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

%I A338899 #6 Nov 20 2020 17:19:18
%S A338899 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 A338899 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 A338899 3,9,1,17,4,7,1,18,2,13,2,14,4,8,1,19,2,15
%N A338899 Concatenated sequence of prime indices of squarefree semiprimes (A006881).
%C A338899 This is a triangle with two columns and strictly increasing rows, namely {A270650(n), A270652(n)}.
%C A338899 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 A338899 The sequence of terms together with their prime indices begins:
%e A338899       6: {1,2}     57: {2,8}     106: {1,16}    155: {3,11}
%e A338899      10: {1,3}     58: {1,10}    111: {2,12}    158: {1,22}
%e A338899      14: {1,4}     62: {1,11}    115: {3,9}     159: {2,16}
%e A338899      15: {2,3}     65: {3,6}     118: {1,17}    161: {4,9}
%e A338899      21: {2,4}     69: {2,9}     119: {4,7}     166: {1,23}
%e A338899      22: {1,5}     74: {1,12}    122: {1,18}    177: {2,17}
%e A338899      26: {1,6}     77: {4,5}     123: {2,13}    178: {1,24}
%e A338899      33: {2,5}     82: {1,13}    129: {2,14}    183: {2,18}
%e A338899      34: {1,7}     85: {3,7}     133: {4,8}     185: {3,12}
%e A338899      35: {3,4}     86: {1,14}    134: {1,19}    187: {5,7}
%e A338899      38: {1,8}     87: {2,10}    141: {2,15}    194: {1,25}
%e A338899      39: {2,6}     91: {4,6}     142: {1,20}    201: {2,19}
%e A338899      46: {1,9}     93: {2,11}    143: {5,6}     202: {1,26}
%e A338899      51: {2,7}     94: {1,15}    145: {3,10}    203: {4,10}
%e A338899      55: {3,5}     95: {3,8}     146: {1,21}    205: {3,13}
%t A338899 Join@@Cases[Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&],k_:>PrimePi/@First/@FactorInteger[k]]
%Y A338899 A270650 is the first column.
%Y A338899 A270652 is the second column.
%Y A338899 A320656 counts multiset partitions using these rows, or factorizations into squarefree semiprimes.
%Y A338899 A338898 is the version including squares, with columns A338912 and A338913.
%Y A338899 A338900 gives row differences.
%Y A338899 A338901 gives the row numbers for first appearances.
%Y A338899 A001221 and A001222 count distinct/all prime indices.
%Y A338899 A001358 lists semiprimes.
%Y A338899 A004526 counts 2-part partitions, with strict case shifted right once.
%Y A338899 A005117 lists squarefree numbers.
%Y A338899 A006881 lists squarefree semiprimes.
%Y A338899 A046315 and A100484 list odd and even semiprimes.
%Y A338899 A046388 lists odd squarefree semiprimes.
%Y A338899 A166237 gives first differences of squarefree semiprimes.
%Y A338899 Cf. A030229, A056239, A065516, A112798, A115392, A167171, A176506, A320893, A338904, A338906, A338907, A338910, A338911.
%K A338899 nonn,hear,tabf
%O A338899 1,2
%A A338899 _Gus Wiseman_, Nov 16 2020