A083500 -id:A083500 - cp's OEIS frontend

A083501 Cube roots arising in A083500.

2, 3, 4, 5, 6, 7, 4, 9, 7, 11, 12, 13, 9, 9, 16, 17, 18, 7, 11, 21, 16, 23, 24, 25, 26, 9, 10, 25, 30, 31, 25, 33, 34, 35, 11, 13, 26, 39, 16, 41, 42, 25, 36, 45, 16, 47, 48, 49, 18, 51, 52, 29, 54, 19, 56, 25, 49, 59, 60, 61, 47, 25, 16, 65, 61, 67, 29, 69, 70, 51, 72, 25, 64, 47

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003

nonn

a(n) < n+1 and a(n) = n+1 over half of the time.

Cf. A083500.

Do[k = 0; While[i = n(n + k) + 1; ! IntegerQ[i^(1/3)], k++ ]; Print[i^(1/3)], {n, 1, 75}]

Edited and extended by Robert G. Wilson v, May 11 2003

A102306 Numbers k such that A083500(k) differs from A102305(k).

Original entry on oeis.org

7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 31, 35, 36, 37, 39, 42, 43, 45, 49, 52, 54, 56, 57, 61, 62, 63, 65, 67, 70, 72, 73, 74, 76, 77, 78, 79, 81, 84, 86, 90, 91, 93, 95, 97, 98, 99, 103, 104, 105, 108, 109, 111, 112, 114, 117, 119, 122, 124, 126, 127, 129, 130, 133

Offset: 1

Ralf Stephan, Jan 03 2005

nonn

What is the cardinality of a(n) with respect to its complement?

A083502 Smallest k such that n*(n+k) + 1 is an n-th power.

Original entry on oeis.org

1, 2, 18, 16, 1550, 2598, 299586, 812, 29118, 348678430, 67546215506, 20345040, 61054982557998, 281241170407078, 76861433640456450, 2690404, 128583032925805678334, 211927625850, 275941052631578947368402, 174339200

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003

nonn

Sequence is obviously infinite.
If the sequence is restricted to only prime n's, the sequence increases absolutely. See comment in A083503.
[Since there is actually no comment in A083503: this probably means to say that (conjectural!) A083503(prime(n)) = A008864(n) which leads to a(p) = Sum_{s=2..p} binomial(p,s)*p^(s-1) for primes p, an increasing subsequence. - R. J. Mathar, Aug 01 2025]
a(n) = (x^n-1)/n - n, where x is the least integer > 1 with x^n == 1 (mod n). - Robert Israel, Aug 01 2025

Robert Israel, Table of n, a(n) for n = 1..388

The i's in the above Mathematica coding, except for a(1), give A055670.

Cf. A083500, A083503.

A083502 := proc(n)
    local a,b ;
    if n = 1 then
        1 ;
    else
        for b from 2 do
            a := (b^n-1)/n-n ;
            if type( a,'integer') then
                return  a;
            end if;
        end do:
    end if;
end proc:
seq(A083502(n),n=1..20) ; # R. J. Mathar, Aug 01 2025
# alternative
f:= proc(n) local X,S;
  S:= min(map(t -> subs(t,X), {msolve(X^n = 1, n)} minus {{X=1}}));
  if S = infinity then ((n+1)^n - 1)/n - n else (S^n-1)/n - n fi
end proc:
f(1):= 1:
map(f, [$1..50]); # Robert Israel, Aug 01 2025

Do[i = 2; While[k = (i^n - 1)/n - n; !IntegerQ[k], i++ ]; Print[k], {n, 1, 20}]

Edited and extended by Robert G. Wilson v, May 11 2003

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A083501 Cube roots arising in A083500.

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A102306 Numbers k such that A083500(k) differs from A102305(k).

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A083502 Smallest k such that n*(n+k) + 1 is an n-th power.

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