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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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6
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,12
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COMMENTS
| a(n) = A038548(n+1) - 1.
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. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), 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)
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REFERENCES
| 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.
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LINKS
| 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.
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FORMULA
| 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). - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Feb 01 2008
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EXAMPLE
| a(11)=2: 11 = 1*5+1+5 = 2*3+2+3.
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)
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MATHEMATICA
| p2[n_]:= 1/2 (Length[Divisors[n]]-2+((-1)^(Length[Divisors[n]]+1)+1)/2); Table[p2[n+1], {n, 0, 104}] - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Feb 01 2008
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CROSSREFS
| Cf. A067432, A066938, A072668, A006093, A072671.
Sequence in context: A178146 A114708 A084927 * A087624 A085122 A083715
Adjacent sequences: A072667 A072668 A072669 * A072671 A072672 A072673
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KEYWORD
| nonn
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AUTHOR
| Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 30 2002
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