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Semiprimes whose prime indices sum to an even number.
16

%I #9 Nov 29 2020 21:56:11

%S 4,9,10,21,22,25,34,39,46,49,55,57,62,82,85,87,91,94,111,115,118,121,

%T 129,133,134,146,155,159,166,169,183,187,194,203,205,206,213,218,235,

%U 237,247,253,254,259,267,274,289,295,298,301,303,314,321,334,335,339

%N Semiprimes whose prime indices sum to an even number.

%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.

%F A338906 \/ A338907 = A001358.

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

%e 4: {1,1} 87: {2,10} 183: {2,18} 274: {1,33}

%e 9: {2,2} 91: {4,6} 187: {5,7} 289: {7,7}

%e 10: {1,3} 94: {1,15} 194: {1,25} 295: {3,17}

%e 21: {2,4} 111: {2,12} 203: {4,10} 298: {1,35}

%e 22: {1,5} 115: {3,9} 205: {3,13} 301: {4,14}

%e 25: {3,3} 118: {1,17} 206: {1,27} 303: {2,26}

%e 34: {1,7} 121: {5,5} 213: {2,20} 314: {1,37}

%e 39: {2,6} 129: {2,14} 218: {1,29} 321: {2,28}

%e 46: {1,9} 133: {4,8} 235: {3,15} 334: {1,39}

%e 49: {4,4} 134: {1,19} 237: {2,22} 335: {3,19}

%e 55: {3,5} 146: {1,21} 247: {6,8} 339: {2,30}

%e 57: {2,8} 155: {3,11} 253: {5,9} 341: {5,11}

%e 62: {1,11} 159: {2,16} 254: {1,31} 358: {1,41}

%e 82: {1,13} 166: {1,23} 259: {4,12} 361: {8,8}

%e 85: {3,7} 169: {6,6} 267: {2,24} 365: {3,21}

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

%t Select[Range[100],PrimeOmega[#]==2&&EvenQ[Total[primeMS[#]]]&]

%Y A031215 looks at primes instead of semiprimes.

%Y A098350 has this as union of even-indexed antidiagonals.

%Y A300061 looks at all numbers (not just semiprimes).

%Y A338904 has this as union of even-indexed rows.

%Y A338907 is the odd version.

%Y A338908 is the squarefree case.

%Y A001358 lists semiprimes, with odd/even terms A046315/A100484.

%Y A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.

%Y A056239 gives the sum of prime indices (Heinz weight).

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

%Y A087112 groups semiprimes by greater factor.

%Y A289182/A115392 list the positions of odd/even terms in A001358.

%Y A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.

%Y A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.

%Y A338911 lists products of pairs of primes both of even index.

%Y A339114/A339115 give the least/greatest semiprime of weight n.

%Y Cf. A000040, A001222, A024697, A037143, A112798, A300063, A319242, A320655, A332765, A338910, A339004.

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 28 2020