cp's OEIS Frontend

A367097 Least positive integer whose multiset of prime indices has exactly n distinct semi-sums.

%I A367097 #28 Nov 20 2023 11:53:55
%S A367097 1,4,12,30,60,210,330,660,2730,3570,6270,12540,53130,79170,110670,
%T A367097 221340,514140,1799490,2284590,4196010,6750870,13501740,37532220,
%U A367097 97350330,131362770,189620970,379241940,735844830,1471689660
%N A367097 Least positive integer whose multiset of prime indices has exactly n distinct semi-sums.
%C A367097 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 A367097 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 A367097 From _David A. Corneth_, Nov 15 2023: (Start)
%C A367097 Terms are cubefree.
%C A367097 bigomega(a(n)) = A001222(a(n)) >= A002024(n) + 1 = floor(sqrt(2n) + 1/2) + 1 for n > 0. (End)
%F A367097 2 | a(n) for n > 0. - _David A. Corneth_, Nov 13 2023
%e A367097 The prime indices of 60 are {1,1,2,3}, with four semi-sums {2,3,4,5}, and 60 is the first number whose prime indices have four semi-sums, so a(4) = 60.
%e A367097 The terms together with their prime indices begin:
%e A367097        1: {}
%e A367097        4: {1,1}
%e A367097       12: {1,1,2}
%e A367097       30: {1,2,3}
%e A367097       60: {1,1,2,3}
%e A367097      210: {1,2,3,4}
%e A367097      330: {1,2,3,5}
%e A367097      660: {1,1,2,3,5}
%e A367097     2730: {1,2,3,4,6}
%e A367097     3570: {1,2,3,4,7}
%e A367097     6270: {1,2,3,5,8}
%e A367097    12540: {1,1,2,3,5,8}
%e A367097    53130: {1,2,3,4,5,9}
%e A367097    79170: {1,2,3,4,6,10}
%e A367097   110670: {1,2,3,4,7,11}
%e A367097   221340: {1,1,2,3,4,7,11}
%e A367097   514140: {1,1,2,3,5,8,13}
%t A367097 nn=1000;
%t A367097 w=Table[Length[Union[Total/@Subsets[prix[n],{2}]]],{n,nn}];
%t A367097 spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
%t A367097 v=Table[Position[w,k][[1,1]],{k,0,spnm[w]}]
%o A367097 (Python)
%o A367097 from itertools import count
%o A367097 from sympy import factorint, primepi
%o A367097 from sympy.utilities.iterables import multiset_combinations
%o A367097 def A367097(n): return next(k for k in count(1) if len({sum(s) for s in multiset_combinations({primepi(i):j for i,j in factorint(k).items()},2)}) == n) # _Chai Wah Wu_, Nov 13 2023
%Y A367097 The non-binary version is A259941, firsts of A299701.
%Y A367097 These are the positions of first appearances in A366739.
%Y A367097 A001222 counts prime factors (or prime indices), distinct A001221.
%Y A367097 A001358 lists semiprimes, squarefree A006881, complement A100959.
%Y A367097 A056239 adds up prime indices, row sums of A112798.
%Y A367097 A299702 ranks knapsack partitions, counted by A108917.
%Y A367097 A366738 counts semi-sums of partitions, strict A366741.
%Y A367097 Semiprime divisors are listed by A367096 and have:
%Y A367097 - square count: A056170
%Y A367097 - sum: A076290
%Y A367097 - squarefree count: A079275
%Y A367097 - count: A086971
%Y A367097 - firsts: A220264
%Y A367097 Cf. A000720, A001248, A002024, A004709, A304793, A365920, A366740, A366753, A367093, A367095.
%K A367097 nonn,more
%O A367097 0,2
%A A367097 _Gus Wiseman_, Nov 09 2023
%E A367097 a(17)-a(22) from _Chai Wah Wu_, Nov 13 2023
%E A367097 a(23)-a(28) from _David A. Corneth_, Nov 13 2023