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A338898
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Concatenated sequence of prime indices of semiprimes (A001358).
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44
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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, 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, 15, 3, 8, 1, 16, 2, 12, 3, 9, 1, 17, 4, 7, 5, 5, 1, 18, 2
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OFFSET
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1,4
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COMMENTS
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This is a triangle with two columns and weakly increasing rows, namely {A338912(n), A338913(n)}.
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.
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LINKS
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EXAMPLE
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The sequence of semiprimes together with their prime indices begins:
4: {1,1} 46: {1,9} 91: {4,6} 141: {2,15}
6: {1,2} 49: {4,4} 93: {2,11} 142: {1,20}
9: {2,2} 51: {2,7} 94: {1,15} 143: {5,6}
10: {1,3} 55: {3,5} 95: {3,8} 145: {3,10}
14: {1,4} 57: {2,8} 106: {1,16} 146: {1,21}
15: {2,3} 58: {1,10} 111: {2,12} 155: {3,11}
21: {2,4} 62: {1,11} 115: {3,9} 158: {1,22}
22: {1,5} 65: {3,6} 118: {1,17} 159: {2,16}
25: {3,3} 69: {2,9} 119: {4,7} 161: {4,9}
26: {1,6} 74: {1,12} 121: {5,5} 166: {1,23}
33: {2,5} 77: {4,5} 122: {1,18} 169: {6,6}
34: {1,7} 82: {1,13} 123: {2,13} 177: {2,17}
35: {3,4} 85: {3,7} 129: {2,14} 178: {1,24}
38: {1,8} 86: {1,14} 133: {4,8} 183: {2,18}
39: {2,6} 87: {2,10} 134: {1,19} 185: {3,12}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Join@@primeMS/@Select[Range[100], PrimeOmega[#]==2&]
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CROSSREFS
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A112798 restricted to rows of length 2 gives this triangle.
A115392 is the row number for the first appearance of each positive integer.
A001221 counts a number's distinct prime indices.
A001222 counts a number's prime indices.
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes.
A065516 gives first differences of semiprimes.
A320655 counts factorizations into semiprimes.
Cf. A056239, A101048, A320892, A320912, A338900, A338901, A338904, A338906, A338907, A338910, A338911.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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