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

1,2

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.

Links

Table of n, a(n) for n=1..66.

Example

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}

Mathematica

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

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.

Sequence in context: A253567 A048197 A253784 * A251726 A362981 A193671

Adjacent sequences: A342188 A342189 A342190 * A342192 A342193 A342194

Keyword

nonn

Author

Gus Wiseman, Mar 05 2021

Status

approved