%I #8 Mar 05 2021 21:46:45

%S 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 35,36,37,41,42,43,45,47,48,49,53,54,55,59,60,61,63,64,65,67,71,72,73,

%U 75,77,79,81,83,84,89,90,91,96,97,101,103,105,107,108,109

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

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

%e 1: {} 18: {1,2,2} 42: {1,2,4}

%e 2: {1} 19: {8} 43: {14}

%e 3: {2} 21: {2,4} 45: {2,2,3}

%e 4: {1,1} 23: {9} 47: {15}

%e 5: {3} 24: {1,1,1,2} 48: {1,1,1,1,2}

%e 6: {1,2} 25: {3,3} 49: {4,4}

%e 7: {4} 27: {2,2,2} 53: {16}

%e 8: {1,1,1} 29: {10} 54: {1,2,2,2}

%e 9: {2,2} 30: {1,2,3} 55: {3,5}

%e 11: {5} 31: {11} 59: {17}

%e 12: {1,1,2} 32: {1,1,1,1,1} 60: {1,1,2,3}

%e 13: {6} 35: {3,4} 61: {18}

%e 15: {2,3} 36: {1,1,2,2} 63: {2,2,4}

%e 16: {1,1,1,1} 37: {12} 64: {1,1,1,1,1,1}

%e 17: {7} 41: {13} 65: {3,6}

%t Select[Range[100],Min[Divide@@@Partition[PrimePi/@First/@FactorInteger[#],2,1]]>=1/2&]

%Y The multiplicative version (squared instead of doubled) for prime factors is A253784.

%Y These are the Heinz numbers of the partitions counted by A342094.

%Y A003114 counts partitions with adjacent parts differing by more than 1.

%Y A034296 counts partitions with adjacent parts differing by at most 1.

%Y A038548 counts inferior or superior divisors, listed by A161906 or A161908.

%Y Cf. A000929, A003242, A056239, A056924, A112798, A154402, A167606, A337135, A342085, A342096, A342098.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 05 2021