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A342191 Numbers with no adjacent prime indices having quotient < 1/2.

%I A342191 #8 Mar 05 2021 21:46:45
%S A342191 1,2,3,4,5,6,7,8,9,11,12,13,15,16,17,18,19,21,23,24,25,27,29,30,31,32,
%T A342191 35,36,37,41,42,43,45,47,48,49,53,54,55,59,60,61,63,64,65,67,71,72,73,
%U A342191 75,77,79,81,83,84,89,90,91,96,97,101,103,105,107,108,109
%N A342191 Numbers with no adjacent prime indices having quotient < 1/2.
%C A342191 Also Heinz numbers of integer partitions with no adjacent parts having quotient > 2 (counted by A342094). 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 A342191 The sequence of terms together with their prime indices begins:
%e A342191       1: {}            18: {1,2,2}         42: {1,2,4}
%e A342191       2: {1}           19: {8}             43: {14}
%e A342191       3: {2}           21: {2,4}           45: {2,2,3}
%e A342191       4: {1,1}         23: {9}             47: {15}
%e A342191       5: {3}           24: {1,1,1,2}       48: {1,1,1,1,2}
%e A342191       6: {1,2}         25: {3,3}           49: {4,4}
%e A342191       7: {4}           27: {2,2,2}         53: {16}
%e A342191       8: {1,1,1}       29: {10}            54: {1,2,2,2}
%e A342191       9: {2,2}         30: {1,2,3}         55: {3,5}
%e A342191      11: {5}           31: {11}            59: {17}
%e A342191      12: {1,1,2}       32: {1,1,1,1,1}     60: {1,1,2,3}
%e A342191      13: {6}           35: {3,4}           61: {18}
%e A342191      15: {2,3}         36: {1,1,2,2}       63: {2,2,4}
%e A342191      16: {1,1,1,1}     37: {12}            64: {1,1,1,1,1,1}
%e A342191      17: {7}           41: {13}            65: {3,6}
%t A342191 Select[Range[100],Min[Divide@@@Partition[PrimePi/@First/@FactorInteger[#],2,1]]>=1/2&]
%Y A342191 The multiplicative version (squared instead of doubled) for prime factors is A253784.
%Y A342191 These are the Heinz numbers of the partitions counted by A342094.
%Y A342191 A003114 counts partitions with adjacent parts differing by more than 1.
%Y A342191 A034296 counts partitions with adjacent parts differing by at most 1.
%Y A342191 A038548 counts inferior or superior divisors, listed by A161906 or A161908.
%Y A342191 Cf. A000929, A003242, A056239, A056924, A112798, A154402, A167606, A337135, A342085, A342096, A342098.
%K A342191 nonn
%O A342191 1,2
%A A342191 _Gus Wiseman_, Mar 05 2021