A082637 -id:A082637 - cp's OEIS frontend

A082647 Number of ways n can be expressed as the sum of d consecutive positive integers where d>0 is a divisor of n.

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 3, 1, 1, 2, 1, 3, 2, 1, 1, 2, 2, 1, 3, 1, 1, 4, 1, 1, 2, 2, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 3, 1, 1, 4, 1, 2, 3, 1, 1, 2, 3, 1, 3, 1, 1, 3, 1, 3, 2, 1, 2, 3, 1, 1, 3, 2, 1, 2, 2, 1, 4, 3, 1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 1, 2, 4

Offset: 1

Naohiro Nomoto, May 15 2003

Number of ways to write n as the sum of an odd number of consecutive integers. - Vladeta Jovovic, Aug 28 2007
Number of odd divisors of n less than sqrt(2*n). - Vladeta Jovovic, Sep 16 2007
Conjecture: a(n) is also the number of subparts in an octant of the symmetric representation of sigma(n). - Omar E. Pol, Feb 22 2017

			For n=6: 6 has two ways -- (d=3; 3|6), 1+2+3=6; and (d=1; 1|6), 6=6 -- so a(6)=2.

Peter Kagey, Table of n, a(n) for n = 1..10000
M. D. Hirschhorn and P. M. Hirschhorn, Partitions into Consecutive Parts, Mathematics Magazine: 2003, Volume 76, Number 4, pp. 306-308.
William Lowell Putnam Competition, Function A(k) in Problem B6, 2015.

Cf. A001227, A054843, A082637, A082662, A237593, A285901.

N:= 1000: # to get a(1) to a(N)
g:= add(x^(k*(2*k-1))/(1-x^(2*k-1)), k=1..floor(sqrt(N/2))):
S:= series(g,x,N+1):
seq(coeff(S,x,n),n=1..N); # Robert Israel, Dec 08 2015

a(n) = my(q = sqrt(2*n)); sumdiv(n, d, (d%2) && (d < q)); \\ Michel Marcus, Jul 04 2014

G.f.: Sum_{k>0} x^(k*(2*k-1))/(1-x^(2*k-1)). - Vladeta Jovovic, Aug 25 2004
Conjecture: a(n) = A067742(n) + A131576(n). - Omar E. Pol, Feb 22 2017
Conjecture: a(n) = A001227(n) - A131576(n). - Omar E. Pol, Apr 18 2017

A082636 a(n) = smallest number which can be expressed as sum of d consecutive primes in exactly n ways (where d>0 is a divisor of the number).

Original entry on oeis.org

2, 36, 240, 10320, 227304, 287787528

Offset: 1

Naohiro Nomoto, May 14 2003

			For n=2: 36 is in the sequence because 36 has two ways: e.g. (d=2; 2|36) and 17+19=36, (d=4; 4|36) and 5+7+11+13=36. 36 is the smallest number. so a(2)=36.

Cf. A082637.

a(5)-a(6) from Donovan Johnson, Nov 11 2008

A082634 Numbers n which can be expressed as sum of d consecutive primes (where d>0 is a divisor of n).

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 12, 13, 15, 17, 18, 19, 23, 24, 29, 30, 31, 36, 37, 41, 42, 43, 47, 48, 52, 53, 59, 60, 61, 67, 68, 71, 72, 73, 78, 79, 83, 84, 88, 89, 90, 97, 100, 101, 103, 107, 109, 112, 113, 119, 120, 127, 128, 131, 132, 137, 138

Offset: 1

Naohiro Nomoto, May 14 2003

			30 is in the sequence because (d=2; 2|30) and 13+17=30.

Cf. A082636, A082637.

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A082647 Number of ways n can be expressed as the sum of d consecutive positive integers where d>0 is a divisor of n.

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A082636 a(n) = smallest number which can be expressed as sum of d consecutive primes in exactly n ways (where d>0 is a divisor of the number).

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A082634 Numbers n which can be expressed as sum of d consecutive primes (where d>0 is a divisor of n).

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