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 A247935 Number of integer partitions of n whose distinct parts have no binary carries. 19
 1, 1, 2, 3, 4, 5, 8, 10, 11, 14, 18, 21, 26, 30, 38, 49, 47, 55, 66, 74, 84, 96, 110, 126, 134, 151, 171, 195, 209, 235, 272, 318, 307, 349, 377, 422, 448, 491, 534, 595, 617, 674, 734, 801, 841, 925, 998, 1098, 1118, 1219, 1299, 1418, 1476, 1591, 1711, 1865 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From Gus Wiseman, Mar 31 2019: (Start) A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. For example, the reversed binary expansions of 2, 5, and 8 are {0,1} {1,0,1} {0,0,0,1} and since there are no columns with more than one 1, the partition (8,5,2) is counted under a(15). The Heinz numbers of these partitions are given by A325097. (End) LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 EXAMPLE From Gus Wiseman, Mar 30 2019: (Start) The a(1) = 1 through a(8) = 11 partitions: (1) (2) (3) (4) (5) (6) (7) (8) (11) (21) (22) (41) (33) (43) (44) (111) (211) (221) (42) (52) (422) (1111) (2111) (222) (61) (611) (11111) (411) (421) (2222) (2211) (2221) (4211) (21111) (4111) (22211) (111111) (22111) (41111) (211111) (221111) (1111111) (2111111) (11111111) (End) MAPLE with(Bits): b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t) +`if`(i>n or And(t, i)>0, 0, add(b(n-i*j, i-1, Or(t, i)), j=1..n/i)))) end: a:= n-> b(n\$2, 0): seq(a(n), n=0..80); # Alois P. Heinz, Dec 28 2014 MATHEMATICA binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1]; stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}]; Table[Length[Select[IntegerPartitions[n], stableQ[#, Intersection[binpos[#1], binpos[#2]]!={}&]&]], {n, 0, 20}] (* Gus Wiseman, Mar 30 2019 *) b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, t] + If[i > n || BitAnd[t, i] > 0, 0, Sum[b[n - i*j, i - 1, BitOr[t, i]], {j, 1, n/i}]]]]; a[n_] := b[n, n, 0]; a /@ Range[0, 80] (* Jean-François Alcover, May 23 2021, after Alois P. Heinz *) CROSSREFS Cf. A000110, A000120, A050315, A070939, A080572, A248605, A267610. Cf. A325095, A325096, A325097, A325098, A325102, A325103, A325109. Sequence in context: A101547 A047597 A309960 * A005233 A155736 A074897 Adjacent sequences: A247932 A247933 A247934 * A247936 A247937 A247938 KEYWORD nonn AUTHOR David S. Newman, Sep 26 2014 EXTENSIONS More terms from Alois P. Heinz, Oct 15 2014 Name edited by Gus Wiseman, Mar 31 2019 STATUS approved

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Last modified September 29 19:43 EDT 2023. Contains 365776 sequences. (Running on oeis4.)