cp's OEIS Frontend

A343338 Numbers with no prime index dividing or divisible by all the other prime indices.

%I A343338 #6 Apr 15 2021 21:42:22
%S A343338 1,15,33,35,45,51,55,69,75,77,85,91,93,95,99,105,119,123,135,141,143,
%T A343338 145,153,155,161,165,175,177,187,201,203,205,207,209,215,217,219,221,
%U A343338 225,231,245,247,249,253,255,265,275,279,285,287,291,295,297,299,301
%N A343338 Numbers with no prime index dividing or divisible by all the other prime indices.
%C A343338 Alternative name: 1 and numbers whose smallest prime index does not divide all the other prime indices, nor whose greatest prime index is divisible by all the other prime indices.
%C A343338 First differs from A302697 in having 91.
%C A343338 First differs from A337987 in having 91.
%C A343338 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 A343338 Also Heinz numbers of partitions with greatest part not divisible by all the others and smallest part not dividing all the others (counted by A343342). 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 A343338 Intersection of A342193 and A343337.
%e A343338 The sequence of terms together with their prime indices begins:
%e A343338       1: {}         105: {2,3,4}      203: {4,10}
%e A343338      15: {2,3}      119: {4,7}        205: {3,13}
%e A343338      33: {2,5}      123: {2,13}       207: {2,2,9}
%e A343338      35: {3,4}      135: {2,2,2,3}    209: {5,8}
%e A343338      45: {2,2,3}    141: {2,15}       215: {3,14}
%e A343338      51: {2,7}      143: {5,6}        217: {4,11}
%e A343338      55: {3,5}      145: {3,10}       219: {2,21}
%e A343338      69: {2,9}      153: {2,2,7}      221: {6,7}
%e A343338      75: {2,3,3}    155: {3,11}       225: {2,2,3,3}
%e A343338      77: {4,5}      161: {4,9}        231: {2,4,5}
%e A343338      85: {3,7}      165: {2,3,5}      245: {3,4,4}
%e A343338      91: {4,6}      175: {3,3,4}      247: {6,8}
%e A343338      93: {2,11}     177: {2,17}       249: {2,23}
%e A343338      95: {3,8}      187: {5,7}        253: {5,9}
%e A343338      99: {2,2,5}    201: {2,19}       255: {2,3,7}
%e A343338 For example, the prime indices of 975 are {2,3,3,6}, all of which divide 6, but not all of which are multiples of 2, so 975 is not in the sequence.
%t A343338 Select[Range[100],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)&&!And@@IntegerQ/@(p/Min@@p)]&]
%Y A343338 The first condition alone gives A342193.
%Y A343338 The second condition alone gives A343337.
%Y A343338 The half-opposite versions are A343339 and A343340.
%Y A343338 The partitions with these Heinz numbers are counted by A343342.
%Y A343338 The opposite version is the complement of A343343.
%Y A343338 A000005 counts divisors.
%Y A343338 A000070 counts partitions with a selected part.
%Y A343338 A001055 counts factorizations.
%Y A343338 A056239 adds up prime indices, row sums of A112798.
%Y A343338 A067824 counts strict chains of divisors starting with n.
%Y A343338 A253249 counts strict chains of divisors.
%Y A343338 A339564 counts factorizations with a selected factor.
%Y A343338 Cf. A083710, A130689, A338470, A339562, A341450, A343341, A343346, A343347, A343348, A343377, A343379, A343382.
%K A343338 nonn
%O A343338 1,2
%A A343338 _Gus Wiseman_, Apr 13 2021