cp's OEIS Frontend

A343340 Numbers with a prime index dividing all the other prime indices, but with no prime index divisible by all the other prime indices.

%I A343340 #6 Apr 15 2021 21:42:39
%S A343340 30,60,66,70,90,102,110,120,132,138,140,150,154,170,180,182,186,190,
%T A343340 198,204,210,220,238,240,246,264,270,273,276,280,282,286,290,300,306,
%U A343340 308,310,322,330,340,350,354,360,364,372,374,380,396,402,406,408,410,414
%N A343340 Numbers with a prime index dividing all the other prime indices, but with no prime index divisible by all the other prime indices.
%C A343340 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 A343340 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 A343340 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 A343340 Complement of A342193 in A343337.
%e A343340 The sequence of terms together with their prime indices begins:
%e A343340      30: {1,2,3}        182: {1,4,6}          282: {1,2,15}
%e A343340      60: {1,1,2,3}      186: {1,2,11}         286: {1,5,6}
%e A343340      66: {1,2,5}        190: {1,3,8}          290: {1,3,10}
%e A343340      70: {1,3,4}        198: {1,2,2,5}        300: {1,1,2,3,3}
%e A343340      90: {1,2,2,3}      204: {1,1,2,7}        306: {1,2,2,7}
%e A343340     102: {1,2,7}        210: {1,2,3,4}        308: {1,1,4,5}
%e A343340     110: {1,3,5}        220: {1,1,3,5}        310: {1,3,11}
%e A343340     120: {1,1,1,2,3}    238: {1,4,7}          322: {1,4,9}
%e A343340     132: {1,1,2,5}      240: {1,1,1,1,2,3}    330: {1,2,3,5}
%e A343340     138: {1,2,9}        246: {1,2,13}         340: {1,1,3,7}
%e A343340     140: {1,1,3,4}      264: {1,1,1,2,5}      350: {1,3,3,4}
%e A343340     150: {1,2,3,3}      270: {1,2,2,2,3}      354: {1,2,17}
%e A343340     154: {1,4,5}        273: {2,4,6}          360: {1,1,1,2,2,3}
%e A343340     170: {1,3,7}        276: {1,1,2,9}        364: {1,1,4,6}
%e A343340     180: {1,1,2,2,3}    280: {1,1,1,3,4}      372: {1,1,2,11}
%t A343340 Select[Range[2,100],With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)&&And@@IntegerQ/@(p/Min@@p)]&]
%Y A343340 The first condition alone gives the complement of A342193.
%Y A343340 The second condition alone gives A343337.
%Y A343340 The partitions with these Heinz numbers are counted by A343345.
%Y A343340 A000005 counts divisors.
%Y A343340 A000070 counts partitions with a selected part.
%Y A343340 A001055 counts factorizations.
%Y A343340 A056239 adds up prime indices, row sums of A112798.
%Y A343340 A067824 counts strict chains of divisors starting with n.
%Y A343340 A253249 counts strict chains of divisors.
%Y A343340 Cf. A083710, A083711, A097986, A098965, A130714, A339562, A339563, A343341, A343377, A343381.
%K A343340 nonn
%O A343340 1,1
%A A343340 _Gus Wiseman_, Apr 15 2021