A243223 -id:A243223 - cp's OEIS frontend

A243224 Number of odd divisors d of n such that d > 1 and d(1+d/3)/2 <= n <= 3d(d-1)/2.

0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0

Offset: 1

Jean-Christophe Hervé, Jun 01 2014

nonn

This sequence is useful for computing A243223, the number of partitions of n into summands in arithmetic progression with common difference 3. The definition follows Nyblom and Evans 2003 (see LINK) with slight modifications and corrections.

			a(6) = 1 because 3, the unique odd divisor > 1 of 6 satisfies 3(1+3/3)/2 <= 6 <= 3.3(3-1)/2.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000
M. A. Nyblom and C. Evans, On the enumeration of partitions with summands in arithmetic progression, Australasian Journal of Combinatorics, Vol. 28 (2003), pp. 149-159.

Cf. A243223.

a(n) = sumdiv(n, d, (d > 1) && (d % 2) && (d*(1+d/3)/2 <= n) && (n <= 3*d*(d-1)/2)); \\ Michel Marcus, Jun 02 2014

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

Original entry on oeis.org

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

Jean-Christophe Hervé, Jun 01 2014

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).

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

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.

Cf. A243223.

A243223(a(n)) = 0.

cp's OEIS Frontend

A243224 Number of odd divisors d of n such that d > 1 and d(1+d/3)/2 <= n <= 3d(d-1)/2.

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