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A338898 Concatenated sequence of prime indices of semiprimes (A001358).

%I A338898 #8 Nov 20 2020 17:19:09
%S A338898 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 A338898 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 A338898 15,3,8,1,16,2,12,3,9,1,17,4,7,5,5,1,18,2
%N A338898 Concatenated sequence of prime indices of semiprimes (A001358).
%C A338898 This is a triangle with two columns and weakly increasing rows, namely {A338912(n), A338913(n)}.
%C A338898 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 A338898 The sequence of semiprimes together with their prime indices begins:
%e A338898       4: {1,1}     46: {1,9}      91: {4,6}     141: {2,15}
%e A338898       6: {1,2}     49: {4,4}      93: {2,11}    142: {1,20}
%e A338898       9: {2,2}     51: {2,7}      94: {1,15}    143: {5,6}
%e A338898      10: {1,3}     55: {3,5}      95: {3,8}     145: {3,10}
%e A338898      14: {1,4}     57: {2,8}     106: {1,16}    146: {1,21}
%e A338898      15: {2,3}     58: {1,10}    111: {2,12}    155: {3,11}
%e A338898      21: {2,4}     62: {1,11}    115: {3,9}     158: {1,22}
%e A338898      22: {1,5}     65: {3,6}     118: {1,17}    159: {2,16}
%e A338898      25: {3,3}     69: {2,9}     119: {4,7}     161: {4,9}
%e A338898      26: {1,6}     74: {1,12}    121: {5,5}     166: {1,23}
%e A338898      33: {2,5}     77: {4,5}     122: {1,18}    169: {6,6}
%e A338898      34: {1,7}     82: {1,13}    123: {2,13}    177: {2,17}
%e A338898      35: {3,4}     85: {3,7}     129: {2,14}    178: {1,24}
%e A338898      38: {1,8}     86: {1,14}    133: {4,8}     183: {2,18}
%e A338898      39: {2,6}     87: {2,10}    134: {1,19}    185: {3,12}
%t A338898 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A338898 Join@@primeMS/@Select[Range[100],PrimeOmega[#]==2&]
%Y A338898 A112798 restricted to rows of length 2 gives this triangle.
%Y A338898 A115392 is the row number for the first appearance of each positive integer.
%Y A338898 A176506 gives row differences.
%Y A338898 A338899 is the squarefree version.
%Y A338898 A338912 is column 1.
%Y A338898 A338913 is column 2.
%Y A338898 A001221 counts a number's distinct prime indices.
%Y A338898 A001222 counts a number's prime indices.
%Y A338898 A001358 lists semiprimes.
%Y A338898 A004526 counts 2-part partitions.
%Y A338898 A006881 lists squarefree semiprimes.
%Y A338898 A037143 lists primes and semiprimes.
%Y A338898 A046315 and A100484 list odd and even semiprimes.
%Y A338898 A046388 and A100484 list odd and even squarefree semiprimes.
%Y A338898 A065516 gives first differences of semiprimes.
%Y A338898 A084126 and A084127 give the prime factors of semiprimes.
%Y A338898 A270650 and A270652 give the prime indices of squarefree semiprimes.
%Y A338898 A320655 counts factorizations into semiprimes.
%Y A338898 Cf. A056239, A101048, A320892, A320912, A338900, A338901, A338904, A338906, A338907, A338910, A338911.
%K A338898 nonn,tabf
%O A338898 1,4
%A A338898 _Gus Wiseman_, Nov 15 2020