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