A325160 Products of distinct, non-consecutive primes. Squarefree numbers not divisible by any two consecutive primes.

1, 2, 3, 5, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 110, 111, 113, 115, 118, 119

Offset

1,2

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions into distinct non-consecutive parts (counted by A003114). The nonsquarefree case is A319630, which gives the Heinz numbers of integer partitions with no consecutive parts (counted by A116931).

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 6, 52, 515, 5146, 51435, 514416, 5144232, 51442384, ... . Apparently, the asymptotic density of this sequence exists and equals 0.51442... . - Amiram Eldar, Sep 24 2022

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  21: {2,4}
  22: {1,5}
  23: {9}
  26: {1,6}
  29: {10}
  31: {11}
  33: {2,5}
  34: {1,7}
  37: {12}

Mathematica

Select[Range[100], Min@@Differences[Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]]>1&]

Prog

(PARI) isok(k) = {if (issquarefree(k), my(v = apply(primepi, factor(k)[, 1])); ! #select(x->(v[x+1]-v[x] == 1), [1..#v-1])); } \\ Michel Marcus, Jan 09 2021

Crossrefs

Cf. A001227, A003114, A005117, A025157, A034296, A056239, A073485, A073491, A089995, A112798, A116931, A319630, A325161, A325162.

Sequence in context: A325405 A118241 A356237 * A258613 A325460 A002269

Adjacent sequences: A325157 A325158 A325159 * A325161 A325162 A325163

Keyword

nonn

Author

Gus Wiseman, Apr 05 2019

Status

approved