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

1,1

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.

Links

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

Formula

Intersection of A007916 and A055932.

Example

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}

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A071707 A069087 A367504 * A085345 A348774 A265843

Adjacent sequences: A304447 A304448 A304449 * A304451 A304452 A304453

Keyword

nonn

Author

Gus Wiseman, May 12 2018

Status

approved