A077552 -id:A077552 - cp's OEIS frontend

A352823 Number of nonfixed points y(i) != i, where y is the weakly increasing sequence of prime indices of n.

0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 2, 1, 0, 1, 4, 2, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 3, 1, 1, 2, 5, 2, 1, 1, 2, 2, 2, 1, 4, 1, 1, 2, 2, 2, 1, 1, 4, 3, 1, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 2, 1, 2, 5, 1, 2, 2, 2, 1, 1, 1, 3, 3

Offset: 1

Gus Wiseman, Apr 05 2022

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.

			The prime indices of 6500 are {1,1,3,3,3,6}, with nonfixed points at positions {2,4,5}, so a(6500) = 3.

Index entries for sequences computed from indices in prime factorization

* = unproved
Positions of zeros are A002110
Positions of first appearances are A077552.
The complement triangle version is A238352.
A version for compositions is A352513, complement A352512.
The complement is A352822.
The reverse version is A352825, complement A352824.
Complement positions of 1's are A352831, counted by A352832.
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
*A001522 counts partitions with a fixed point, ranked by A352827.
A056239 adds up prime indices, row sums of A112798 and A296150.
*A064428 counts partitions without a fixed point, ranked by A352826.
A122111 represents partition conjugation using Heinz numbers.
A124010 gives prime signature, sorted A118914, conjugate rank A238745.
A115720 and A115994 count partitions by their Durfee square.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352833 counts partitions by fixed points, rank statistic A352824.
Cf. A065770, A093641, A114088, A252464, A257990, A325163, A325164, A325165, A325169, A342192, A352486-A352491, A352828, A352829.

pnq[y_]:=Length[Select[Range[Length[y]],#!=y[[#]]&]];
Table[pnq[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,100}]

A352823(n) = { my(f=factor(n),i=0,c=0); for(k=1,#f~,while(f[k,2], f[k,2]--; i++; c += (i!=primepi(f[k,1])))); (c); }; \\ Antti Karttunen, Apr 11 2022

a(n) = A001222(n) - A352822(n). - Antti Karttunen, Apr 11 2022

Data section extended up to 105 terms by Antti Karttunen, Apr 11 2022

A198633 Total number of round trips, each of length 2*n on the graph P_3 (o-o-o).

Original entry on oeis.org

3, 4, 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

Offset: 0

Wolfdieter Lang, Nov 02 2011

See the array and triangle A198632 for the general case for the graph P_N (there N is n and the length is l=2*k).

			With the graph P_3 as 1-2-3:
n=0: 3, from the length 0 walks starting at 1, 2 and 3.
n=2: 8, from the walks of length 4, namely 12121, 12321, 21212, 23232, 21232, 23212, 32323 and 32123.

Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A198632, 2*A005248, A198635.

Essentially the same as A000079, A020707, A077552 etc.

Join[{3},NestList[2#&,4,30]] (* Harvey P. Dale, Nov 07 2020 *)

a(n)=if(n,2<Charles R Greathouse IV, Jan 02 2012

a(n) = w(3,2*n), n>=0, with w(3,l) the total number of closed walks on the graph P_3 (the simple path with 3 points (vertices) and 2 lines (or edges)).
O.g.f. for w(3,l) (with zeros for odd l): y*(d/dy)S(3,y)/S(3,y) with y=1/x and Chebyshev S-polynomials (coefficients A049310). See A198632, also for a rewritten form.
Empirical g.f.: (3-2*x)/(1-2*x). - Colin Barker, Jan 02 2012
This g.f. follows from the Chebyshev o.g.f. given above with x -> sqrt(x). Therefore a(0) = 3 and a(n) = 2^(n+1), n >= 1. - Wolfdieter Lang, Feb 18 2013.

A146541 Binomial transform of A010688.

Original entry on oeis.org

1, 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: 0

Philippe Deléham, Oct 31 2008

Hankel transform is := 1,-48,0,0,0,0,0,0,0,...

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000007, A011782, A000079, A003945, A046055, A146523, A082505, A135092.

Essentially the same as A132479, A077552, A020707.

Join[{1},2^Range[3,40]] (* Harvey P. Dale, Feb 28 2016 *)

Vec((1+6*x)/(1-2*x) + O(x^50)) \\ Colin Barker, Mar 17 2016

a(0)=1, a(n) = 2^(n+2) for n>0.
a(n) = Sum_{k, 0..n} A109466(n,k)*A146534(k).
a(n) = A132479(n), n>1. - R. J. Mathar, Nov 02 2008
G.f.: (1+6*x) / (1-2*x). - Colin Barker, Mar 17 2016

Corrected and extended by Harvey P. Dale, Feb 28 2016

A132307 2*A007318^(2) - A000012.

Original entry on oeis.org

1, 3, 1, 7, 7, 1, 15, 23, 11, 1, 31, 63, 47, 15, 1, 63, 159, 159, 79, 19, 1, 127, 383, 479, 319, 119, 23, 1, 255, 895, 1343, 1119, 559, 167, 27, 1, 511, 2047, 3583, 3583, 2239, 895, 223, 31, 1, 1023, 4607, 9215, 10751, 8063, 4031, 1343, 287, 35, 1

Offset: 0

Gary W. Adamson, Aug 18 2007

Row sums = A132308: (1, 4, 15, 50, 157, 480, 1451, ...). Inverse binomial transform of A132307 = triangle A132309 (having row sums A077552).

			First few rows of the triangle:
   1;
   3,   1;
   7,   7,   1;
  15,  23,  11,  1;
  31,  63,  47, 15,  1;
  63, 159, 159, 79, 19, 1;
  ...

Cf. A132308, A132309, A077552.

2*A007318^(2) - A000012 as infinite lower triangular matrices.

A132308 a(n) = 2*3^n - n - 1.

Original entry on oeis.org

1, 4, 15, 50, 157, 480, 1451, 4366, 13113, 39356, 118087, 354282, 1062869, 3188632, 9565923, 28697798, 86093425, 258280308, 774840959, 2324522914, 6973568781, 20920706384, 62762119195, 188286357630, 564859072937, 1694577218860

Offset: 0

Gary W. Adamson, Aug 18 2007

nonn

Row sums of triangle A132307.

			a(2) = 15 = sum of row 2 terms of triangle A132307: (7 + 7 + 1).
a(2) = 15 = (1, 2, 1) dot (1, 3, 8) = (1 + 6 + 8); where A077552 = (1, 3, 8, 16, 32, 64, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A132307, A132309, A077552.

A132308:=n->2*3^n-n-1; seq(A132308(n), n=0..30); # Wesley Ivan Hurt, Mar 26 2014

s=1;lst={s};Do[s+=(n+=s++)+n;AppendTo[lst, s], {n, 0, 5!, 1}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
Table[2*3^n-n-1,{n,0,30}] (* or *) LinearRecurrence[{5,-7,3},{1,4,15},30] (* Harvey P. Dale, Aug 06 2013 *)
CoefficientList[Series[(1 - x + 2 x^2)/((1 - 3 x) (1 - x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 27 2014 *)

Binomial transform of A077552.
a(n) = 2*3^n - n - 1. - Rolf Pleisch, Sep 26 2010
G.f.: (1-x+2*x^2)/((1-3*x)*(1-x)^2). - Bruno Berselli, Mar 31 2011

More terms from Vladimir Joseph Stephan Orlovsky, Nov 15 2008
Wrong formula 2*n^3-n-1 removed by Rolf Pleisch, Oct 19 2010
Simpler definition from Wesley Ivan Hurt, Mar 26 2014

A132309 A007318^(-1) * A132307.

Original entry on oeis.org

1, 2, 1, 2, 5, 1, 2, 5, 8, 1, 2, 9, 9, 11, 1, 2, 9, 24, 14, 14, 1, 2, 13, 25, 50, 20, 17, 1, 2, 13, 48, 55, 90, 27, 20, 1, 2, 17, 49, 133, 105, 147, 35, 23, 1, 2, 17, 80, 140, 308, 182, 224, 44, 26, 1

Offset: 0

Gary W. Adamson, Aug 18 2007

Row sums = A077552: (1, 3, 8, 16, 32, 64, 128, ...).

			First few rows of the triangle:
  1;
  2,  1;
  2,  5,  1;
  2,  5,  8,  1;
  2,  9,  9, 11,  1;
  2,  9, 24, 14, 14,  1;
  2, 13, 25, 50, 20, 17,  1;
  2, 13, 48, 55, 90, 27, 20,  1;
  ...

Cf. A132307, A077552.

Inverse binomial transform of triangle A132307.

A258935 Independence number of Keller graphs.

Original entry on oeis.org

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

cp's OEIS Frontend

A352823 Number of nonfixed points y(i) != i, where y is the weakly increasing sequence of prime indices of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A198633 Total number of round trips, each of length 2*n on the graph P_3 (o-o-o).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A146541 Binomial transform of A010688.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A132307 2*A007318^(2) - A000012.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A132308 a(n) = 2*3^n - n - 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A132309 A007318^(-1) * A132307.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A258935 Independence number of Keller graphs.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula