OFFSET
1,1
COMMENTS
Alternative name: Numbers > 1 whose smallest prime index does not divide all the other prime indices, but whose greatest prime index is divisible by all the other prime indices.
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.
Also Heinz numbers of partitions with greatest part divisible by all the others, but smallest part not dividing all the others (counted by A343344). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The sequence of terms together with their prime indices begins:
195: {2,3,6} 2585: {3,5,15} 4575: {2,3,3,18}
555: {2,3,12} 2715: {2,3,42} 4755: {2,3,66}
585: {2,2,3,6} 2745: {2,2,3,18} 4875: {2,3,3,3,6}
915: {2,3,18} 2775: {2,3,3,12} 4995: {2,2,2,3,12}
957: {2,5,10} 2871: {2,2,5,10} 5085: {2,2,3,30}
975: {2,3,3,6} 2925: {2,2,3,3,6} 5265: {2,2,2,2,3,6}
1295: {3,4,12} 3115: {3,4,24} 5285: {3,4,36}
1335: {2,3,24} 3345: {2,3,48} 5385: {2,3,72}
1665: {2,2,3,12} 3367: {4,6,12} 5457: {2,7,28}
1695: {2,3,30} 3729: {2,5,30} 5467: {4,5,20}
1755: {2,2,2,3,6} 3765: {2,3,54} 5709: {2,5,40}
2193: {2,7,14} 3885: {2,3,4,12} 5955: {2,3,78}
2265: {2,3,36} 4005: {2,2,3,24} 6205: {3,7,21}
2343: {2,5,20} 4209: {2,9,18} 6215: {3,5,30}
2535: {2,3,6,6} 4215: {2,3,60} 6475: {3,3,4,12}
MATHEMATICA
Select[Range[2, 1000], With[{p=PrimePi/@First/@FactorInteger[#]}, And@@IntegerQ/@(Max@@p/p)&&!And@@IntegerQ/@(p/Min@@p)]&]
CROSSREFS
The first condition alone gives A342193.
The second condition alone gives the complement of A343337.
The partitions with these Heinz numbers are counted by A343344.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A067824 counts strict chains of divisors starting with n.
A253249 counts strict chains of divisors.
A339564 counts factorizations with a selected factor.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 15 2021
STATUS
approved