A387890 First difference of consecutive even perfect powers (no perfect power between them).

4, 4, 16, 20, 28, 24, 36, 68, 40, 56, 32, 100, 92, 168, 100, 152, 48, 104, 356, 100, 116, 496, 104, 80, 8, 144, 368, 316, 28, 732, 648, 784, 252, 704, 184, 828, 964, 200, 832, 396, 1148, 728, 732, 1208, 656, 112, 932, 804, 196, 748, 804, 316, 764, 1160, 1136, 292, 308, 1264, 1448

Offset

1,1

Comments

The minimum descent value a(n)>a(n+1) is conjectured to be 8, as in Pillai's Conjecture.

Links

Table of n, a(n) for n=1..59.

Gergely Földvári, Coefficial Divisibility (with terms up to the first 1000 perfect powers).

Eric Weisstein's World of Mathematics, Pillai's Conjecture

Example

The first pair of consecutive even perfect powers (no perfect power in between) are 4 and 8, their difference (8-4=4) gives the first term of the integer sequence, 4.
The second pair is 32 and 36, their difference (36-32=4) gives the second term of the integer sequence which is also 4.

Prog

(Python)
from math import isqrt
def even_pp_diffs(n):
    p = {1}
    for a in range(2, isqrt(n)+1):
        x = a * a
        while x <= n:
            p.add(x)
            x *= a
    p = sorted(p)
    return [p[i+1] - p[i] for i in range(len(p)-1) if p[i] % 2 == 0 and p[i+1] % 2 == 0]
print(even_pp_diffs(2 * 10**6))

Crossrefs

Cf. A001597, A387889, A074981, A076427. Based on subsequence of A075090. Subsequence of A053289.

Sequence in context: A129884 A137725 A240035 * A321677 A223819 A082649

Adjacent sequences: A387887 A387888 A387889 * A387891 A387892 A387893

Keyword

nonn

Author

Gergely Földvári, Sep 11 2025

Status

approved