OFFSET
0,2
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.
FORMULA
2 | a(n) for n > 0. - David A. Corneth, Nov 13 2023
EXAMPLE
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}
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]}]
PROG
(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
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 09 2023
EXTENSIONS
a(17)-a(22) from Chai Wah Wu, Nov 13 2023
a(23)-a(28) from David A. Corneth, Nov 13 2023
STATUS
approved