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

Author

Gus Wiseman, Apr 05 2019

Keywords

• nonn

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

Examples

			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}
		

Links

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

Crossrefs

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

Programs

Mathematica

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

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