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Numbers with no prime index dividing or divisible by all the other prime indices.
11

%I #6 Apr 15 2021 21:42:22

%S 1,15,33,35,45,51,55,69,75,77,85,91,93,95,99,105,119,123,135,141,143,

%T 145,153,155,161,165,175,177,187,201,203,205,207,209,215,217,219,221,

%U 225,231,245,247,249,253,255,265,275,279,285,287,291,295,297,299,301

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

%C 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 First differs from A302697 in having 91.

%C First differs from A337987 in having 91.

%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 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 Intersection of A342193 and A343337.

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

%e 1: {} 105: {2,3,4} 203: {4,10}

%e 15: {2,3} 119: {4,7} 205: {3,13}

%e 33: {2,5} 123: {2,13} 207: {2,2,9}

%e 35: {3,4} 135: {2,2,2,3} 209: {5,8}

%e 45: {2,2,3} 141: {2,15} 215: {3,14}

%e 51: {2,7} 143: {5,6} 217: {4,11}

%e 55: {3,5} 145: {3,10} 219: {2,21}

%e 69: {2,9} 153: {2,2,7} 221: {6,7}

%e 75: {2,3,3} 155: {3,11} 225: {2,2,3,3}

%e 77: {4,5} 161: {4,9} 231: {2,4,5}

%e 85: {3,7} 165: {2,3,5} 245: {3,4,4}

%e 91: {4,6} 175: {3,3,4} 247: {6,8}

%e 93: {2,11} 177: {2,17} 249: {2,23}

%e 95: {3,8} 187: {5,7} 253: {5,9}

%e 99: {2,2,5} 201: {2,19} 255: {2,3,7}

%e 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 Select[Range[100],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)&&!And@@IntegerQ/@(p/Min@@p)]&]

%Y The first condition alone gives A342193.

%Y The second condition alone gives A343337.

%Y The half-opposite versions are A343339 and A343340.

%Y The partitions with these Heinz numbers are counted by A343342.

%Y The opposite version is the complement of A343343.

%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 A339564 counts factorizations with a selected factor.

%Y Cf. A083710, A130689, A338470, A339562, A341450, A343341, A343346, A343347, A343348, A343377, A343379, A343382.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 13 2021