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

1, 4, 12, 30, 60, 210, 330, 660, 2730, 3570, 6270, 12540, 53130, 79170, 110670, 221340, 514140, 1799490, 2284590, 4196010, 6750870, 13501740, 37532220, 97350330, 131362770, 189620970, 379241940, 735844830, 1471689660

Offset: 0

Author

Gus Wiseman, Nov 09 2023

Keywords

• nonn
• more

Comments

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.

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.

From David A. Corneth, Nov 15 2023: (Start)

Terms are cubefree.

bigomega(a(n)) = A001222(a(n)) >= A002024(n) + 1 = floor(sqrt(2n) + 1/2) + 1 for n > 0. (End)

Examples

			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.
The terms together with their prime indices begin:
       1: {}
       4: {1,1}
      12: {1,1,2}
      30: {1,2,3}
      60: {1,1,2,3}
     210: {1,2,3,4}
     330: {1,2,3,5}
     660: {1,1,2,3,5}
    2730: {1,2,3,4,6}
    3570: {1,2,3,4,7}
    6270: {1,2,3,5,8}
   12540: {1,1,2,3,5,8}
   53130: {1,2,3,4,5,9}
   79170: {1,2,3,4,6,10}
  110670: {1,2,3,4,7,11}
  221340: {1,1,2,3,4,7,11}
  514140: {1,1,2,3,5,8,13}
		

Crossrefs

The non-binary version is A259941, firsts of A299701.

These are the positions of first appearances in A366739.

A001222 counts prime factors (or prime indices), distinct A001221.

A001358 lists semiprimes, squarefree A006881, complement A100959.

A056239 adds up prime indices, row sums of A112798.

A299702 ranks knapsack partitions, counted by A108917.

A366738 counts semi-sums of partitions, strict A366741.

Semiprime divisors are listed by A367096 and have:

- square count: A056170

- sum: A076290

- squarefree count: A079275

- count: A086971

- firsts: A220264

Cf. A000720, A001248, A002024, A004709, A304793, A365920, A366740, A366753, A367093, A367095.

Programs

Mathematica

nn=1000;
w=Table[Length[Union[Total/@Subsets[prix[n],{2}]]],{n,nn}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
v=Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

Python

from itertools import count
from sympy import factorint, primepi
from sympy.utilities.iterables import multiset_combinations
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

Formula

2 | a(n) for n > 0. - David A. Corneth, Nov 13 2023

Extensions

a(17)-a(22) from Chai Wah Wu, Nov 13 2023

a(23)-a(28) from David A. Corneth, Nov 13 2023