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A338906 Semiprimes whose prime indices sum to an even number.

%I A338906 #18 Apr 22 2025 05:08:56
%S A338906 4,9,10,21,22,25,34,39,46,49,55,57,62,82,85,87,91,94,111,115,118,121,
%T A338906 129,133,134,146,155,159,166,169,183,187,194,203,205,206,213,218,235,
%U A338906 237,247,253,254,259,267,274,289,295,298,301,303,314,321,334,335,339
%N A338906 Semiprimes whose prime indices sum to an even number.
%C A338906 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 A338906 \/ A338907 = A001358.
%e A338906 The sequence of terms together with their prime indices begins:
%e A338906       4: {1,1}      87: {2,10}    183: {2,18}    274: {1,33}
%e A338906       9: {2,2}      91: {4,6}     187: {5,7}     289: {7,7}
%e A338906      10: {1,3}      94: {1,15}    194: {1,25}    295: {3,17}
%e A338906      21: {2,4}     111: {2,12}    203: {4,10}    298: {1,35}
%e A338906      22: {1,5}     115: {3,9}     205: {3,13}    301: {4,14}
%e A338906      25: {3,3}     118: {1,17}    206: {1,27}    303: {2,26}
%e A338906      34: {1,7}     121: {5,5}     213: {2,20}    314: {1,37}
%e A338906      39: {2,6}     129: {2,14}    218: {1,29}    321: {2,28}
%e A338906      46: {1,9}     133: {4,8}     235: {3,15}    334: {1,39}
%e A338906      49: {4,4}     134: {1,19}    237: {2,22}    335: {3,19}
%e A338906      55: {3,5}     146: {1,21}    247: {6,8}     339: {2,30}
%e A338906      57: {2,8}     155: {3,11}    253: {5,9}     341: {5,11}
%e A338906      62: {1,11}    159: {2,16}    254: {1,31}    358: {1,41}
%e A338906      82: {1,13}    166: {1,23}    259: {4,12}    361: {8,8}
%e A338906      85: {3,7}     169: {6,6}     267: {2,24}    365: {3,21}
%t A338906 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A338906 Select[Range[100],PrimeOmega[#]==2&&EvenQ[Total[primeMS[#]]]&]
%o A338906 (Python)
%o A338906 from math import isqrt
%o A338906 from sympy import primepi, primerange
%o A338906 def A338906(n):
%o A338906     def bisection(f,kmin=0,kmax=1):
%o A338906         while f(kmax) > kmax: kmax <<= 1
%o A338906         kmin = kmax >> 1
%o A338906         while kmax-kmin > 1:
%o A338906             kmid = kmax+kmin>>1
%o A338906             if f(kmid) <= kmid:
%o A338906                 kmax = kmid
%o A338906             else:
%o A338906                 kmin = kmid
%o A338906         return kmax
%o A338906     def f(x): return n+x-sum((primepi(x//p)-a>>1) for a,p in enumerate(primerange(isqrt(x)+1),-1))
%o A338906     return bisection(f,n,n) # _Chai Wah Wu_, Apr 03 2025
%Y A338906 A031215 looks at primes instead of semiprimes.
%Y A338906 A098350 has this as union of even-indexed antidiagonals.
%Y A338906 A300061 looks at all numbers (not just semiprimes).
%Y A338906 A338904 has this as union of even-indexed rows.
%Y A338906 A338907 is the odd version.
%Y A338906 A338908 is the squarefree case.
%Y A338906 A001358 lists semiprimes, with odd/even terms A046315/A100484.
%Y A338906 A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
%Y A338906 A056239 gives the sum of prime indices (Heinz weight).
%Y A338906 A084126 and A084127 give the prime factors of semiprimes.
%Y A338906 A087112 groups semiprimes by greater factor.
%Y A338906 A289182/A115392 list the positions of odd/even terms in A001358.
%Y A338906 A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
%Y A338906 A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
%Y A338906 A338911 lists products of pairs of primes both of even index.
%Y A338906 A339114/A339115 give the least/greatest semiprime of weight n.
%Y A338906 Cf. A000040, A001222, A024697, A037143, A112798, A300063, A319242, A320655, A332765, A338910, A339004.
%K A338906 nonn
%O A338906 1,1
%A A338906 _Gus Wiseman_, Nov 28 2020