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A072670
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Number of ways to write n as i*j + i + j, 0 < i <= j.
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30
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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
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OFFSET
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0,12
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COMMENTS
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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
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)
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
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LINKS
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FORMULA
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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). (End)
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
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EXAMPLE
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a(11)=2: 11 = 1*5 + 1 + 5 = 2*3 + 2 + 3.
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)
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MAPLE
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0, seq(ceil(numtheory:-tau(n+1)/2)-1, n=1..100); # Robert Israel, Aug 04 2015
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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