

A048888


a(n) = Sum_{m=1..n} T(m,n+1m), array T as in A048887.


15



0, 1, 2, 4, 7, 13, 23, 42, 76, 139, 255, 471, 873, 1627, 3044, 5718, 10779, 20387, 38673, 73561, 140267, 268065, 513349, 984910, 1892874, 3643569, 7023561, 13557019, 26200181, 50691977, 98182665, 190353369, 369393465, 717457655
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OFFSET

0,3


COMMENTS

Define a "numbral arithmetic" by replacing addition with binary bitwise inclusiveOR (so that [3] + [5] = [7] etc.) and multiplication becomes shift&OR instead of shift&add (so that [3] * [3] = [7] etc.). [d] divides [n] means there exists an [e] with [d] * [e] = [n]. For example the six divisors of [14] are [1], [2], [3], [6], [7] and [14]. Then it appears that this sequence gives the number of proper divisors of [2^n1]. Conjecture confirmed by Richard C. Schroeppel, Dec 14 2001. (End)
The number of "prime endofunctions" on n points, meaning the cardinality of the subset of the A001372(n) mappings (or mapping patterns) up to isomorphism from n (unlabeled) points to themselves (endofunctions) which are neither the sum of prime endofunctions (i.e., whose disjoint connected components are prime endofunctions) nor the categorical product of prime endofunctions. The n for which a(n) is prime (n such that the number of prime endofunctions on n points is itself prime) are 2, 4, 5, 6, 9, 13, 19, ...  Jonathan Vos Post, Nov 19 2006
For n>=1, compositions p(1)+p(2)+...+p(m)=n such that p(k)<=p(1)+1, see example.  Joerg Arndt, Dec 28 2012


LINKS

D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.


FORMULA

G.f.: Sum_{k>0} x^k*(1x^k)/(12*x+x^(k+1)).  Vladeta Jovovic, Feb 25 2003


EXAMPLE

There are a(6)=23 compositions p(1)+p(2)+...+p(m)=6 such that p(k)<=p(1)+1:
[ 1] [ 1 1 1 1 1 1 ]
[ 2] [ 1 1 1 1 2 ]
[ 3] [ 1 1 1 2 1 ]
[ 4] [ 1 1 2 1 1 ]
[ 5] [ 1 1 2 2 ]
[ 6] [ 1 2 1 1 1 ]
[ 7] [ 1 2 1 2 ]
[ 8] [ 1 2 2 1 ]
[ 9] [ 2 1 1 1 1 ]
[10] [ 2 1 1 2 ]
[11] [ 2 1 2 1 ]
[12] [ 2 1 3 ]
[13] [ 2 2 1 1 ]
[14] [ 2 2 2 ]
[15] [ 2 3 1 ]
[16] [ 3 1 1 1 ]
[17] [ 3 1 2 ]
[18] [ 3 2 1 ]
[19] [ 3 3 ]
[20] [ 4 1 1 ]
[21] [ 4 2 ]
[22] [ 5 1 ]
[23] [ 6 ]
(End)


PROG

(PARI)
N = 66; x = 'x + O('x^N);
gf = sum(n=0, N, (1x^n)*x^n/(12*x+x^(n+1)) ) + 'c0;
v = Vec(gf); v[1]='c0; v


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



