This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A342193 #13 Apr 17 2021 01:57:16
%S A342193 1,15,33,35,45,51,55,69,75,77,85,91,93,95,99,105,119,123,135,141,143,
%T A342193 145,153,155,161,165,175,177,187,195,201,203,205,207,209,215,217,219,
%U A342193 221,225,231,245,247,249,253,255,265,275,279,285,287,291,295,297,299
%N A342193 Numbers with no prime index dividing all the other prime indices.
%C A342193 Alternative name: 1 and numbers with smallest prime index not dividing all the other prime indices.
%C A342193 First differs from A339562 in having 45.
%C A342193 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 A342193 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 A342193 The sequence of terms together with their prime indices begins:
%e A342193 1: {} 105: {2,3,4} 201: {2,19}
%e A342193 15: {2,3} 119: {4,7} 203: {4,10}
%e A342193 33: {2,5} 123: {2,13} 205: {3,13}
%e A342193 35: {3,4} 135: {2,2,2,3} 207: {2,2,9}
%e A342193 45: {2,2,3} 141: {2,15} 209: {5,8}
%e A342193 51: {2,7} 143: {5,6} 215: {3,14}
%e A342193 55: {3,5} 145: {3,10} 217: {4,11}
%e A342193 69: {2,9} 153: {2,2,7} 219: {2,21}
%e A342193 75: {2,3,3} 155: {3,11} 221: {6,7}
%e A342193 77: {4,5} 161: {4,9} 225: {2,2,3,3}
%e A342193 85: {3,7} 165: {2,3,5} 231: {2,4,5}
%e A342193 91: {4,6} 175: {3,3,4} 245: {3,4,4}
%e A342193 93: {2,11} 177: {2,17} 247: {6,8}
%e A342193 95: {3,8} 187: {5,7} 249: {2,23}
%e A342193 99: {2,2,5} 195: {2,3,6} 253: {5,9}
%t A342193 Select[Range[100],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(p/Min@@p)]&]
%Y A342193 The complement is counted by A083710 (strict: A097986).
%Y A342193 The complement with no 1's is A083711 (strict: A098965).
%Y A342193 These partitions are counted by A338470 (strict: A341450).
%Y A342193 The squarefree case is A339562, with squarefree complement A339563.
%Y A342193 The case with maximum prime index not divisible by all others is A343338.
%Y A342193 The case with maximum prime index divisible by all others is A343339.
%Y A342193 A000005 counts divisors.
%Y A342193 A000070 counts partitions with a selected part.
%Y A342193 A001221 counts distinct prime factors.
%Y A342193 A006128 counts partitions with a selected position (strict: A015723).
%Y A342193 A056239 adds up prime indices, row sums of A112798.
%Y A342193 A299702 lists Heinz numbers of knapsack partitions.
%Y A342193 A339564 counts factorizations with a selected factor.
%Y A342193 Cf. A066637, A072774, A098743, A253249, A264401, A257993, A342050, A342051, A343344.
%K A342193 nonn
%O A342193 1,2
%A A342193 _Gus Wiseman_, Apr 11 2021