A397126 - OEIS

A397126

Lexicographically earliest sequence of distinct powerful numbers such that for any prime p, any run of consecutive multiples of p has length exactly 2.

1, 4, 8, 9, 27, 16, 32, 25, 100, 36, 81, 49, 196, 64, 121, 484, 72, 225, 125, 108, 144, 169, 676, 128, 243, 216, 200, 625, 256, 288, 441, 343, 289, 1156, 324, 675, 400, 392, 1225, 500, 432, 729, 361, 1444, 512, 529, 2116, 576, 1089, 968, 648, 1125, 800, 784, 1323

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OFFSET

1,2

COMMENTS

Like A280864, this sequence can be partitioned into blocks on the occasion of coprime a(j) and a(j+1). This sequence also maintains the effect that powers of primes (here, strictly proper prime powers in A246547) may only appear at a(j) or a(j+1), terms that are coprime to a neighboring term.

Unlike A280864, blocks in this sequence occasionally begin with numbers that are not powers of primes (in A024619, specifically A286708).

Oftentimes, perfect powers of squarefree composites (in A303606) follow proper prime powers (in A246547) at the start of a block. This is analogous to even squarefree semiprimes (in A100484) following primes at the start of a block in A280864. However, 256 starts the 11th block but 288 follows in this sequence, 44^2 follows 11^3.

More strictly, blocks that begin with p^2 (in A001248) for prime p > 3, are succeeded by (2*p)^2 at least within the smallest 256 blocks.

Terms in the middle of blocks are strictly in A286708, that is, they are never powers of primes.

This variant of A280864 admits multiple duplex blocks wherein both terms share the same squarefree kernel, akin to block 2 of A280864, {2, 4}, likely the only such block in that sequence. Examples here include {4, 8}, {9, 27}, and {108, 144}.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..16384

EXAMPLE

Table of the first 16 terms:

n a(n) Prime power factors

---------------------------------

1: 1

2: 4 = 2^2

3: 8 = 2^3

4: 9 = 3^2

5: 27 = 3^3

6: 16 = 2^4

7: 32 = 2^5

8: 25 = 5^2

9: 100 = 2^2 5^2

10: 36 = 2^2 3^2

11: 81 = 3^4

12: 49 = 7^2

13: 196 = 2^2 7^2

14: 64 = 2^6

15: 121 = 11^2

16: 484 = 2^2 11^2

Table of the first 12 blocks:

m\i 1 2 3 4 5 6 7 8 9 10

-----------------------------------------------------------------

1: 1

2: 4 8

3: 9 27

4: 16 32

5: 25 100 36 81

6: 49 196 64

7: 121 484 72 225 125

8: 108 144

9: 169 676 128

10: 243 216 200 625

11: 256 288 441 343

12: 289 1156 324 675 400 392 1225 500 432 729

-----------------------------------------------------------------

m\i 1 2 3 4 5 6 7 8 9 10

MATHEMATICA

Block[{c, k, m, r, s, w, nn, rad},

nn = 120; c[_] := False; r = s = 1;

rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]];

Reap[Do[w = GCD[r, s]; m = r/w; k = m;

While[Or[c[k], ! Divisible[k, rad[k]^2], ! CoprimeQ[w, k] ], k += m];

Sow[k]; Set[{c[k], s, r}, {True, r, rad[k] } ],

{i, nn}] ][[-1, 1]] ]

CROSSREFS

Cf. A001694, A246547, A280864, A286708, A303606, A338444 (squarefree version).

Sequence in context: A338325 A168363 A396876 * A397368 A182046 A171468

Adjacent sequences: A397120 A397124 A397125 * A397127 A397128 A397129

KEYWORD

nonn

AUTHOR

Michael De Vlieger, Jun 20 2026

STATUS

approved