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A243225 Numbers which are not the sum of positive integers in an arithmetic progression with common difference 3.

%I A243225 #38 Nov 09 2024 03:16:50
%S A243225 1,2,3,4,6,8,10,14,16,20,28,32,44,52,56,64,68,76,88,104,128,136,152,
%T A243225 184,208,232,248,256,272,296,304,328,344,368,464,496,512,592,656,688,
%U A243225 736,752,848,928,944,976,992,1024,1072,1136,1168,1184,1264,1312,1328,1376,1424,1504,1696,1888
%N A243225 Numbers which are not the sum of positive integers in an arithmetic progression with common difference 3.
%C A243225 Also numbers which are not of the form n = (r+1)(2a+3r)/2 for any positive integers r and a >= 1.
%C A243225 Except a(3) = 3, these are the powers of 2 and the products of a power of two 2^k with an odd prime p such that 1+2^(k+1)/3 <= p <= 3(2^(k+1)-1). For example, 20 is in the sequence as 20 = 2^2*5 and 1+2^3/3 <= 5 <= 3(2^3-1).
%C A243225 The equivalent sequence for arithmetic progressions with a common difference of 2 is A000040, the prime numbers (i.e., the numbers > 1 which are not sum of positive integers in arithmetic progression with a common difference 2 are exactly the primes).
%H A243225 Alois P. Heinz, <a href="/A243225/b243225.txt">Table of n, a(n) for n = 1..10000</a> (first 121 terms from Jean-Christophe Hervé)
%H A243225 J. W. Andrushkiw, R. I. Andrushkiw and C. E. Corzatt, <a href="http://www.jstor.org/stable/2689456">Representations of Positive Integers as Sums of Arithmetic Progressions</a>, Mathematics Magazine, Vol. 49, No. 5 (Nov., 1976), pp. 245-248.
%H A243225 Francisco Javier de Vega, <a href="https://doi.org/10.3390/axioms12100905">Some Variants of Integer Multiplication</a>, Axioms (2023) Vol. 12, 905. See p. 8.
%H A243225 M. A. Nyblom and C. Evans, <a href="http://ajc.maths.uq.edu.au/pdf/28/ajc_v28_p149.pdf">On the enumeration of partitions with summands in arithmetic progression</a>, Australian Journal of Combinatorics, Vol. 28 (2003), pp. 149-159.
%F A243225 A243223(a(n)) = 0.
%e A243225 5 is not in the sequence because 5 = 1+4.
%Y A243225 Cf. A243223.
%K A243225 nonn,look
%O A243225 1,2
%A A243225 _Jean-Christophe Hervé_, Jun 01 2014