Amritpal Singh - cp's OEIS frontend

User: Amritpal Singh

Amritpal Singh has authored 2 sequences.

A253388 Numbers n such that the number of divisors of n is the product of two distinct primes.

12, 18, 20, 28, 32, 44, 45, 48, 50, 52, 63, 68, 75, 76, 80, 92, 98, 99, 112, 116, 117, 124, 144, 147, 148, 153, 162, 164, 171, 172, 175, 176, 188, 192, 207, 208, 212, 236, 242, 243, 244, 245, 261, 268, 272, 275, 279, 284, 292, 304, 316, 320, 324, 325, 332, 333

Offset: 1

Amritpal Singh, Dec 31 2014

nonn

n such that A000005(n) is in A006881.

n is either of the form p^k where p is prime and k+1 is in A006881 or p1^k1*p2^k2 where p1 and p2 are distinct primes and k1+1 and k2+1 are distinct primes. - Robert Israel, Dec 31 2014

			12 has 6 divisors, and 6 is the product of two distinct primes, 2 and 3.

Robert Israel, Table of n, a(n) for n = 1..10000
Amritpal Singh, c++ program for generating the sequence

Cf. A000005, A006881. Contains A030515.

filter:= proc(n) local F;
  F:= ifactors(numtheory:-tau(n))[2];
  nops(F)=2 and F[1,2]=1 and F[2,2]=1;
end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 31 2014

a253388Q[x_] := Block[{d = FactorInteger[DivisorSigma[0, x]]},
Length[d] == 2 && Max[Last@Transpose@d] == 1]; a253388[n_] := Select[Range@n, a253388Q]; a253388[333] (* Michael De Vlieger, Jan 02 2015 *)
fQ[x_] := PrimeOmega@ x == 2 == PrimeNu@ x; Select[ Range@ 250, fQ[ DivisorSigma[0, #]] &] (* Robert G. Wilson v, Jan 13 2015 *)

isok(n) = (nbd = numdiv(n)) && (omega(nbd) == 2) && (bigomega(nbd) == 2); \\ Michel Marcus, Feb 07 2015

A218034 Number of ways to seat 4 types of people in n labeled seats around a circle such that no two adjacent people are of the same type.

Original entry on oeis.org

1, 4, 12, 24, 84, 240, 732, 2184, 6564, 19680, 59052, 177144, 531444, 1594320, 4782972, 14348904, 43046724, 129140160, 387420492, 1162261464, 3486784404, 10460353200, 31381059612, 94143178824, 282429536484, 847288609440, 2541865828332, 7625597484984, 22876792454964

Offset: 0

Amritpal Singh, Oct 19 2012

Number of length-n words with 4 letters and no two adjacent identical letters (including, for n >= 2, the first and last letter). - Joerg Arndt, Oct 21 2012
a(n), for n > 1, apparently is the trace of the n-th power of the adjacency matrix of the complete 4-graph, a 4 X 4 matrix with diagonal elements all zeros and off-diagonal all ones (cf. A054878). - Tom Copeland, Nov 06 2012
The corrected formula by Geoffrey Critzer below (for a general k) is a special case of Theorem 2 in Burstein and Wilf (1997). See also Edlin and Zeilberger (2000), Corollary 5.5 in Taylor (2014), and Section 5 in Hadjicostas and Zhang (2018). - Petros Hadjicostas, Jul 09 2018

K. Böhmová, C. Dalfó, C. Huemer, On cyclic Kautz digraphs, Preprint 2016.
A. Burstein and H. S. Wilf, On cyclic strings without long constant blocks, Fibonacci Quarterly, 35 (1997), 240-247.
Cristina Dalfó, From subKautz digraphs to cyclic Kautz digraphs, arXiv:1709.01882 [math.CO], 2017.
C. Dalfó, The spectra of subKautz and cyclic Kautz digraphs, Linear Algebra and its Applications, 531 (2017), pp. 210-219.
A. E. Edlin and D. Zeilberger, The Goulden-Jackson cluster method for cyclic words, Adv. Appl. Math., 25 (2000), 228-232.
Petros Hadjicostas and Lingyun Zhang, On cyclic strings avoiding a pattern, Discrete Mathematics, 341 (2018), 1662-1674.
Jair Taylor, Counting Words with Laguerre Series, Electron. J. Combin., 21 (2014), P2.1.
Index entries for linear recurrences with constant coefficients, signature (2,3).

Cf. A092297.

nn=28; CoefficientList[Series[1+4x +12x^2/(1+x)^2/(1-4x/(1+x)),{x,0,nn}],x] (* Geoffrey Critzer, Apr 05 2014 *)

a[0]:1$ a[1]:4$ a[n]:=3^n + 3*(-1)^n$ makelist(a[n],n,0,40); /* Martin Ettl, Oct 21 2012 */

a(n) = if( n<2, [1,4][n+1], 3^n + 3*(-1)^n ); /* Joerg Arndt, Oct 21 2012 */

a(0) = 1, a(1) = 4, a(n) = 3^n + 3*(-1)^n for n >= 2.
a(n) = 4 * A054878(n) for n >= 2. - Joerg Arndt, Oct 21 2012
G.f.: (1 + 2*x + x^2 - 12*x^3)/((1 + x)*(1 - 3*x)). - Colin Barker, Oct 22 2012
Generally for such words on k letters, g.f.: 1 + k*x + (k^2-k)*x^2/(1 + x)^2/(1 - k*x/(1 + x)). - Geoffrey Critzer, Apr 05 2014 [Corrected by Petros Hadjicostas, Jul 08 2018]
a(n+m) = a(n)*a(m)/4 + a(n+1)*a(m+1)/12. - Yuchun Ji, Sep 12 2017

a(0) = 1 prepended and more terms added by Joerg Arndt, Oct 21 2012

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User: Amritpal Singh

A253388 Numbers n such that the number of divisors of n is the product of two distinct primes.

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