A181800 Powerful numbers that are the least integer of their prime signature.

1, 4, 8, 16, 32, 36, 64, 72, 128, 144, 216, 256, 288, 432, 512, 576, 864, 900, 1024, 1152, 1296, 1728, 1800, 2048, 2304, 2592, 3456, 3600, 4096, 4608, 5184, 5400, 6912, 7200, 7776, 8192, 9216, 10368, 10800, 13824, 14400, 15552, 16384, 18432, 20736, 21600, 27000

Offset: 1

Author

Matthew Vandermast, Dec 02 2010

Keywords

• nonn

Comments

Apart from 1, all terms in this sequence are multiples of 4. - Charles R Greathouse IV, Dec 13 2016

Apart from 1, the prime factor exponents of the terms in this sequence represent exactly every factorization of every integer k >= 2. E.g. the two factorizations of 4 are represented by 16 (2^4 -> {4}) and 36 (2^2 * 3^2 -> {2, 2}; 2 * 2 = 4). The smallest such term for each k is subsequence A085629. - Charles L. Hohn, Apr 04 2025

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Charles R Greathouse IV)

Crossrefs

Cf. A001694 (powerful numbers), A025487 (least integer of each prime signature).

Cf. A085629.

Programs

PARI

is(n)=if(n<4,return(n==1)); my(f=factor(n)); vecmin(f[,2])>1 && f[,1]~ == primes(#f~) && f[,2] == vecsort(f[,2],,4) \\ Charles R Greathouse IV, Sep 14 2015

Formula

Intersection of A001694 and A025487.