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A338899 Concatenated sequence of prime indices of squarefree semiprimes (A006881). 45

%I

%S 1,2,1,3,1,4,2,3,2,4,1,5,1,6,2,5,1,7,3,4,1,8,2,6,1,9,2,7,3,5,2,8,1,10,

%T 1,11,3,6,2,9,1,12,4,5,1,13,3,7,1,14,2,10,4,6,2,11,1,15,3,8,1,16,2,12,

%U 3,9,1,17,4,7,1,18,2,13,2,14,4,8,1,19,2,15

%N Concatenated sequence of prime indices of squarefree semiprimes (A006881).

%C This is a triangle with two columns and strictly increasing rows, namely {A270650(n), A270652(n)}.

%C A squarefree semiprime is a product of any two distinct 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 terms together with their prime indices begins:

%e 6: {1,2} 57: {2,8} 106: {1,16} 155: {3,11}

%e 10: {1,3} 58: {1,10} 111: {2,12} 158: {1,22}

%e 14: {1,4} 62: {1,11} 115: {3,9} 159: {2,16}

%e 15: {2,3} 65: {3,6} 118: {1,17} 161: {4,9}

%e 21: {2,4} 69: {2,9} 119: {4,7} 166: {1,23}

%e 22: {1,5} 74: {1,12} 122: {1,18} 177: {2,17}

%e 26: {1,6} 77: {4,5} 123: {2,13} 178: {1,24}

%e 33: {2,5} 82: {1,13} 129: {2,14} 183: {2,18}

%e 34: {1,7} 85: {3,7} 133: {4,8} 185: {3,12}

%e 35: {3,4} 86: {1,14} 134: {1,19} 187: {5,7}

%e 38: {1,8} 87: {2,10} 141: {2,15} 194: {1,25}

%e 39: {2,6} 91: {4,6} 142: {1,20} 201: {2,19}

%e 46: {1,9} 93: {2,11} 143: {5,6} 202: {1,26}

%e 51: {2,7} 94: {1,15} 145: {3,10} 203: {4,10}

%e 55: {3,5} 95: {3,8} 146: {1,21} 205: {3,13}

%t Join@@Cases[Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&],k_:>PrimePi/@First/@FactorInteger[k]]

%Y A270650 is the first column.

%Y A270652 is the second column.

%Y A320656 counts multiset partitions using these rows, or factorizations into squarefree semiprimes.

%Y A338898 is the version including squares, with columns A338912 and A338913.

%Y A338900 gives row differences.

%Y A338901 gives the row numbers for first appearances.

%Y A001221 and A001222 count distinct/all prime indices.

%Y A001358 lists semiprimes.

%Y A004526 counts 2-part partitions, with strict case shifted right once.

%Y A005117 lists squarefree numbers.

%Y A006881 lists squarefree semiprimes.

%Y A046315 and A100484 list odd and even semiprimes.

%Y A046388 lists odd squarefree semiprimes.

%Y A166237 gives first differences of squarefree semiprimes.

%Y Cf. A030229, A056239, A065516, A112798, A115392, A167171, A176506, A320893, A338904, A338906, A338907, A338910, A338911.

%K nonn,hear,tabf

%O 1,2

%A _Gus Wiseman_, Nov 16 2020

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Last modified August 3 06:43 EDT 2021. Contains 346435 sequences. (Running on oeis4.)