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A108348
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Numbers of the form p^k + p^(k-1) + ... + p + 1 (where p is a prime and k>=0) in ascending order.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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a(2)=3=2+1 since a(1)=1 and 2 is not expressible in the required form.
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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