login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Numbers with a prime index dividing all the other prime indices, but with no prime index divisible by all the other prime indices.
7

%I #6 Apr 15 2021 21:42:39

%S 30,60,66,70,90,102,110,120,132,138,140,150,154,170,180,182,186,190,

%T 198,204,210,220,238,240,246,264,270,273,276,280,282,286,290,300,306,

%U 308,310,322,330,340,350,354,360,364,372,374,380,396,402,406,408,410,414

%N Numbers with a prime index dividing all the other prime indices, but with no prime index divisible by all the other prime indices.

%C 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.

%C 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.

%C 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.

%F Complement of A342193 in A343337.

%e The sequence of terms together with their prime indices begins:

%e 30: {1,2,3} 182: {1,4,6} 282: {1,2,15}

%e 60: {1,1,2,3} 186: {1,2,11} 286: {1,5,6}

%e 66: {1,2,5} 190: {1,3,8} 290: {1,3,10}

%e 70: {1,3,4} 198: {1,2,2,5} 300: {1,1,2,3,3}

%e 90: {1,2,2,3} 204: {1,1,2,7} 306: {1,2,2,7}

%e 102: {1,2,7} 210: {1,2,3,4} 308: {1,1,4,5}

%e 110: {1,3,5} 220: {1,1,3,5} 310: {1,3,11}

%e 120: {1,1,1,2,3} 238: {1,4,7} 322: {1,4,9}

%e 132: {1,1,2,5} 240: {1,1,1,1,2,3} 330: {1,2,3,5}

%e 138: {1,2,9} 246: {1,2,13} 340: {1,1,3,7}

%e 140: {1,1,3,4} 264: {1,1,1,2,5} 350: {1,3,3,4}

%e 150: {1,2,3,3} 270: {1,2,2,2,3} 354: {1,2,17}

%e 154: {1,4,5} 273: {2,4,6} 360: {1,1,1,2,2,3}

%e 170: {1,3,7} 276: {1,1,2,9} 364: {1,1,4,6}

%e 180: {1,1,2,2,3} 280: {1,1,1,3,4} 372: {1,1,2,11}

%t Select[Range[2,100],With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)&&And@@IntegerQ/@(p/Min@@p)]&]

%Y The first condition alone gives the complement of A342193.

%Y The second condition alone gives A343337.

%Y The partitions with these Heinz numbers are counted by A343345.

%Y A000005 counts divisors.

%Y A000070 counts partitions with a selected part.

%Y A001055 counts factorizations.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A067824 counts strict chains of divisors starting with n.

%Y A253249 counts strict chains of divisors.

%Y Cf. A083710, A083711, A097986, A098965, A130714, A339562, A339563, A343341, A343377, A343381.

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 15 2021