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Numbers with no prime index divisible by all the other prime indices.
18

%I #12 Aug 27 2021 05:58:40

%S 1,15,30,33,35,45,51,55,60,66,69,70,75,77,85,90,91,93,95,99,102,105,

%T 110,119,120,123,132,135,138,140,141,143,145,150,153,154,155,161,165,

%U 170,175,177,180,182,186,187,190,198,201,203,204,205,207,209,210,215

%N Numbers with no prime index divisible by all the other prime indices.

%C Alternative name: 1 and numbers whose greatest prime index is not divisible by all the other prime indices.

%C First differs from A318992 in lacking 195.

%C First differs from A343343 in lacking 195.

%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 (counted by A343341). 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.

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

%e 1: {} 90: {1,2,2,3} 141: {2,15}

%e 15: {2,3} 91: {4,6} 143: {5,6}

%e 30: {1,2,3} 93: {2,11} 145: {3,10}

%e 33: {2,5} 95: {3,8} 150: {1,2,3,3}

%e 35: {3,4} 99: {2,2,5} 153: {2,2,7}

%e 45: {2,2,3} 102: {1,2,7} 154: {1,4,5}

%e 51: {2,7} 105: {2,3,4} 155: {3,11}

%e 55: {3,5} 110: {1,3,5} 161: {4,9}

%e 60: {1,1,2,3} 119: {4,7} 165: {2,3,5}

%e 66: {1,2,5} 120: {1,1,1,2,3} 170: {1,3,7}

%e 69: {2,9} 123: {2,13} 175: {3,3,4}

%e 70: {1,3,4} 132: {1,1,2,5} 177: {2,17}

%e 75: {2,3,3} 135: {2,2,2,3} 180: {1,1,2,2,3}

%e 77: {4,5} 138: {1,2,9} 182: {1,4,6}

%e 85: {3,7} 140: {1,1,3,4} 186: {1,2,11}

%e For example, 195 has prime indices {2,3,6}, and 6 is divisible by both 2 and 3, so 195 does not belong to the sequence.

%t Select[Range[1000],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)]&]

%Y The complement is counted by A130689.

%Y The dual version is A342193.

%Y The case with smallest prime index not dividing all the others is A343338.

%Y The case with smallest prime index dividing by all the others is A343340.

%Y These are the Heinz numbers of the partitions counted by A343341.

%Y Including the dual version gives A343343.

%Y A000005 counts divisors.

%Y A000070 counts partitions with a selected part.

%Y A006128 counts partitions with a selected position.

%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 A339564 counts factorizations with a selected factor.

%Y Cf. A083710, A257993, A338470, A341450, A343342, A343345, A343346, A343377, A343379, A343381, A343382.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 13 2021