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

1, 2, 3, 4, 6, 8, 10, 14, 16, 20, 28, 32, 44, 52, 56, 64, 68, 76, 88, 104, 128, 136, 152, 184, 208, 232, 248, 256, 272, 296, 304, 328, 344, 368, 464, 496, 512, 592, 656, 688, 736, 752, 848, 928, 944, 976, 992, 1024, 1072, 1136, 1168, 1184, 1264, 1312, 1328, 1376, 1424, 1504, 1696, 1888

Offset: 1

Author

Jean-Christophe Hervé, Jun 01 2014

Keywords

• nonn
• look

Comments

Also numbers which are not of the form n = (r+1)(2a+3r)/2 for any positive integers r and a >= 1.

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).

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).

Examples

			5 is not in the sequence because 5 = 1+4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 121 terms from Jean-Christophe Hervé)
• J. W. Andrushkiw, R. I. Andrushkiw and C. E. Corzatt, Representations of Positive Integers as Sums of Arithmetic Progressions, Mathematics Magazine, Vol. 49, No. 5 (Nov., 1976), pp. 245-248.
• Francisco Javier de Vega, Some Variants of Integer Multiplication, Axioms (2023) Vol. 12, 905. See p. 8.
• M. A. Nyblom and C. Evans, On the enumeration of partitions with summands in arithmetic progression, Australian Journal of Combinatorics, Vol. 28 (2003), pp. 149-159.

Crossrefs

Cf. A243223.

Formula

A243223(a(n)) = 0.