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

2, 3, 7, 43, 1811, 654149, 27082315109, 153694141992520880899, 337110658273917297268061074384231117039, 8424197597064114319193772925959967322398440121059128471513803869133407474043

Offset: 1

Author

T. D. Noe, Sep 16 2002

Keywords

• nice
• nonn

Comments

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.

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

{a(n)} = 2, 3, 7, ..., so A225671(1) = 3. - Jonathan Sondow, May 13 2013

References

• R. K. Guy, Unsolved Problems in Number Theory, D11.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..14
• K. S. Brown, Odd, Greedy and Stubborn (Unit Fractions)
• Eric Weisstein's World of Mathematics, Egyptian Fraction

Crossrefs

Cf. A000058, A225669, A225671.

Programs

Mathematica

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
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 *)

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

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