This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A381767 #40 May 25 2025 21:26:21
%S A381767 1,4,6,7,8,9,10,11,12,13,14,16,17,18,19,20,21,22,23,24,25,26,27,28,29,
%T A381767 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,47,48,49,50,51,52,53,
%U A381767 54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71
%N A381767 a(n) = ceiling(n^(n/(n-1))) with a(1)=1.
%C A381767 a(n) is the smallest positive integer expressible as both the sum and product of the same set of n positive real numbers.
%C A381767 For a given n>1, the smallest possible sum (which is also the product) is achieved when all n real numbers are equal. Restricting to positive real numbers, let each number be c, then: n*c = c^n. Solving for c, we get: c=n^(1/(n-1)). Substituting this back into the sum (product) equation, n*c=c^n = n^(n/(n-1)). Thus, the smallest integer satisfying this condition is a(n) = ceiling(n^(n/(n-1))).
%C A381767 The growth of n^(n/(n-1)) means some integers are skipped in the sequence, see A382131 for those.
%F A381767 n^(n/(n-1)) = n + log(n) + O(log(n)^2)/n. - _Yifan Xie_, May 20 2025
%e A381767 For n=1 the equation 1*c=c^1 is true for all values of c. We define a(1)=1 as for any positive integer equal to or greater than 1 we have x=x (singular sum equals singular product).
%e A381767 For n=4, solving 4*c=c^4 with positive c yields c~1.5874, giving 4*c=c^4~6.3496. Any integer greater than 6.3496 allows for a multiset of n positive real numbers that sum to and multiply to that integer. Hence, the smallest integer satisfying the condition is a(4)=7.
%e A381767 A constructive procedure is to let {c1,c2} be two numbers such that c1 appears once and c2 appears (n-1) times, then c1+(n-1)*c2 = c1*c2^(n-1)=d can be solved numerically for {c1,c2} for a given {n,d}. Multiple real solutions appear.
%e A381767 a(4)=7
%e A381767 7 : {3.0373, 1.3209, 1.3209, 1.3209} or {0.788707, 2.07043, 2.07043, 2.07043}
%e A381767 100 : {96.9691, 1.01031, 1.01031, 1.01031} or {0.00270022, 33.3324, 33.3324, 33.3324}
%t A381767 Prepend[Table[Ceiling[n^(n/(n - 1))], {n, 2, 100}], 1]
%o A381767 (Python)
%o A381767 from sympy import integer_nthroot
%o A381767 def A381767(n): return (lambda x: x[0]+(not x[1]))(integer_nthroot(n**n,n-1)) if n>1 else 1 # _Chai Wah Wu_, Apr 01 2025
%o A381767 (PARI) a(n) = if (n==1, 1, ceil(n^(n/(n-1)))); \\ _Michel Marcus_, Apr 05 2025
%Y A381767 Cf. A382131 (complement).
%K A381767 nonn
%O A381767 1,2
%A A381767 _Mike Sheppard_, Mar 06 2025