cp's OEIS Frontend

A367093 Least positive integer with n more semiprime divisors than semi-sums of prime indices.

%I A367093 #16 Nov 18 2023 13:12:45
%S A367093 1,90,630,2310,6930,34650,30030,90090,450450,570570,510510,1531530,
%T A367093 7657650,14804790,11741730,9699690,29099070,145495350
%N A367093 Least positive integer with n more semiprime divisors than semi-sums of prime indices.
%C A367093 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 A367093 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.
%C A367093 Are all primorials after 210 included?
%F A367093 a(n) is the least positive integer such that A086971(a(n)) - A366739(a(n)) = n.
%e A367093 The terms together with their prime indices begin:
%e A367093        1: {}
%e A367093       90: {1,2,2,3}
%e A367093      630: {1,2,2,3,4}
%e A367093     2310: {1,2,3,4,5}
%e A367093     6930: {1,2,2,3,4,5}
%e A367093    34650: {1,2,2,3,3,4,5}
%e A367093    30030: {1,2,3,4,5,6}
%e A367093    90090: {1,2,2,3,4,5,6}
%e A367093   450450: {1,2,2,3,3,4,5,6}
%e A367093   570570: {1,2,3,4,5,6,8}
%e A367093   510510: {1,2,3,4,5,6,7}
%t A367093 nn=10000;
%t A367093 w=Table[Length[Union[Subsets[prix[n],{2}]]]-Length[Union[Total/@Subsets[prix[n],{2}]]],{n,nn}];
%t A367093 spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
%t A367093 Table[Position[w,k][[1,1]],{k,0,spnm[w]}]
%o A367093 (Python)
%o A367093 from itertools import count
%o A367093 from sympy import factorint, primepi
%o A367093 from sympy.utilities.iterables import multiset_combinations
%o A367093 def A367093(n):
%o A367093     for k in count(1):
%o A367093         c, a = 0, set()
%o A367093         for s in (sum(p) for p in multiset_combinations({primepi(i):j for i,j in factorint(k).items()},2)):
%o A367093             if s not in a:
%o A367093                 a.add(s)
%o A367093             else:
%o A367093                 c += 1
%o A367093             if c > n:
%o A367093                 break
%o A367093         if c == n:
%o A367093             return k # _Chai Wah Wu_, Nov 13 2023
%Y A367093 The first part (semiprime divisors) is A086971, firsts A220264.
%Y A367093 The second part (semi-sums of prime indices) is A366739, firsts A367097.
%Y A367093 All sums of pairs of prime indices are counted by A367095.
%Y A367093 The non-binary version is A367105.
%Y A367093 A001222 counts prime factors (or prime indices), distinct A001221.
%Y A367093 A001358 lists semiprimes, squarefree A006881, conjugate A065119.
%Y A367093 A056239 adds up prime indices, row sums of A112798.
%Y A367093 A299701 counts subset-sums of prime indices, positive A304793.
%Y A367093 Semiprime divisors are listed by A367096 and have:
%Y A367093 - square count: A056170
%Y A367093 - sum: A076290
%Y A367093 - squarefree count: A079275
%Y A367093 - count: A086971
%Y A367093 - firsts: A220264
%Y A367093 Cf. A000720, A001248, A008967, A117855, A304792, A365541, A365920, A366738, A366740, A366753.
%K A367093 nonn,more
%O A367093 0,2
%A A367093 _Gus Wiseman_, Nov 05 2023
%E A367093 a(12)-a(16) from _Chai Wah Wu_, Nov 13 2023
%E A367093 a(17) from _Chai Wah Wu_, Nov 18 2023