OFFSET
1,1
COMMENTS
Alternative name: Numbers > 1 whose smallest prime index divides all the other prime indices, but whose greatest prime index is not 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 not divisible by all the others, but smallest part dividing all the others (counted by A343345). 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:
30: {1,2,3} 182: {1,4,6} 282: {1,2,15}
60: {1,1,2,3} 186: {1,2,11} 286: {1,5,6}
66: {1,2,5} 190: {1,3,8} 290: {1,3,10}
70: {1,3,4} 198: {1,2,2,5} 300: {1,1,2,3,3}
90: {1,2,2,3} 204: {1,1,2,7} 306: {1,2,2,7}
102: {1,2,7} 210: {1,2,3,4} 308: {1,1,4,5}
110: {1,3,5} 220: {1,1,3,5} 310: {1,3,11}
120: {1,1,1,2,3} 238: {1,4,7} 322: {1,4,9}
132: {1,1,2,5} 240: {1,1,1,1,2,3} 330: {1,2,3,5}
138: {1,2,9} 246: {1,2,13} 340: {1,1,3,7}
140: {1,1,3,4} 264: {1,1,1,2,5} 350: {1,3,3,4}
150: {1,2,3,3} 270: {1,2,2,2,3} 354: {1,2,17}
154: {1,4,5} 273: {2,4,6} 360: {1,1,1,2,2,3}
170: {1,3,7} 276: {1,1,2,9} 364: {1,1,4,6}
180: {1,1,2,2,3} 280: {1,1,1,3,4} 372: {1,1,2,11}
MATHEMATICA
Select[Range[2, 100], With[{p=PrimePi/@First/@FactorInteger[#]}, !And@@IntegerQ/@(Max@@p/p)&&And@@IntegerQ/@(p/Min@@p)]&]
CROSSREFS
The first condition alone gives the complement of A342193.
The second condition alone gives A343337.
The partitions with these Heinz numbers are counted by A343345.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A001055 counts factorizations.
A067824 counts strict chains of divisors starting with n.
A253249 counts strict chains of divisors.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 15 2021
STATUS
approved