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A366739 Number of distinct semi-sums of the multiset of prime indices of n. Number of distinct sums of prime indices of semiprime divisors of n (counted by A086971).

%I A366739 #20 Jan 20 2025 10:22:00
%S A366739 0,0,0,1,0,1,0,1,1,1,0,2,0,1,1,1,0,2,0,2,1,1,0,2,1,1,1,2,0,3,0,1,1,1,
%T A366739 1,3,0,1,1,2,0,3,0,2,2,1,0,2,1,2,1,2,0,2,1,2,1,1,0,4,0,1,2,1,1,3,0,2,
%U A366739 1,3,0,3,0,1,2,2,1,3,0,2,1,1,0,4,1,1,1,2,0,3,1,2,1,1,1,2,0,2,2,3,0,3,0,2,3
%N A366739 Number of distinct semi-sums of the multiset of prime indices of n. Number of distinct sums of prime indices of semiprime divisors of n (counted by A086971).
%C A366739 First differs from A086971 at a(90) = 3, A086971(90) = 4.
%C A366739 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A366739 We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
%H A366739 Antti Karttunen, <a href="/A366739/b366739.txt">Table of n, a(n) for n = 1..65537</a>
%H A366739 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%F A366739 a(n) <= A086971(n). - _Antti Karttunen_, Jan 20 2025
%e A366739 The prime indices of 90 are {1,2,2,3}, with semi-sums
%e A366739   3 = 1+2
%e A366739   4 = 1+3 (or 2+2)
%e A366739   5 = 2+3
%e A366739 so a(90) = 3.
%e A366739 Alternatively, the semiprime divisors of 90 are (6,9,10,15), with prime indices ({1,2},{2,2},{1,3},{2,3}) with sums (3,4,4,5) so a(90) = 3.
%t A366739 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A366739 Table[Length[Union[Total/@Subsets[prix[n],{2}]]],{n,100}]
%o A366739 (PARI) A366739(n) = #Set(apply(d->((f)->sum(i=1,#f~,f[i,2]*primepi(f[i,1])))(factor(d)), select(d->2==bigomega(d), divisors(n)))); \\ _Antti Karttunen_, Jan 20 2025
%Y A366739 The non-binary version is A299701.
%Y A366739 Summing over partitions gives A366738, strict A366741.
%Y A366739 For all sums of pairs of elements we have A367095.
%Y A366739 Positions of first appearances are A367097.
%Y A366739 A001222 counts prime factors (or prime indices), distinct A001221.
%Y A366739 A001358 lists semiprimes, squarefree A006881, conjugate A065119.
%Y A366739 A056239 adds up prime indices, row sums of A112798.
%Y A366739 A299702 ranks knapsack partitions, counted by A108917.
%Y A366739 Semiprime divisors are listed by A367096 and have:
%Y A366739 - square count: A056170
%Y A366739 - sum: A076290
%Y A366739 - squarefree count: A079275
%Y A366739 - count: A086971
%Y A366739 - firsts: A220264
%Y A366739 Cf. A000005, A000720, A001248, A008967, A117855, A304792, A304793, A365541, A365920, A366740, A366753, A367093.
%K A366739 nonn
%O A366739 1,12
%A A366739 _Gus Wiseman_, Nov 04 2023
%E A366739 Data section extended to a(105) by _Antti Karttunen_, Jan 20 2025