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Concatenated sequence of prime indices of semiprimes (A001358).
44

%I #8 Nov 20 2020 17:19:09

%S 1,1,1,2,2,2,1,3,1,4,2,3,2,4,1,5,3,3,1,6,2,5,1,7,3,4,1,8,2,6,1,9,4,4,

%T 2,7,3,5,2,8,1,10,1,11,3,6,2,9,1,12,4,5,1,13,3,7,1,14,2,10,4,6,2,11,1,

%U 15,3,8,1,16,2,12,3,9,1,17,4,7,5,5,1,18,2

%N Concatenated sequence of prime indices of semiprimes (A001358).

%C This is a triangle with two columns and weakly increasing rows, namely {A338912(n), A338913(n)}.

%C A semiprime is a product of any two prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

%e The sequence of semiprimes together with their prime indices begins:

%e 4: {1,1} 46: {1,9} 91: {4,6} 141: {2,15}

%e 6: {1,2} 49: {4,4} 93: {2,11} 142: {1,20}

%e 9: {2,2} 51: {2,7} 94: {1,15} 143: {5,6}

%e 10: {1,3} 55: {3,5} 95: {3,8} 145: {3,10}

%e 14: {1,4} 57: {2,8} 106: {1,16} 146: {1,21}

%e 15: {2,3} 58: {1,10} 111: {2,12} 155: {3,11}

%e 21: {2,4} 62: {1,11} 115: {3,9} 158: {1,22}

%e 22: {1,5} 65: {3,6} 118: {1,17} 159: {2,16}

%e 25: {3,3} 69: {2,9} 119: {4,7} 161: {4,9}

%e 26: {1,6} 74: {1,12} 121: {5,5} 166: {1,23}

%e 33: {2,5} 77: {4,5} 122: {1,18} 169: {6,6}

%e 34: {1,7} 82: {1,13} 123: {2,13} 177: {2,17}

%e 35: {3,4} 85: {3,7} 129: {2,14} 178: {1,24}

%e 38: {1,8} 86: {1,14} 133: {4,8} 183: {2,18}

%e 39: {2,6} 87: {2,10} 134: {1,19} 185: {3,12}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Join@@primeMS/@Select[Range[100],PrimeOmega[#]==2&]

%Y A112798 restricted to rows of length 2 gives this triangle.

%Y A115392 is the row number for the first appearance of each positive integer.

%Y A176506 gives row differences.

%Y A338899 is the squarefree version.

%Y A338912 is column 1.

%Y A338913 is column 2.

%Y A001221 counts a number's distinct prime indices.

%Y A001222 counts a number's prime indices.

%Y A001358 lists semiprimes.

%Y A004526 counts 2-part partitions.

%Y A006881 lists squarefree semiprimes.

%Y A037143 lists primes and semiprimes.

%Y A046315 and A100484 list odd and even semiprimes.

%Y A046388 and A100484 list odd and even squarefree semiprimes.

%Y A065516 gives first differences of semiprimes.

%Y A084126 and A084127 give the prime factors of semiprimes.

%Y A270650 and A270652 give the prime indices of squarefree semiprimes.

%Y A320655 counts factorizations into semiprimes.

%Y Cf. A056239, A101048, A320892, A320912, A338900, A338901, A338904, A338906, A338907, A338910, A338911.

%K nonn,tabf

%O 1,4

%A _Gus Wiseman_, Nov 15 2020