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

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, 35, 36, 37, 41, 42, 43, 45, 47, 48, 49, 53, 54, 55, 59, 60, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 84, 89, 90, 91, 96, 97, 101, 103, 105, 107, 108, 109

Offset: 1

Author

Gus Wiseman, Mar 05 2021

Keywords

• nonn

Comments

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.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}            18: {1,2,2}         42: {1,2,4}
      2: {1}           19: {8}             43: {14}
      3: {2}           21: {2,4}           45: {2,2,3}
      4: {1,1}         23: {9}             47: {15}
      5: {3}           24: {1,1,1,2}       48: {1,1,1,1,2}
      6: {1,2}         25: {3,3}           49: {4,4}
      7: {4}           27: {2,2,2}         53: {16}
      8: {1,1,1}       29: {10}            54: {1,2,2,2}
      9: {2,2}         30: {1,2,3}         55: {3,5}
     11: {5}           31: {11}            59: {17}
     12: {1,1,2}       32: {1,1,1,1,1}     60: {1,1,2,3}
     13: {6}           35: {3,4}           61: {18}
     15: {2,3}         36: {1,1,2,2}       63: {2,2,4}
     16: {1,1,1,1}     37: {12}            64: {1,1,1,1,1,1}
     17: {7}           41: {13}            65: {3,6}
		

Crossrefs

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

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

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

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

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

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

Programs

Mathematica

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