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 A334438 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically. 28
 1, 2, 3, 4, 5, 6, 8, 7, 10, 9, 12, 16, 11, 14, 15, 20, 18, 24, 32, 13, 22, 21, 25, 28, 30, 27, 40, 36, 48, 64, 17, 26, 33, 35, 44, 42, 50, 45, 56, 60, 54, 80, 72, 96, 128, 19, 34, 39, 55, 49, 52, 66, 70, 63, 75, 88, 84, 100, 90, 81, 112, 120, 108, 160, 144, 192, 256 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS First differs from A185974 shifted left once at a(76) = 99, A185974(75) = 98. A permutation of the positive integers. This is the Abramowitz-Stegun ordering of integer partitions (A334433) except that the finer order is reverse-lexicographic instead of lexicographic. The version for reversed partitions is A334435. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions. As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0. LINKS Wikiversity, Lexicographic and colexicographic order FORMULA A001221(a(n)) = A103921(n). A001222(a(n)) = A036043(n). EXAMPLE The sequence of terms together with their prime indices begins: 1: {} 32: {1,1,1,1,1} 50: {1,3,3} 2: {1} 13: {6} 45: {2,2,3} 3: {2} 22: {1,5} 56: {1,1,1,4} 4: {1,1} 21: {2,4} 60: {1,1,2,3} 5: {3} 25: {3,3} 54: {1,2,2,2} 6: {1,2} 28: {1,1,4} 80: {1,1,1,1,3} 8: {1,1,1} 30: {1,2,3} 72: {1,1,1,2,2} 7: {4} 27: {2,2,2} 96: {1,1,1,1,1,2} 10: {1,3} 40: {1,1,1,3} 128: {1,1,1,1,1,1,1} 9: {2,2} 36: {1,1,2,2} 19: {8} 12: {1,1,2} 48: {1,1,1,1,2} 34: {1,7} 16: {1,1,1,1} 64: {1,1,1,1,1,1} 39: {2,6} 11: {5} 17: {7} 55: {3,5} 14: {1,4} 26: {1,6} 49: {4,4} 15: {2,3} 33: {2,5} 52: {1,1,6} 20: {1,1,3} 35: {3,4} 66: {1,2,5} 18: {1,2,2} 44: {1,1,5} 70: {1,3,4} 24: {1,1,1,2} 42: {1,2,4} 63: {2,2,4} Triangle begins: 1 2 3 4 5 6 8 7 10 9 12 16 11 14 15 20 18 24 32 13 22 21 25 28 30 27 40 36 48 64 17 26 33 35 44 42 50 45 56 60 54 80 72 96 128 This corresponds to the following tetrangle: 0 (1) (2)(11) (3)(21)(111) (4)(31)(22)(211)(1111) (5)(41)(32)(311)(221)(2111)(11111) MATHEMATICA revlensort[f_, c_]:=If[Length[f]!=Length[c], Length[f]

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Last modified December 7 21:25 EST 2022. Contains 358669 sequences. (Running on oeis4.)