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A332097 Maximum of s^n - Sum_{0 < x < s} x^n. 4
1, 1, 4, 28, 317, 4606, 84477, 1919575, 47891482, 1512466345, 48627032377, 1930020260416, 77986967769593, 3624337209819538, 178110510699972510, 9381158756438306167, 548676565488760277878, 31900481466759651567625, 2189463436999785648552851, 144075114432622269076465962 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

For small values of s, we have Sum_{0 < x < s} x^n ~ (s-1)^n, but for s > n/log(2) + 1.5 (cf. A332101) the difference E(s) = s^n - Sum_{0 < x < s} x^n becomes negative. Just before, the difference has its maximum: We have E(s) < E(s+1) <=> 2*s^n < (s+1)^n <=> s < 1/(2^(1/n)-1), so E takes its maximum at s = A078607(n), the least integer larger than this limiting value. This appears to be almost always equal to A332101(n) - 2.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..367

FORMULA

a(n) = s^n - Sum_{0 < x < s} x^n for s = ceiling(1/(2^(1/n)-1)) = A078607(n).

MAPLE

a:= proc(n) option remember; `if`(n=0, 1, (s->

      s^n-add(x^n, x=1..s-1))(ceil(1/(2^(1/n)-1))))

    end:

seq(a(n), n=0..20);  # Alois P. Heinz, May 09 2020

MATHEMATICA

a[0] = 1; a[n_] := (s = Ceiling[1/(2^(1/n) - 1)])^n - Sum[k^n, {k, 1, s - 1}]; Array[a, 20, 0] (* Amiram Eldar, May 09 2020 *)

PROG

(PARI) {apply( A332097(n, s=1\(sqrtn(2, n-!n)-1))=(s+1)^n-sum(k=1, s, k^n)}, [0..20])

CROSSREFS

Cf. A030052 (least k such that k^n = sum of distinct n-th powers).

Cf. A078607 (s for which E(s) = a(n) <=> least k such that 2*k^n > (k+1)^n).

Cf. A332065 (all k such that k^n is a sum of distinct n-th powers).

Cf. A332101 (least k such that k^n <= sum of all smaller n-th powers).

Sequence in context: A177554 A316212 A192485 * A198513 A113371 A280570

Adjacent sequences:  A332072 A332073 A332074 * A332098 A332099 A332101

KEYWORD

nonn

AUTHOR

M. F. Hasler, May 07 2020

STATUS

approved

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Last modified July 5 19:02 EDT 2020. Contains 335473 sequences. (Running on oeis4.)