A387889 First difference of consecutive odd perfect powers (no perfect power between them).

2, 4, 18, 18, 38, 10, 12, 100, 106, 128, 30, 148, 154, 166, 198, 260, 138, 216, 26, 28, 270, 248, 284, 546, 524, 506, 574, 652, 250, 726, 94, 568, 170, 548, 954, 1000, 638, 180, 890, 412, 888, 674, 908, 1170, 1262, 1036, 782, 1048, 1590, 734, 252, 894, 1750, 648, 74, 76, 702, 2000

Offset

1,1

Comments

The first and smallest term 2 is conjectured to be a singular occurrence in the integer sequence, with 10 conjectured to be the minimum descent value a(n)>a(n+1), as in Pillai's Conjecture.

Links

Table of n, a(n) for n=1..58.

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

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

Example

The first two consecutive odd perfect powers (no perfect power between them) are 25 and 27, thus their difference (27-25) gives the first term, 2.

Prog

(Python)
from math import isqrt
def odd_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] & 1 and p[i+1] & 1]
print(odd_pp_diffs(2 * 10**6))

Crossrefs

Cf. A001597, A387890, A074981, A076427. Based on subsequence of A075109. Subsequence of A053289.

Sequence in context: A242528 A137933 A143116 * A304856 A115987 A076692

Adjacent sequences: A387886 A387887 A387888 * A387890 A387891 A387892

Keyword

nonn

Author

Gergely Földvári, Sep 11 2025

Status

approved