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A342339 Heinz numbers of the integer partitions counted by A342337, which have all adjacent parts (x, y) satisfying either x = y or x = 2y.

%I A342339 #6 Mar 12 2021 09:13:21
%S A342339 1,2,3,4,5,6,7,8,9,11,12,13,16,17,18,19,21,23,24,25,27,29,31,32,36,37,
%T A342339 41,42,43,47,48,49,53,54,59,61,63,64,65,67,71,72,73,79,81,83,84,89,96,
%U A342339 97,101,103,107,108,109,113,121,125,126,127,128,131,133,137
%N A342339 Heinz numbers of the integer partitions counted by A342337, which have all adjacent parts (x, y) satisfying either x = y or x = 2y.
%C A342339 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
%e A342339 The sequence of terms together with their prime indices begins:
%e A342339       1: {}            19: {8}             48: {1,1,1,1,2}
%e A342339       2: {1}           21: {2,4}           49: {4,4}
%e A342339       3: {2}           23: {9}             53: {16}
%e A342339       4: {1,1}         24: {1,1,1,2}       54: {1,2,2,2}
%e A342339       5: {3}           25: {3,3}           59: {17}
%e A342339       6: {1,2}         27: {2,2,2}         61: {18}
%e A342339       7: {4}           29: {10}            63: {2,2,4}
%e A342339       8: {1,1,1}       31: {11}            64: {1,1,1,1,1,1}
%e A342339       9: {2,2}         32: {1,1,1,1,1}     65: {3,6}
%e A342339      11: {5}           36: {1,1,2,2}       67: {19}
%e A342339      12: {1,1,2}       37: {12}            71: {20}
%e A342339      13: {6}           41: {13}            72: {1,1,1,2,2}
%e A342339      16: {1,1,1,1}     42: {1,2,4}         73: {21}
%e A342339      17: {7}           43: {14}            79: {22}
%e A342339      18: {1,2,2}       47: {15}            81: {2,2,2,2}
%t A342339 Select[Range[100],With[{y=PrimePi/@First/@FactorInteger[#]},And@@Table[y[[i]]==y[[i-1]]||y[[i]]==2*y[[i-1]],{i,2,Length[y]}]]&]
%Y A342339 The first condition alone gives A000961 (perfect powers).
%Y A342339 The second condition alone is counted by A154402.
%Y A342339 These partitions are counted by A342337.
%Y A342339 A018819 counts partitions into powers of 2.
%Y A342339 A000929 counts partitions with adjacent parts x >= 2y.
%Y A342339 A002843 counts compositions with adjacent parts x <= 2y.
%Y A342339 A045690 counts sets with maximum n in with adjacent elements y < 2x.
%Y A342339 A224957 counts compositions with x <= 2y and y <= 2x (strict: A342342).
%Y A342339 A274199 counts compositions with adjacent parts x < 2y.
%Y A342339 A342094 counts partitions with adjacent x <= 2y (strict: A342095).
%Y A342339 A342096 counts partitions without adjacent x >= 2y (strict: A342097).
%Y A342339 A342098 counts partitions with adjacent parts x > 2y.
%Y A342339 A342330 counts compositions with x < 2y and y < 2x (strict: A342341).
%Y A342339 A342331 counts compositions with adjacent parts x = 2y or y = 2x.
%Y A342339 A342332 counts compositions with adjacent parts x > 2y or y > 2x.
%Y A342339 A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.
%Y A342339 A342334 counts compositions with adjacent parts x >= 2y or y > 2x.
%Y A342339 A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
%Y A342339 A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
%Y A342339 A342342 counts strict compositions with adjacent parts x <= 2y and y <= 2x.
%Y A342339 Cf. A003114, A003242, A034296, A040039, A167606. A342083, A342084, A342087, A342191, A342336, A342339, A342340.
%K A342339 nonn
%O A342339 1,2
%A A342339 _Gus Wiseman_, Mar 11 2021