1,2

A proper subset of A002191 (e.g. 28 is in A002191, but not in this sequence). a(15)=31 admits two representations: 31=2^4+2^3+2^2+2+1=5^2+5+1. Are there other numbers with two or more representation?

I have checked all the sums of primes up to prime number 56873 to a sum total >= 10^100 and have not come across another number that has multiple representations. - Patrick Schutte (patrick(AT)onyxsa.co.za), Mar 28 2007

M. F. Hasler, Table of n, a(n) for n = 1..1000

a(2)=3=2+1 since a(1)=1 and 2 is not expressible in the required form.

(PARI) A108348(n)={ local(m=1, a=[m]); while( #a<n, m++; forprime(p=2, m, if( m%p==1 && m*(p-1)==p^round(log(m*(p-1))/log(p))-1, a=concat(a, m); next(2)) )); a }; a=A108348(1000) [M. F. Hasler]

(GAP) SumNum := function ( FNum) local a, ap, b, bp, at, bt; a := 2; repeat at := 1; ap := 1; repeat at := at + a^ap; b := 2; repeat bt := 1; bp := 1; repeat bt := bt + b^bp; if at = bt and bp > 1 and a <> b then Print("a ", a, " ap ", ap, " at ", at, " "); Print("b ", b, " bp ", bp, " bt ", bt, " "); Print("---------------- "); fi; bp := bp + 1; until bt > at; b := NextPrime(b); until b >=a; ap := ap + 1; until at > 10^100; a := NextPrime(a); until a >FNum; end; - Patrick Schutte (patrick(AT)onyxsa.co.za), Mar 28 2007

Cf. A002191.

Sequence in context: A066650 A007609 A002191 this_sequence A085149 A007370 A044813

Adjacent sequences: A108345 A108346 A108347 this_sequence A108349 A108350 A108351

nonn

Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Jul 01 2005