A083503 Basis of the n-th power arising in A083502.

3, 3, 4, 3, 6, 5, 8, 3, 4, 9, 12, 5, 14, 13, 16, 3, 18, 5, 20, 3, 4, 21, 24, 5, 6, 25, 4, 13, 30, 11, 32, 3, 34, 33, 36, 5, 38, 37, 16, 3, 42, 5, 44, 21, 16, 45, 48, 5, 8, 9, 52, 5, 54, 5, 16, 13, 7, 57, 60, 7, 62, 61, 4, 3, 66, 23, 68, 13, 70, 29, 72, 5, 74, 73, 16, 37, 78, 17, 80, 3, 4

Offset: 1

Author

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

Keywords

• nonn

Comments

For n > 1, a(n) = A074792(n) is the least solution > 1 of x^n == 1 (mod n). - Robert Israel, Aug 01 2025

Examples

			2*(2+2)+1=3^2; 3*(3+18)+1=4^3; 4*(4+16)+1=3^4; 5*(5+1550)+1=6^5; 6*(6+2598)+1=5^6; 7*(7+299586)+1=8^7; 8*(8+812)+1=3^8; 9*(9+29118)+1=4^9; 10*(10+348678430)+1=9^10. - _R. J. Mathar_, Aug 01 2025
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A083502, A074792.

Programs

Maple

A083503 := proc(n)
    local a,b ;
    if n = 1 then
        3 ;
    else
        for b from 2 do
            a := (b^n-1)/n-n ;
            if type( a,'integer') then
                return  b;
            end if;
        end do:
    end if;
end proc:
seq(A083503(n),n=1..80) ; # 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 else S fi
end proc:
f(1):= 3:
map(f, [$1..100]); # Robert Israel, Aug 01 2025

Mathematica

Do[i = 2; While[k = (i^n - 1)/n - n; !IntegerQ[k], i++ ]; Print[i], {n, 2, 81}]

Formula

n*(n + A083502(n)) + 1 = a(n)^n. - R. J. Mathar, Aug 01 2025

Extensions

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