A342193 - OEIS

%I #13 Apr 17 2021 01:57:16

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

%T 145,153,155,161,165,175,177,187,195,201,203,205,207,209,215,217,219,

%U 221,225,231,245,247,249,253,255,265,275,279,285,287,291,295,297,299

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

%C Alternative name: 1 and numbers with smallest prime index not dividing all the other prime indices.

%C First differs from A339562 in having 45.

%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 1 and Heinz numbers of integer partitions with smallest part not dividing all the others (counted by A338470). 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: {}         105: {2,3,4}      201: {2,19}

%e      15: {2,3}      119: {4,7}        203: {4,10}

%e      33: {2,5}      123: {2,13}       205: {3,13}

%e      35: {3,4}      135: {2,2,2,3}    207: {2,2,9}

%e      45: {2,2,3}    141: {2,15}       209: {5,8}

%e      51: {2,7}      143: {5,6}        215: {3,14}

%e      55: {3,5}      145: {3,10}       217: {4,11}

%e      69: {2,9}      153: {2,2,7}      219: {2,21}

%e      75: {2,3,3}    155: {3,11}       221: {6,7}

%e      77: {4,5}      161: {4,9}        225: {2,2,3,3}

%e      85: {3,7}      165: {2,3,5}      231: {2,4,5}

%e      91: {4,6}      175: {3,3,4}      245: {3,4,4}

%e      93: {2,11}     177: {2,17}       247: {6,8}

%e      95: {3,8}      187: {5,7}        249: {2,23}

%e      99: {2,2,5}    195: {2,3,6}      253: {5,9}

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

%Y The complement is counted by A083710 (strict: A097986).

%Y The complement with no 1's is A083711 (strict: A098965).

%Y These partitions are counted by A338470 (strict: A341450).

%Y The squarefree case is A339562, with squarefree complement A339563.

%Y The case with maximum prime index not divisible by all others is A343338.

%Y The case with maximum prime index divisible by all others is A343339.

%Y A000005 counts divisors.

%Y A000070 counts partitions with a selected part.

%Y A001221 counts distinct prime factors.

%Y A006128 counts partitions with a selected position (strict: A015723).

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

%Y A299702 lists Heinz numbers of knapsack partitions.

%Y A339564 counts factorizations with a selected factor.

%Y Cf. A066637, A072774, A098743, A253249, A264401, A257993, A342050, A342051, A343344.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 11 2021