cp's OEIS Frontend

A075442 Slowest-growing sequence of primes whose reciprocals sum to 1.

%I A075442 #24 Feb 16 2025 08:32:47
%S A075442 2,3,7,43,1811,654149,27082315109,153694141992520880899,
%T A075442 337110658273917297268061074384231117039,
%U A075442 8424197597064114319193772925959967322398440121059128471513803869133407474043
%N A075442 Slowest-growing sequence of primes whose reciprocals sum to 1.
%C A075442 This sequence was mentioned by K. S. Brown. The sequence is generated by a greedy algorithm given by the Mathematica program. The sum converges quadratically.
%C A075442 It is easily shown that this sequence is infinite. For suppose there was a finite representation of unity as a sum of unit fractions with distinct prime denominators. Multiply the equation by the product of all denominators to obtain this product of prime numbers on one side of the equation and a sum of products consisting of this product with always exactly one of the prime numbers removed on the other side. Then each of the prime numbers divides one side of the equation but not the other, since it divides all the products added except exactly one. Contradiction. - Peter C. Heinig (algorithms(AT)gmx.de), Sep 22 2006
%C A075442 {a(n)} = 2, 3, 7, ..., so A225671(1) = 3. - _Jonathan Sondow_, May 13 2013
%D A075442 R. K. Guy, Unsolved Problems in Number Theory, D11.
%H A075442 Robert G. Wilson v, <a href="/A075442/b075442.txt"> Table of n, a(n) for n = 1..14 </a>
%H A075442 K. S. Brown, <a href="http://www.mathpages.com/home/kmath454.htm">Odd, Greedy and Stubborn (Unit Fractions)</a>
%H A075442 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%t A075442 x=1; lst={}; Do[n=Ceiling[1/x]; If[PrimeQ[n], n++ ]; While[ !PrimeQ[n], n++ ]; x=x-1/n; AppendTo[lst, n], {10}]; lst
%t A075442 a[n_] := a[n] = Block[{sm = Sum[1/(a[i]), {i, n - 1}]}, NextPrime[ Max[ a[n - 1], 1/(1 - sm)]]]; a[0] = 1; Array[a, 10] (* _Robert G. Wilson v_, Oct 28 2010 *)
%o A075442 (PARI) a(n)=if(n<3, return(prime(n))); my(x=1.); for(i=1,n-1,x-=1/a(i)); nextprime(1/x) \\ _Charles R Greathouse IV_, Apr 29 2015
%o A075442 (PARI) a_vector(N=10)= my(r=1, v=vector(N)); for(i=1, N, v[i]= nextprime(1+1/r); r-= 1/v[i]); v; \\ Ruud H.G. van Tol, Jul 29 2023
%Y A075442 Cf. A000058, A225669, A225671.
%K A075442 nice,nonn
%O A075442 1,1
%A A075442 _T. D. Noe_, Sep 16 2002