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A066740 Number of distinct partitions of A007504(n) which can be obtained by merging parts in the partition 2+3+5+...+prime(n), where prime(n) is the n-th prime.

%I A066740 #13 Nov 05 2023 14:59:57
%S A066740 1,1,2,5,13,44,151,614,2446,11066,53368,253927,1316375,7213979,
%T A066740 38175696,213766427
%N A066740 Number of distinct partitions of A007504(n) which can be obtained by merging parts in the partition 2+3+5+...+prime(n), where prime(n) is the n-th prime.
%e A066740 For n=4, the 13 partitions are 17, 2+15, 3+14, 5+12, 7+10, 8+9, 2+3+12, 2+5+10, 2+7+8, 3+5+9, 3+7+7, 5+5+7, 2+3+5+7. 5+12 and 7+10 can be obtained in two ways each: 5+12 = (5)+(2+3+7) = (2+3)+(5+7), 7+10 = (7)+(2+3+5) = (2+5)+(3+7).
%p A066740 b:= proc(n) local p; p:= `if`(n=0, 1, ithprime(n));
%p A066740       b(n):= `if`(n<2, {[p$n]}, map(x-> [sort([x[], p]),
%p A066740       seq(sort(subsop(i=x[i]+p, x)), i=1..nops(x))][], b(n-1)))
%p A066740     end:
%p A066740 a:= n-> nops(b(n)):
%p A066740 seq(a(n), n=0..10);  # _Alois P. Heinz_, May 31 2013
%t A066740 addto[ p_, k_ ] := Module[ {}, lth=Length[ p ]; Union[ Sort/@Append[ Table[ Join[ Take[ p, i-1 ], {p[ [ i ] ]+k}, Take[ p, i-lth ] ], {i, 1, lth} ], Append[ p, k ] ] ] ]; addtolist[ plist_, k_ ] := Union[ Join@@(addto[ #, k ]&/@plist) ]; l[ 0 ]={{}}; l[ n_ ] := l[ n ]=addtolist[ l[ n-1 ], Prime[ n ] ]; a[ n_ ] := Length[ l[ n ] ]
%Y A066740 Cf. A007504, A066723.
%K A066740 more,nonn
%O A066740 0,3
%A A066740 _Naohiro Nomoto_, Jan 16 2002
%E A066740 Edited by _Dean Hickerson_, Jan 18 2002
%E A066740 a(14)-a(15) from _Sean A. Irvine_, Nov 05 2023