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

1,2

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).

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.

Formula

A243223(a(n))) = 0.

Example

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

Crossrefs

Cf. A243223.

Sequence in context: A239100 A337046 A341031 * A220851 A028290 A003107

Adjacent sequences: A243222 A243223 A243224 * A243226 A243227 A243228

Keyword

nonn,look

Author

Jean-Christophe Hervé, Jun 01 2014

Status

approved