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%I #8 Dec 05 2013 19:55:58
%S 1,5,15,45,495,16830,4358970,1159486020,196818113950920,
%T 3151092455396895169036800,136084696980410308844836382925537725529600,
%U 9996588705394796239042140065772174939073840705818917941136700639014745600
%N Partition of positive integers into shortest possible groups, starting with (1), (2,3), (4,5,6), (7,8,9,10,11), such that a(n) = the sum of the terms of the n-th group is a multiple of a(n-1) and a(n) > a(n-1).
%C Dropping requirement a(n) > a(n-1) leads to a different partition: (1), (2, 3), (4, 5, 6), (7, 8), ... - see A160275.
%C For partition starting with (1), (2), (3,4,5), see A075631.
%F a(n) = A000217(A079801(n)) - A000217(A079801(n-1)) [From R. J. Mathar and Max Alekseyev]
%o (PARI) A000217(n)= { return(n*(n+1)/2) ; } upto(first,osum,strict)= { local(trifirst,tstsu) ; trifirst=A000217(first-1) ; for(lst=first+1,first+100000000, tstsu=A000217(lst)-trifirst ; if(strict==1 && tstsu<= osum, next ; ) ; if( tstsu % osum == 0, return(lst) ; ) ; ) ; return(-1) ; } { a=1 ; first=2 ; for(n=2,40, last=upto(first,a,1) ; a=A000217(last)-A000217(first-1) ; print(a,",") ; first=last+1 ; ) ; - _R. J. Mathar_, May 06 2006
%Y Cf. A079799, A079800, A079801, A075631.
%K nonn
%O 1,2
%A _Amarnath Murthy_, Feb 05 2003
%E More terms from _R. J. Mathar_, May 06 2006
%E Edited and extended by _Max Alekseyev_, May 08 2009
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