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%I #5 Nov 19 2019 16:36:13
%S 1,2,3,1,2,4,1,3,5,6,1,4,2,3,7,8,2,4,1,5,9,1,6,10,1,2,3,11,2,5,1,7,3,
%T 4,12,1,8,2,6,13,1,2,4,14,1,9,15,2,7,16,3,5,2,8,1,10,17,18,1,11,3,6,1,
%U 2,5,19,2,9,1,3,4,20,21,1,12,4,5,1,2,6
%N Irregular triangle read by rows where row n lists the prime indices of the n-th squarefree number.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%e Triangle begins:
%e 1: {} 33: {2,5} 66: {1,2,5} 97: {25}
%e 2: {1} 34: {1,7} 67: {19} 101: {26}
%e 3: {2} 35: {3,4} 69: {2,9} 102: {1,2,7}
%e 5: {3} 37: {12} 70: {1,3,4} 103: {27}
%e 6: {1,2} 38: {1,8} 71: {20} 105: {2,3,4}
%e 7: {4} 39: {2,6} 73: {21} 106: {1,16}
%e 10: {1,3} 41: {13} 74: {1,12} 107: {28}
%e 11: {5} 42: {1,2,4} 77: {4,5} 109: {29}
%e 13: {6} 43: {14} 78: {1,2,6} 110: {1,3,5}
%e 14: {1,4} 46: {1,9} 79: {22} 111: {2,12}
%e 15: {2,3} 47: {15} 82: {1,13} 113: {30}
%e 17: {7} 51: {2,7} 83: {23} 114: {1,2,8}
%e 19: {8} 53: {16} 85: {3,7} 115: {3,9}
%e 21: {2,4} 55: {3,5} 86: {1,14} 118: {1,17}
%e 22: {1,5} 57: {2,8} 87: {2,10} 119: {4,7}
%e 23: {9} 58: {1,10} 89: {24} 122: {1,18}
%e 26: {1,6} 59: {17} 91: {4,6} 123: {2,13}
%e 29: {10} 61: {18} 93: {2,11} 127: {31}
%e 30: {1,2,3} 62: {1,11} 94: {1,15} 129: {2,14}
%e 31: {11} 65: {3,6} 95: {3,8} 130: {1,3,6}
%t Table[PrimePi/@First/@If[k==1,{},FactorInteger[k]],{k,Select[Range[30],SquareFreeQ]}]
%Y Row sums are A319246.
%Y Row lengths are A072047.
%Y Same as A319247 with rows reversed.
%Y Composition of A000720 and A265668.
%Y Looking at all numbers instead of just squarefree numbers gives A112798.
%Y Cf. A000009, A005117, A048672, A056239, A299755, A302590.
%K nonn,tabf
%O 1,2
%A _Gus Wiseman_, Nov 18 2019