0,10

Number of nontrivial ways to write n as sum of at least 2 consecutive integers. That is, we are not counting the trivial solution n=n. E.g. a(9)=2 because 9 = 4 + 5 and 9 = 2 + 3 + 4. a(8)=0 because there are no integers m and k such that m + (m+1) + ... + (m+k-1) = 8 apart from k=1, m=8. - Alfred Heiligenbrunner (alfred.heiligenbrunner(AT)gmx.at), Jun 07 2004

Comment from Michael Gilleland, Dec 29, 2002: Also number of sums of sequences of consecutive positive integers excluding sequences of length 1 (e.g. 9 = 2+3+4 or 4+5 so a(9)=2). (Useful for cribbage players.)

Let M be any positive integer. Then a(n) = number of proper divisors of M^n+1 of the form M^k+1.

This sequence gives the distinct differences of triangular numbers Ti giving n : n = Ti - Tj; none if n=2^k. If factor a = n or a > (n/a - 1)/2 : i = n/a + (a-1)/2; j = n/a - (a+1)/2. Else : i = n/2a + (2a-1)/2; j = n/2a - (2a-1)/2. Examples: 7 is prime; 7 = T4 - T2 = (1+2+3+4) - (1+2) (a=7; n/a=1). The odd factors of 35 are 35, 7 and 5; 35 = T18 - T16 (a=35) = T8 - T1 (a=7) = T5 - T7 (a=5). 144 = T20 - T11 (a=9) = T49 - T46 (a=3). - M. Dauchez (mdzzdm(AT)yahoo.fr), Oct 31 2005

Also number of partitions of n into the form 1+2+...(k-1)+k+k+...+k for some k>=2. Example: a(9)=2 because we have [2,2,2,2,1] and [3,3,2,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2006

Graham, Knuth and Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), see exercise 2.30 on p. 65.

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

a(n) = 0 iff n=2^k.

G.f.=sum(x^(k(k+1)/2)/(1-x^k), k=2..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2006

a(21)=3 as 10^k+1 is a divisor of 10^21+1 for k in {1,3,7}:

10^21+1 = 7 * 7 * 11 * 13 * 127 * 2689 * 459691 * 909091] =

=10000001*99999990000001=(11*909091)*(7*7*13*127*2689*459691)

=1001*999000999000999001=(7*11*13)*(7*127*2689*459691*909091)

=11*90909090909090909091=(11)*(7*7*13*127*2689*459691*909091)

g:=sum(x^(k*(k+1)/2)/(1-x^k), k=2..20): gser:=series(g, x=0, 115): seq(coeff(gser, x, n), n=0..100); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2006

g[n_]:=Module[{dL=Divisors[2n], dP}, dP=Transpose[{dL, 2n/dL}]; Select[dP, ((1<#[[1]]<#[[2]]) && (Mod[ #[[1]]-#[[2]], 2]==1))&] ]; Table[Length[g[n]], {n, 1, 100}]

Cf. A001227, A062397, A057934. Equals A001227(n) - 1.

Sequence in context: A062093 A046214 A115413 this_sequence A033630 A101446 A062760

Adjacent sequences: A069280 A069281 A069282 this_sequence A069284 A069285 A069286

nonn

Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 13 2002

Edited by Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 25 2002