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A111775 Number of ways n can be written as a sum of at least three consecutive integers. 4
0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 1, 1, 1, 2, 1, 0, 3, 0, 0, 2, 1, 2, 2, 0, 1, 2, 1, 0, 3, 0, 1, 4, 1, 0, 1, 1, 2, 2, 1, 0, 3, 2, 1, 2, 1, 0, 3, 0, 1, 4, 0, 2, 3, 0, 1, 2, 3, 0, 2, 0, 1, 4, 1, 2, 3, 0, 1, 3, 1, 0, 3, 2, 1, 2, 1, 0, 5, 2, 1, 2, 1, 2, 1, 0, 2, 4, 2, 0, 3, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,15

COMMENTS

Powers of 2 and (odd) primes cannot be written as a sum of at least three consecutive integers. a(n) strongly depends on the number of odd divisors of n (A001227): Suppose n is to be written as sum of k consecutive integers starting with m, then 2n = k(2m + k - 1). Only one of the factors is odd. For each odd divisor of n there is a unique corresponding k, k=1 and k=2 must be excluded.

When the initial 0 term is a(1), a(n) is the number of times n occurs after the second column in the square array of A049777. - Bob Selcoe, Feb 14 2014

For nonnegative integers x,y where x-y>=3: a(n) equals the number of ways n can be expressed as a function of (x*(x+1)-y*(y+1))/2 when the initial 0 term is a(1). - Bob Selcoe, Feb 14 2014

REFERENCES

Nieuw Archief voor Wiskunde 5/6 nr. 2 Problems/UWC Problem C part 4, Jun 2005, p. 181-182

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

K. S. Brown's Mathpages, Partitions into Consecutive Integers

A. Heiligenbrunner, Sum of adjacent numbers (in German)

Nieuw Archief voor Wiskunde 5/6 nr. 2 Problems/UWC, Problem C: solution of this Problem

J. Spies, Sage program for computing A111775

FORMULA

For n > 1, if n is even then a(n)=A001227(n)-1=A069283(n) or else a(n)=A001227(n)-2.

EXAMPLE

a(15) = 2 because 15 = 4+5+6 and 15 = 1+2+3+4+5. The number of odd divisors of 15 is 4.

G.f. = x^6 + x^9 + x^10 + x^12 + x^14 + 2*x^15 + 2*x^18 + x^20 + 2*x^21 + x^22 + ...

a(30) = 3 because there are 3 ways to satisfy (x*(x+1)-y*(y+1))/2 = 30 when x-y>=3: x=8, y=3; x=9, y=5; and x=11, y=8. - Bob Selcoe, Feb 14 2014

MAPLE

A001227:= proc(n) local d, s; s := 0: for d from 1 by 2 to n do if n mod d = 0 then s:=s+1 fi: end do: return(s); end proc; A111775:= proc(n) local k; if n=1 then return(0) fi: k := A001227(n): if type(n, even) then k:=k-1 else k:=k-2 fi: return k; end proc; seq(A111775(i), i=1..150);

PROG

(PARI) {a(n) = local(m); if( n<1, 0, sum( i=0, n, m=0; if( issquare( 1 + 8*(n + i * (i + 1)/2), &m), m\2 > i+2)))}; /* Michael Somos, Aug 27 2012 */

CROSSREFS

Cf. A111774, A001227 (number of odd divisors), A069283.

Sequence in context: A161116 A262726 A112605 * A025844 A035461 A118508

Adjacent sequences:  A111772 A111773 A111774 * A111776 A111777 A111778

KEYWORD

easy,nonn

AUTHOR

Jaap Spies, Aug 16 2005

STATUS

approved

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Last modified November 17 21:13 EST 2018. Contains 317279 sequences. (Running on oeis4.)