A389960 Powers p^m, with prime p and m > 1, such that neither p^m-1 nor p^m+1 are squarefree.

49, 125, 243, 343, 1681, 1849, 4913, 6859, 11449, 24389, 24649, 29791, 37249, 57121, 59049, 63001, 66049, 68921, 79507, 85849, 94249, 117649, 148877, 161051, 196249, 208849, 300763, 310249, 332929, 351649, 368449, 413449, 493039, 552049, 573049, 687241, 704969

Offset

1,1

Comments

Superset of 2^m for m in A187965.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Example

Table of n, a(n) for select n, showing prime decomposition of a(n)-1, and a(n)+1:
 n     a(n)          a(n)-1              a(n)+1
-----------------------------------------------------------
 1       49 =   7^2   2^4 * 3             2 * 5^2
 2      125 =   5^3   2^2 * 31            2 * 3^2 * 7
 3      243 =   3^5   2 * 11^2            2^2 * 61
 4      343 =   7^3   2 * 3^2 * 19        2^3 * 43
 5     1681 =  41^2   2^4 * 3 * 5 * 7     2 * 29^2
 6     1849 =  43^2   2^3 * 3 * 7 * 11    2 * 5^2 * 37
 7     4913 =  17^3   2^4 * 307           2 * 3^3 * 7 * 13
 8     6859 =  19^3   2 * 3^3 * 127       2^2 * 5 * 7^3
 9    11449 = 107^2   2^3 * 3^3 * 53      2 * 5^2 * 229
10    24389 =  29^3   2^2 * 7 * 13 * 67   2 * 3^2 * 5 * 271
11    24649 = 157^2   2^3 * 3 * 13 * 79   2 * 5^2 * 17 * 29
51  2097152 =   2^21  7^2 * 127 * 337     3^2 * 43 * 5419

Mathematica

nn = 2^20; s = Sqrt[nn]; i = 1; Union@ Reap[While[Set[{p, j}, {Prime[i], 2}]; While[p^j <= nn, If[AllTrue[# + {-1, 1}, MoebiusMu[#] == 0], Sow[#]] &[p^j]; j++]; p <= s, i++] ][[-1, 1]]

Prog

(PARI) isok(k) = isprimepower(k) && !isprime(k) && !issquarefree(k-1) && !issquarefree(k+1); \\ Michel Marcus, Oct 20 2025

Crossrefs

Cf. A013929, A126706, A187965, A246547, A249435, A269758.

Sequence in context: A080665 A130007 A378629 * A202331 A044300 A044681

Adjacent sequences: A389957 A389958 A389959 * A389961 A389962 A389963

Keyword

nonn

Author

Michael De Vlieger, Oct 20 2025

Status

approved