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%I #58 Jan 14 2024 02:27:25
%S 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,
%T 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,
%U 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
%N Number of ways to write n as i*j + i + j, 0 < i <= j.
%C 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
%C From _Daniel Forgues_, Sep 20 2011: (Start)
%C a(n) is the number of nontrivial factorizations of n+1, in two factors.
%C 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)
%C a(n) is the number of ways to write n+1 as i*j, 1 < i <= j. - _Arkadiusz Wesolowski_, Nov 18 2012
%C For a generalization, see comment in A260804. - _Vladimir Shevelev_, Aug 04 2015
%C 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
%H Robert Israel, <a href="/A072670/b072670.txt">Table of n, a(n) for n = 0..10000</a>
%H Joseph W. Andrushkiw, Roman I. Andrushkiw and Clifton E. Corzatt, <a href="http://www.jstor.org/stable/2689456">Representations of Positive Integers as Sums of Arithmetic Progressions</a>, Mathematics Magazine, Vol. 49, No. 5 (Nov., 1976), pp. 245-248.
%H M. A. Nyblom and C. Evans, <a href="http://ajc.maths.uq.edu.au/pdf/28/ajc_v28_p149.pdf">On the enumeration of partitions with summands in arithmetic progression</a>, Australian Journal of Combinatorics, Vol. 28 (2003), pp. 149-159.
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1508.03970">Representation of positive integers by the form x1...xk+x1+...+xk</a>, arXiv:1508.03970 [math.NT], 2015.
%F a(n) = A038548(n+1) - 1.
%F From _N-E. Fahssi_, Feb 01 2008: (Start)
%F 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.
%F G.f.: Sum_{n>=1} a(n) x^n = 1/x Sum_{k>=2} x^(k^2)/(1-x^k). (End)
%F lim_{n->infinity} a(A002110(n)-1) = infinity. - _Vladimir Shevelev_, Aug 04 2015
%F a(n) = A161840(n+1)/2. - _Omar E. Pol_, Feb 27 2019
%F Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3) / 2, where gamma is Euler's constant (A001620). - _Amiram Eldar_, Jan 14 2024
%e a(11)=2: 11 = 1*5 + 1 + 5 = 2*3 + 2 + 3.
%e From _Daniel Forgues_, Sep 20 2011 (Start)
%e Number of nontrivial factorizations of n+1 in two factors:
%e 0 for the unit 1 and prime numbers
%e 1 for a square: n^2 = n*n
%e 1 for 6 (2*3), 10 (2*5), 14 (2*7), 15 (3*5)
%e 1 for a cube: n^3 = n*n^2
%e 2 for 12 (2*6, 3*4), for 18 (2*9, 3*6) (End)
%p 0, seq(ceil(numtheory:-tau(n+1)/2)-1, n=1..100); # _Robert Israel_, Aug 04 2015
%t 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 *)
%t Table[Ceiling[DivisorSigma[0, n + 1]/2] - 1, {n, 0, 104}] (* _Arkadiusz Wesolowski_, Nov 18 2012 *)
%o (PARI) is_ok(k,i,j)=0<i&&j>=i&&k===i*j+i+j;
%o 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 */
%o (PARI) a(n)=(numdiv(n+1)+issquare(n+1))/2-1 \\ _Charles R Greathouse IV_, Jul 14 2017
%Y Cf. A001620, A067432, A066938, A072668, A006093, A072671, A161840, A260803, A260804.
%K nonn
%O 0,12
%A _Reinhard Zumkeller_, Jun 30 2002
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