

A072670


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


30



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.  NE. Fahssi, Feb 01 2008
From Daniel Forgues, Sep 20 2011: (Start)
a(n) is the number of nontrivial factorizations of n+1, in two factors.
a(n) is the number of ways to write n+1 as i*j + i + j + 1 = (i+1)(j+1), 0 < i <= j. (End)
a(n) is the number of ways to write n+1 as i*j, 1 < i <= j.  Arkadiusz Wesolowski, Nov 18 2012
For a generalization, see comment in A260804.  Vladimir Shevelev, Aug 04 2015
Number of partitions of n into 3 parts whose largest part is equal to the product of the other two.  Wesley Ivan Hurt, Jan 04 2022


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. 245248.
M. A. Nyblom and C. Evans, On the enumeration of partitions with summands in arithmetic progression, Australian Journal of Combinatorics, Vol. 28 (2003), pp. 149159.
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.
From NE. Fahssi, Feb 01 2008: (Start)
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)/(1x^k). (End)
lim_{n>infinity} a(A002110(n)1) = infinity.  Vladimir Shevelev, Aug 04 2015
a(n) = A161840(n+1)/2.  Omar E. Pol, Feb 27 2019


EXAMPLE

a(11)=2: 11 = 1*5 + 1 + 5 = 2*3 + 2 + 3.
From Daniel Forgues, Sep 20 2011 (Start)
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}] (* NE. 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 */
(PARI) a(n)=(numdiv(n+1)+issquare(n+1))/21 \\ Charles R Greathouse IV, Jul 14 2017


CROSSREFS

Cf. A067432, A066938, A072668, A006093, A072671, A161840, A260803, A260804.
Sequence in context: A114708 A084927 A333750 * A087624 A294891 A294879
Adjacent sequences: A072667 A072668 A072669 * A072671 A072672 A072673


KEYWORD

nonn


AUTHOR

Reinhard Zumkeller, Jun 30 2002


STATUS

approved



