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

1, 90, 630, 2310, 6930, 34650, 30030, 90090, 450450, 570570, 510510, 1531530, 7657650, 14804790, 11741730, 9699690, 29099070, 145495350

Offset: 0

Author

Gus Wiseman, Nov 05 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.

Are all primorials after 210 included?

Examples

			The terms together with their prime indices begin:
       1: {}
      90: {1,2,2,3}
     630: {1,2,2,3,4}
    2310: {1,2,3,4,5}
    6930: {1,2,2,3,4,5}
   34650: {1,2,2,3,3,4,5}
   30030: {1,2,3,4,5,6}
   90090: {1,2,2,3,4,5,6}
  450450: {1,2,2,3,3,4,5,6}
  570570: {1,2,3,4,5,6,8}
  510510: {1,2,3,4,5,6,7}
		

Crossrefs

The first part (semiprime divisors) is A086971, firsts A220264.

The second part (semi-sums of prime indices) is A366739, firsts A367097.

All sums of pairs of prime indices are counted by A367095.

The non-binary version is A367105.

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

A001358 lists semiprimes, squarefree A006881, conjugate A065119.

A056239 adds up prime indices, row sums of A112798.

A299701 counts subset-sums of prime indices, positive A304793.

Semiprime divisors are listed by A367096 and have:

- square count: A056170

- sum: A076290

- squarefree count: A079275

- count: A086971

- firsts: A220264

Cf. A000720, A001248, A008967, A117855, A304792, A365541, A365920, A366738, A366740, A366753.

Programs

Mathematica

nn=10000;
w=Table[Length[Union[Subsets[prix[n],{2}]]]-Length[Union[Total/@Subsets[prix[n],{2}]]],{n,nn}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
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 A367093(n):
    for k in count(1):
        c, a = 0, set()
        for s in (sum(p) for p in multiset_combinations({primepi(i):j for i,j in factorint(k).items()},2)):
            if s not in a:
                a.add(s)
            else:
                c += 1
            if c > n:
                break
        if c == n:
            return k # Chai Wah Wu, Nov 13 2023

Formula

a(n) is the least positive integer such that A086971(a(n)) - A366739(a(n)) = n.

Extensions

a(12)-a(16) from Chai Wah Wu, Nov 13 2023

a(17) from Chai Wah Wu, Nov 18 2023