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A072670 Number of ways to write n as i*j+i+j, 0<i<=j. 18
0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 2, 0, 3, 0, 2, 1, 1, 1, 4, 0, 1, 1, 3, 0, 3, 0, 2, 2, 1, 0, 4, 1, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 5, 0, 1, 2, 3, 1, 3, 0, 2, 1, 3, 0, 5, 0, 1, 2, 2, 1, 3, 0, 4, 2, 1, 0, 5, 1, 1, 1, 3, 0, 5, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 3, 0, 3, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

a(n) is the number of partitions of n+1 with summands in arithmetic progression having common difference 2. For example a(29)=3 because there are 3 partitions of 30 that are in arithmetic progressions : 2+4+6+8+10, 8+10+12 and 14+16. - N-E. Fahssi, Feb 01 2008

Daniel Forgues, Sep 20 2011 (Start)

  a(n) is number of nontrivial factorizations of n+1, in two factors.

  Number of ways to write n+1 as i*j+i+j+1 = (i+1)(j+1), 0 < i <= j. (End)

For a generalization, see comment in A260804. - Vladimir Shevelev, Aug 04 2015

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

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.

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.

Vladimir Shevelev, Representation of positive integers by the form x1...xk+x1+...+xk, arXiv:1508.03970 [math.NT], 2015.

FORMULA

a(n) = A038548(n+1) - 1.

a(n) = p2(n+1), where p2(n)= 1/2(d(n)-2+((-1)^{d(n)+1}+1)/2); d(n) is the number of divisors of n: A000005. G.F.: Sum_{n >= 1} a(n) x^n = 1/x Sum_{k>=2} x^(k^2)/(1-x^k). - N-E. Fahssi, Feb 01 2008

a(n) = number of ways to write n + 1 as i*j, 1 < i <= j. - Arkadiusz Wesolowski, Nov 18 2012

lim_{n->infinity} a(A002110(n)-1) = infinity. - Vladimir Shevelev, Aug 04 2015

EXAMPLE

a(11)=2: 11 = 1*5+1+5 = 2*3+2+3.

From Daniel Forgues, Sep 20 2011 (Start)

a(n) is number of nontrivial factorizations of n+1 in two factors:

  0 for the unit 1 and prime numbers

  1 for a square: n^2 = n*n

  1 for 6 (2*3), 10 (2*5), 14 (2*7), 15 (3*5)

  1 for a cube: n^3 = n*n^2

  2 for 12 (2*6, 3*4), for 18 (2*9, 3*6) (End)

MAPLE

0, seq(ceil(numtheory:-tau(n+1)/2)-1, n=1..100); # Robert Israel, Aug 04 2015

MATHEMATICA

p2[n_] := 1/2 (Length[Divisors[n]] - 2 + ((-1)^(Length[Divisors[n]] + 1) + 1)/2); Table[p2[n + 1], {n, 0, 104}] (* N-E. Fahssi, Feb 01 2008 *)

Table[Ceiling[DivisorSigma[0, n + 1]/2] - 1, {n, 0, 104}] (* Arkadiusz Wesolowski, Nov 18 2012 *)

PROG

(PARI) is_ok(k, i, j)=0<i&&j>=i&&k===i*j+i+j;

first(m)=my(v=vector(m, z, 0)); for(l=1, m, for(j=1, l, for(i=1, j, if(is_ok(l, i, j), v[l]++)))); concat([0], v); /* Anders Hellström, Aug 04 2015 */

CROSSREFS

Cf. A067432, A066938, A072668, A006093, A072671, A260803, A260804.

Sequence in context: A178146 A114708 A084927 * A087624 A085122 A083715

Adjacent sequences:  A072667 A072668 A072669 * A072671 A072672 A072673

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Jun 30 2002

STATUS

approved

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Last modified March 29 03:14 EDT 2017. Contains 284250 sequences.