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Successive coprime numbers with distinct successive differences: gcd(a(k+1),a(k)) = gcd(a(m+1),a(m)) = 1 and a(k+1)-a(k) = a(m+1)-a(m) <==> m=k.
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%I #17 Jan 11 2020 00:33:57

%S 1,2,5,7,11,16,23,29,37,46,57,67,79,92,107,121,137,154,173,191,211,

%T 232,255,277,301,326,353,379,407,436,467,497,529,562,597,631,667,704,

%U 743,781,821,862,905,947,991,1036,1083,1129,1177,1226,1277

%N Successive coprime numbers with distinct successive differences: gcd(a(k+1),a(k)) = gcd(a(m+1),a(m)) = 1 and a(k+1)-a(k) = a(m+1)-a(m) <==> m=k.

%C Conjecture: If a(k+1)-a(k) = n then k < C*n. Can someone find an estimate for the constant C?

%H Reinhard Zumkeller, <a href="/A089781/b089781.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n+1) = a(n) + (smallest number coprime with a(n) and not already added). - _Reinhard Zumkeller_, Aug 15 2015

%e 5 follows 2 as 4 is not coprime to 2 and 5-2 = 3, 2-1 = 1.

%o (Haskell)

%o import Data.List (delete)

%o a089781 n = a089781_list !! (n-1)

%o a089781_list = 1 : f [1..] 1 where

%o f xs y = g xs where

%o g (z:zs) = if gcd y z == 1 then y' : f (delete z xs) y' else g zs

%o where y' = y + z

%o -- _Reinhard Zumkeller_, Aug 15 2015

%Y Cf. A116966 (first differences), A111244.

%K nonn

%O 1,2

%A _Amarnath Murthy_, Nov 24 2003

%E More terms from _Sean A. Irvine_, Jun 01 2011