

Stan Wagon, Nov 06 2015

			For G(2), a maximum independent set is {03,10,12,13,23}.

W. Jarnicki, W. Myrvold, P. Saltzman, S. Wagon, Properties, proved and conjectured, of Keller, queen, and Mycielski graphs, Ars Mathematica Contemporanea 13:2 (2017) 427-460.

Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Keller Graph

Cf. A006946, A135831, A135907.

Essentially the same as A143858, A240951, A198633, A171497, A151821, A146541 and A077552.

Mathematica
```
Join[{4, 5}, 2^Range[3, 10]]
```

a(n)=if(n>2,2^n,n+3) \\ Charles R Greathouse IV, Nov 07 2015

a(n) = 2^n except a(1) = 4 and a(2) = 5.

G.f.: x*(x*(3+2*x)-4)/(2*x-1), e.g.f.: exp(2*x)+x^2/2+2*x-1. - Benedict W. J. Irwin, Jul 15 2016

A260196 1, -3, followed by -1's.

Original entry on oeis.org

1, -3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 0

Paul Curtz, Jul 19 2015

1/(n+1) is the inverse Akiyama-Tanigawa transform of A164555(n)/A027642(n).
For more on the Akiyama-Tanigawa transform, see Links (correction: page 7 read here A164555 instead of A027641) and A177427.
Here:
   1,  -3, -1, -1, -1, -1, ...
   4,  -4,  0,  0,  0,  0, ...
   8,  -8,  0,  0,  0,  0, ...
  16, -16,  0,  0,  0,  0, ...
  etc.
Other process, using signed A130534(n), different of A008275(n):
1,                            1/1,                                      1,
1, 4,                      (  1,     -1)/1,                            -3,
1, 4, 8,                   (  2,     -3,   1)/2,                       -1,
1, 4, 8, 16,          *    (  6,    -11,   6,   -1)/6,             =   -1,
1, 4, 8, 16, 32,           ( 24,    -50,  35,  -10,  1)/24,            -1,
1, 4, 8, 16, 32, 64,       (120,   -274, 225,  -85, 15, -1)/120,       -1,
etc.                       etc.                                        etc.
Via the modified Stirling numbers of the first kind, the second triangle, Iw(n), is the inverse of Worpitzky transform A163626(n).
a(n) is the third sequence of a family beginning with
1,  1,  1,  1,  1,  1,  1,  1, ...              = A000012(n)
1,  0,  0,  0,  0,  0,  0,  0,  0, ...          = A000007(n)
1, -3, -1, -1, -1, -1, -1, -1, -1, -1, ... .
A000012(n) is the inverse Akiyama-Tanigawa transform of A000007(n), with or without its second term.
A000007(n) is the inverse Akiyama-Tanigawa transform of A000012(n), with or without its second term.
a(n) is the inverse Akiyama-Tanigawa transform of 2^n omitting the second term i.e. 2.

Colin Barker, Table of n, a(n) for n = 0..1000
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, Journal of Integer Sequences, 3(2000), article 00.2.9
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000007, A000012, A008275, A027642, A130534, A151821, A163626, A164555, A177427.

first(m)=vector(m,i,i--;if(i>1,-1,if(i==0,1,if(i==1,-3)))) \\ Anders Hellström, Aug 28 2015

Vec(-(2*x^2-4*x+1)/(x-1) + O(x^100)) \\ Colin Barker, Sep 11 2015

Inverse Akiyama-Tanigawa transform of A151821(n).
From Colin Barker, Sep 11 2015: (Start)
a(n) = -1 for n>1.
a(n) = a(n-1) for n>2.
G.f.: -(2*x^2-4*x+1) / (x-1).
(End)

A355580 Powerful 3-smooth numbers: numbers of the form 2^i * 3^j with i, j != 1.

Original entry on oeis.org

1, 4, 8, 9, 16, 27, 32, 36, 64, 72, 81, 108, 128, 144, 216, 243, 256, 288, 324, 432, 512, 576, 648, 729, 864, 972, 1024, 1152, 1296, 1728, 1944, 2048, 2187, 2304, 2592, 2916, 3456, 3888, 4096, 4608, 5184, 5832, 6561, 6912, 7776, 8192, 8748, 9216, 10368, 11664

Offset: 1

Amiram Eldar, Jul 08 2022

This sequence is closed under multiplication.

			a(2) = 4 = 2^2.
a(3) = 8 = 2^3.
a(8)= 36 = 2^2 * 3^2.

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

Intersection of A001694 and A003586.
Union of A151821, (A000244 \ {3}) and 36*A003586.
A003586 \ (Union of A007283 and A008776).
Cf. A355581.

q[n_] := Module[{e = IntegerExponent[n, {2, 3}]}, e[[1]] != 1 && e[[2]] != 1 && Times@@({2, 3}^e) == n]; Select[Range[12000], q]

is(n) = {my(f=factor(n)); n == 1 || (vecmax(f[,1]) <= 3 && vecmin(f[,2]) > 1)};

from itertools import count, takewhile
def aupto(lim):
    pows2 = list(takewhile(lambda x: xMichael S. Branicky, Jul 08 2022

Sum_{n>=1} 1/a(n) = 7/4.

A358554 Least Matula-Goebel number of a rooted tree with n internal (non-leaf) nodes.

Original entry on oeis.org

1, 2, 3, 5, 11, 25, 55, 121, 275, 605, 1331, 3025, 6655, 14641, 33275, 73205

Offset: 1

Gus Wiseman, Nov 27 2022

Positions of first appearances in A342507.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The terms together with their corresponding rooted trees begin:
      1: o
      2: (o)
      3: ((o))
      5: (((o)))
     11: ((((o))))
     25: (((o))((o)))
     55: (((o))(((o))))
    121: ((((o)))(((o))))
    275: (((o))((o))(((o))))
    605: (((o))(((o)))(((o))))
   1331: ((((o)))(((o)))(((o))))
   3025: (((o))((o))(((o)))(((o))))
   6655: (((o))(((o)))(((o)))(((o))))
  14641: ((((o)))(((o)))(((o)))(((o))))
  33275: (((o))((o))(((o)))(((o)))(((o))))
  73205: (((o))(((o)))(((o)))(((o)))(((o))))

For height instead of internals we have A007097, firsts of A109082.
For leaves instead of internals we have A151821, firsts of A109129.
Positions of first appearances in A342507.
The ordered version gives firsts of A358553.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height.
A055277 counts rooted trees by nodes and leaves.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A001263, A206487, A358578, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seq=Table[Count[MGTree[n],[_],{0,Infinity}],{n,1000}];
Table[Position[seq,n][[1,1]],{n,Union[seq]}]

A373901 Numbers m such that (m - k)^k modulo m is not equal to positive k <= m.

Original entry on oeis.org

1, 3, 4, 8, 11, 16, 17, 19, 32, 43, 59, 64, 67, 83, 107, 128, 131, 137, 179, 227, 251, 256, 347, 467, 512, 523, 563, 587, 683, 691, 787, 907, 971, 977, 1019, 1024, 1097, 1187, 1259, 1283, 1307, 1459, 1523, 1579, 1619, 1627, 1699, 1723, 1787, 1867, 1907, 1931, 1987, 2027, 2048

Offset: 1

Juri-Stepan Gerasimov, Jun 22 2024

nonn

The odd composite and the even composite terms of this sequence are respectively 3353, 4213, 9233, 10813, ... and all and only powers of 2 >= 4.

			1 is in this sequence because (1-1)^1 mod 1 = 0 != 1.
3 is in this sequence because
  (3-1)^1 mod 3 = 2 != 1,
  (3-2)^2 mod 3 = 1 != 2,
  (3-3)^3 mod 3 = 0 != 3.

Supersequence of A151821.

[m : m in [1..2^11] | #[k: k in [1..m] | (m-k)^k mod m eq k] eq 0];

isok(m) = for (k=1, m, if ((m - k)^k % m == k, return (0))); return(1); \\ Michel Marcus, Jun 30 2024

A381077 Sorted positions of first appearances in A380986 (product of prime indices minus product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 49, 63, 81, 99, 121, 125, 135, 169, 171, 245, 279, 289, 343, 361, 363, 369, 375, 387, 477, 529, 531, 575, 603, 625, 675, 711, 729, 747, 833, 841, 847, 873, 875, 891, 909, 961, 981, 1029, 1083, 1125, 1127, 1179, 1225, 1251, 1377, 1413, 1445, 1467

Offset: 1

Gus Wiseman, Feb 20 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A position of first appearance in a sequence q is an index k such that q(k) is different from q(j) for all j < k.

All terms are odd.

			The terms together with their prime indices begin:
     1: {}
     9: {2,2}
    25: {3,3}
    49: {4,4}
    63: {2,2,4}
    81: {2,2,2,2}
    99: {2,2,5}
   121: {5,5}
   125: {3,3,3}
   135: {2,2,2,3}
   169: {6,6}
   171: {2,2,8}
   245: {3,4,4}
   279: {2,2,11}

For length instead of product we have A151821, firsts of A046660.
For factors instead of indices we have A381076, sorted firsts of A066503.
For sum of factors instead of product of indices we have A381075 (unsorted A280286), A280292.
For quotient instead of difference we have A380988 (unsorted A380987), firsts of A290106.
For quotient and factors we have A001694 (unsorted A064549), firsts of A003557.
For sum instead of product we have A380957 (unsorted A380956), firsts of A380955.
Sorted firsts of A380986, which has nonzero terms at positions A038838.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists the squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000079, A000720, A001222, A081770, A075255, A116861, A178503, A374248, A379681, A380958.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[Times@@prix[n]-Times@@Union[prix[n]],{n,10000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

A144388 Triangle T(n,k) = binomial(n, k) + ((-1)^(n + k))nbinomial(n - 1, k), T(0,0) = 1, read by rows, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 3, 0, 1, -2, 9, 0, 1, 5, -8, 18, 0, 1, -4, 25, -20, 30, 0, 1, 7, -24, 75, -40, 45, 0, 1, -6, 49, -84, 175, -70, 63, 0, 1, 9, -48, 196, -224, 350, -112, 84, 0, 1, -8, 81, -216, 588, -504, 630, -168, 108, 0, 1, 11, -80, 405, -720, 1470, -1008, 1050, -240, 135, 0, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 01 2008

			Triangle begins:
   1;
   0,   1;
   3,   0,    1;
  -2,   9,    0,    1;
   5,  -8,   18,    0,    1;
  -4,  25,  -20,   30,    0,     1;
   7, -24,   75,  -40,   45,     0,    1;
  -6,  49,  -84,  175,  -70,    63,    0,    1;
   9, -48,  196, -224,  350,  -112,   84,    0,   1;
  -8,  81, -216,  588, -504,   630, -168,  108,   0, 1;
  11, -80,  405, -720, 1470, -1008, 1050, -240, 135, 0, 1;
  ...

Cf. A001787, A151821, A144389, A216973.

p[x_, n_] = (x + 1)^n - n*(x - 1)^(n - 1);
Table[CoefficientList[p[x, n], x], {n, 0, 10}] // Flatten

create_list(binomial(n, k) + ((-1)^(n + k))*n*binomial(n - 1, k),n , 0, 15, k, 0, n); /* Franck Maminirina Ramaharo, Jan 25 2019 */

T(n,k) = [x^k] ((x + 1)^n - n*(x - 1)^(n - 1)).

Sum_{k=0..n} T(n,k) = A151821(n-1), n >= 1.

Edited and offset corrected by Franck Maminirina Ramaharo, Jan 25 2019

A236967 Expansion of (1+3x)^2/(1-3x)^2.

Original entry on oeis.org

1, 12, 72, 324, 1296, 4860, 17496, 61236, 209952, 708588, 2361960, 7794468, 25509168, 82904796, 267846264, 860934420, 2754990144, 8781531084, 27894275208, 88331871492, 278942752080, 878669669052, 2761533245592, 8661172452084, 27113235502176, 84728860944300

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 22 2014

Index entries for linear recurrences with constant coefficients, signature (6,-9)

Cf. Expansion of (1 + k*x)^m/(1 - k*x)^m where the values of k,m are:
......|..m = 1..|..m = 2..|..m = 3..|..m = 4..|..m = 5..|..m = 6..|
k = 1 | A040000 | A008574 | A005899 | A008412 | A008413 | A008414 |
k = 2 | A151821 | A241204 |         |         |         |         |
k = 3 | A099856 | A236967 |         |         |         |         |
k = 4 | A081654 |         |         |         |         |         |
-------------------------------------------------------------------

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)^2/(1-3*x)^2));

For n >= 1, a(n) = 4*n*3^n. - Robert Israel, May 08 2014

Edited by Wolfdieter Lang, May 07 2014

A281261 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 2, 2, 1, 5, 2, 5, 9, 2, 1, 15, 14, 2, 7, 35, 20, 2, 1, 28, 70, 27, 2, 9, 84, 126, 35, 2, 1, 45, 210, 210, 44, 2, 11, 165, 462, 330, 54, 2, 1, 66, 495, 924, 495, 65, 2, 13, 286, 1287, 1716, 715, 77, 2, 1, 91, 1001, 3003, 3003, 1001, 90, 2, 15, 455, 3003, 6435, 5005, 1365, 104, 2, 1, 120, 1820, 8008, 12870, 8008, 1820, 119, 2

Offset: 1

Gheorghe Coserea, Jan 18 2017

Row n>1 contains floor((n+3)/2) terms.

			A(x;t) = x + (2*t+2)*x^2 + (t^2+5*t+2)*x^3 + (5*t^2+9*t+2)*x^4 + ...
Triangle starts:
n\k  [1]      [2]      [3]      [4]      [5]      [6]      [7]      [8]
[1]  1;
[2]  2,       2;
[3]  1,       5,       2;
[4]  5,       9,       2;
[5]  1,       15,      14,      2;
[6]  7,       35,      20,      2;
[7]  1,       28,      70,      27,      2;
[8]  9,       84,      126,     35,      2;
[9]  1,       45,      210,     210,     44,      2;
[10] 11,      165,     462,     330,     54,      2;
[11] 1,       66,      495,     924,     495,     65,      2;
[12] 13,      286,     1287,    1716,    715,     77,      2;
[13] 1,       91,      1001,    3003,    3003,    1001,    90,      2;
[14] 15,      455,     3003,    6435,    5005,    1365,    104,     2;
[15] ...

Gheorghe Coserea, Rows n = 1..202, flattened
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

Reverse[CoefficientList[#, t]]& /@ CoefficientList[x*((1-t)*x^3 + (t^2 - 2*t - 1)*x^2 + (2*t - 1)*x + 1)/((t-1)*x^3 + (3-t)*x^2 - 3*x + 1) + O[x]^16, x] // Rest // Flatten (* Jean-François Alcover, Feb 18 2019 *)

N=16; x='x+O('x^N); concat(apply(p->Vec(p),  Vec(Ser(x*((1-t)*x^3 + (t^2-2*t-1)*x^2 + (2*t-1)*x + 1)/((t-1)*x^3 + (3-t)*x^2 - 3*x + 1)))))

N = 14; concat(1, concat(vector(N, n, Vec(substpol(((1+t)^(n+2) + (1-t)^(n+2))/2 - t^2 + 1, t^2, t)))))

A(x;t) = Sum{n>=1} P_n(t)*x^n = x*((1-t)*x^3 + (t^2-2*t-1)*x^2 + (2*t-1)*x + 1)/((t-1)*x^3 + (3-t)*x^2 - 3*x + 1).
A278457(x;t) = serreverse(A(-x;t))(-x).
A151821(n) = P_n(1), A213667(n) = P_n(2).
P_n(t^2) = ((1+t)^(n+1) + (1-t)^(n+1))/2 - t^2 + 1, for n>1.

A380144 Sum of divisors d | k such that rad(d) = rad(k/d) where k is in A001694 and rad = A007947.

Original entry on oeis.org

1, 2, 6, 3, 14, 5, 12, 30, 6, 7, 62, 18, 39, 10, 24, 11, 30, 126, 42, 13, 14, 30, 72, 15, 120, 254, 90, 17, 78, 56, 19, 42, 70, 168, 21, 22, 60, 510, 23, 186, 155, 234, 60, 26, 363, 98, 150, 29, 360, 30, 31, 66, 240, 180, 1022, 33, 90, 378, 34, 35, 546, 84, 132

Offset: 1

Michael De Vlieger, Jan 15 2025

nonn

Alternatively, sum of divisors d | k such that omega(d) = omega(k/d) = omega(k) for powerful k in A001694.
Not a permutation of natural numbers: 4 does not appear.
This sequence is A364988 without zeros. See A364988 for multiplicative properties.

			a(1) = 1 since s(1) = 1 = 1*1. (We count divisor 1 only once, since it is a square root.)
a(2) = 2 since s(2) = 4 = 2*2, and omega(2) = omega(4). Alternatively, rad(2) = rad(4).
a(3) = 6 since s(3) = 8 = 2*4, omega(2) = omega(4) = omega(8), i.e., rad(2) = rad(4) = rad(8), and 2+4 = 6.
a(4) = 3 since s(4) = 9 = 3*3, and omega(3) = omega(9); rad(3) = rad(9).
a(5) = 14 since s(5) = 16 = 2*8 = 4*4; 2+4+8 = 14.
a(6) = 5 since s(6) = 25 = 5*5.
a(7) = 12 since s(7) = 27 = 3*9; 3+9 = 12.
a(8) = 30 since s(8) = 32 = 2*16 = 4*8; 2+4+8+16 = 30.
a(9) = 6 since s(9) = 36 = 6*6, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^5.

Cf. A001221, A001694, A007947, A062503, A151821, A320966, A364988.

nn = 1500;
s = Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}];
Map[Function[{n, r},
  DivisorSum[n, # &, PrimeNu[#] == PrimeNu[n/#] == r &]] @@
  {#, PrimeNu[#]} &, s]

Let s = A001694.
a(n) = A364988(A001694(n)).
a(n) = m*rad(n) with m >= 1.
a(n) = sqrt(s(n)) = rad(s(n)) for s(n) = rad(s(n))^2 (i.e., s(n) in A062503).
a(n) > sqrt(s(n)) and a(n) = m*rad(s(n)), m > 1, for s(n) in A320966.
The sequence of record setters is A151821.
a(n) = s(n)-2 for s(n) in A151821.

cp's OEIS Frontend

A258935 Independence number of Keller graphs.

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A260196 1, -3, followed by -1's.

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A355580 Powerful 3-smooth numbers: numbers of the form 2^i * 3^j with i, j != 1.

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A358554 Least Matula-Goebel number of a rooted tree with n internal (non-leaf) nodes.

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A373901 Numbers m such that (m - k)^k modulo m is not equal to positive k <= m.

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A381077 Sorted positions of first appearances in A380986 (product of prime indices minus product of distinct prime indices).

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A144388 Triangle T(n,k) = binomial(n, k) + ((-1)^(n + k))*n*binomial(n - 1, k), T(0,0) = 1, read by rows, 0 <= k <= n.

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A236967 Expansion of (1+3*x)^2/(1-3*x)^2.

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A144388 Triangle T(n,k) = binomial(n, k) + ((-1)^(n + k))nbinomial(n - 1, k), T(0,0) = 1, read by rows, 0 <= k <= n.

A236967 Expansion of (1+3x)^2/(1-3x)^2.