A304450 Numbers that are not perfect powers and whose prime factors span an initial interval of prime numbers.

2, 6, 12, 18, 24, 30, 48, 54, 60, 72, 90, 96, 108, 120, 150, 162, 180, 192, 210, 240, 270, 288, 300, 360, 384, 420, 432, 450, 480, 486, 540, 600, 630, 648, 720, 750, 768, 810, 840, 864, 960, 972, 1050, 1080, 1152, 1200, 1260, 1350, 1440, 1458, 1470, 1500, 1536

Offset: 1

Author

Gus Wiseman, May 12 2018

Keywords

• nonn

Comments

The multiset of prime indices of a(n) is the a(n)-th row of A112798. This multiset is normal, meaning it spans an initial interval of positive integers, and aperiodic, meaning its multiplicities are relatively prime.

Examples

			Sequence of all normal aperiodic multisets begins
2:   {1}
6:   {1,2}
12:  {1,1,2}
18:  {1,2,2}
24:  {1,1,1,2}
30:  {1,2,3}
48:  {1,1,1,1,2}
54:  {1,2,2,2}
60:  {1,1,2,3}
72:  {1,1,1,2,2}
90:  {1,2,2,3}
96:  {1,1,1,1,1,2}
108: {1,1,2,2,2}
120: {1,1,1,2,3}
150: {1,2,3,3}
162: {1,2,2,2,2}
180: {1,1,2,2,3}
192: {1,1,1,1,1,1,2}
210: {1,2,3,4}
240: {1,1,1,1,2,3}
270: {1,2,2,2,3}
288: {1,1,1,1,1,2,2}
300: {1,1,2,3,3}
360: {1,1,1,2,2,3}
384: {1,1,1,1,1,1,1,2}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A000740, A000837, A000961, A001597, A001694, A005117, A007916, A052409, A052410, A055932, A056239, A112798, A303431, A303546, A303945.

Programs

Mathematica

Select[Range[1000],FactorInteger[#][[-1,1]]==Prime[Length[FactorInteger[#]]]&&GCD@@FactorInteger[#][[All,2]]===1&]

PARI

ok(n)={my(f=factor(n)[,1]); #f && !ispower(n) && #f==primepi(f[#f])} \\ Andrew Howroyd, Aug 26 2018

Formula

Intersection of A007916 and A055932.