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%I #35 Jan 04 2024 10:53:30
%S 0,1,2,4,7,13,23,42,76,139,255,471,873,1627,3044,5718,10779,20387,
%T 38673,73561,140267,268065,513349,984910,1892874,3643569,7023561,
%U 13557019,26200181,50691977,98182665,190353369,369393465,717457655
%N a(n) = Sum_{m=1..n} T(m,n+1-m), array T as in A048887.
%C From _Marc LeBrun_, Dec 12 2001: (Start)
%C Define a "numbral arithmetic" by replacing addition with binary bitwise inclusive-OR (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^n-1]. Conjecture confirmed by Richard C. Schroeppel, Dec 14 2001. (End)
%C 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
%C 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
%H D. Applegate, M. LeBrun, N. J. A. Sloane, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Sloane/carry2.html">Dismal Arithmetic</a>, J. Int. Seq. 14 (2011) # 11.9.8.
%H A. Frosini and S. Rinaldi, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Frosini/fros2.html">On the Sequence A079500 and Its Combinatorial Interpretations</a>, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.
%F G.f.: Sum_{k>0} x^k*(1-x^k)/(1-2*x+x^(k+1)). - _Vladeta Jovovic_, Feb 25 2003
%F a(m) = Sum_{ n=2..m+1 } Fn(m) where Fn is a Fibonacci n-step number (Fibonacci, tetranacci, etc.) indexed as in A000045, A000073, A000078. - _Gerald McGarvey_, Sep 25 2004
%e From _Joerg Arndt_, Dec 28 2012: (Start)
%e There are a(6)=23 compositions p(1)+p(2)+...+p(m)=6 such that p(k)<=p(1)+1:
%e [ 1] [ 1 1 1 1 1 1 ]
%e [ 2] [ 1 1 1 1 2 ]
%e [ 3] [ 1 1 1 2 1 ]
%e [ 4] [ 1 1 2 1 1 ]
%e [ 5] [ 1 1 2 2 ]
%e [ 6] [ 1 2 1 1 1 ]
%e [ 7] [ 1 2 1 2 ]
%e [ 8] [ 1 2 2 1 ]
%e [ 9] [ 2 1 1 1 1 ]
%e [10] [ 2 1 1 2 ]
%e [11] [ 2 1 2 1 ]
%e [12] [ 2 1 3 ]
%e [13] [ 2 2 1 1 ]
%e [14] [ 2 2 2 ]
%e [15] [ 2 3 1 ]
%e [16] [ 3 1 1 1 ]
%e [17] [ 3 1 2 ]
%e [18] [ 3 2 1 ]
%e [19] [ 3 3 ]
%e [20] [ 4 1 1 ]
%e [21] [ 4 2 ]
%e [22] [ 5 1 ]
%e [23] [ 6 ]
%e (End)
%o (PARI)
%o N = 66; x = 'x + O('x^N);
%o gf = sum(n=0,N, (1-x^n)*x^n/(1-2*x+x^(n+1)) ) + 'c0;
%o v = Vec(gf); v[1]-='c0; v
%o /* _Joerg Arndt_, Apr 14 2013 */
%Y Cf. A007059.
%Y Cf. A000312, A001372, A002861, A006961, A001373, A054050, A054745, A125024.
%K nonn
%O 0,3
%A _Clark Kimberling_
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