A108348 Numbers of the form p^k + p^(k-1) + ... + p + 1 (where p is a prime and k>=0) in ascending order.

1, 3, 4, 6, 7, 8, 12, 13, 14, 15, 18, 20, 24, 30, 31, 32, 38, 40, 42, 44, 48, 54, 57, 60, 62, 63, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 121, 127, 128, 132, 133, 138, 140, 150, 152, 156, 158, 164, 168, 174, 180, 182, 183, 192, 194, 198, 200

Offset: 1

Author

Franz Vrabec, Jul 01 2005

Keywords

• nonn

Comments

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

Goormaghtigh conjecture implies that 31 is the only term with 2 representations; see the Wikipedia link below. - Jianing Song, Nov 22 2022

Examples

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

Links

• M. F. Hasler, Table of n, a(n) for n = 1..1000
• Wikipedia, Goormaghtigh conjecture

Crossrefs

Cf. A002191.

Cf. A000040, A090503.

Programs

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

Haskell

a108348 n = a108348_list !! (n-1)
a108348_list = 1 : f [2..] where
   f (x:xs) = g a000040_list where
     g (p:ps) = h 0 $ map ((`div` (p - 1)) . subtract 1) $
                          iterate (* p) (p ^ 2) where
       h i (pp:pps) | pp > x    = if i == 0 then f xs else g ps
                    | pp < x    = h 1 pps
                    | otherwise = x : f xs
-- Reinhard Zumkeller, Nov 26 2013

PARI

A108348(n)={ local(m=1, a=[m]); while( #aA108348(1000) \\ M. F. Hasler