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A060646 Bonse sequence: a(n) = minimal j such that n-j+1 < prime(j).

%I A060646 #24 Aug 09 2022 14:07:09
%S A060646 1,2,2,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,7,7,8,8,8,9,9,9,9,9,10,10,10,
%T A060646 10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,14,14,14,15,
%U A060646 15,15,15,15,16,16,16,16,16,16,16,17,17,17,17,17,17,17,18,18
%N A060646 Bonse sequence: a(n) = minimal j such that n-j+1 < prime(j).
%C A060646 For 3<n and any a(n-1)<a(n) use a(n)=a(n+1)=a(n+2) to show prime(j+1)^3 < prime(1)*...*prime(j) for j>5.
%D A060646 R. Remak, Archiv d. Math. u. Physik (3) vol. 15 (1908) 186-193
%H A060646 Reinhard Zumkeller, <a href="/A060646/b060646.txt">Table of n, a(n) for n = 1..10000</a>
%H A060646 H. Bonse, <a href="https://archive.org/stream/archivdermathem31unkngoog#page/n307/mode/2up">Über eine bekannte Eigenshaft der Zahl 30 und ihre Verallgemeinerung</a>, Archiv d. Math. u. Physik (3) vol. 12 (1907) 292-295.
%H A060646 H. Rademacher and O. Toeplitz, <a href="http://dx.doi.org/10.1007/978-3-662-36239-6_26">Eine Eigenschaft der Zahl 30</a>, (A property of the number 30), Von Zahlen und Figuren (1930, reprint Springer 1968), ch. 22.
%e A060646 For n=5, j=3 gives 5-3+1 = 3 < prime(3) = 5, true; but if j=2 we get 5-2+1 = 4 which is not < prime(2) = 3; hence a(5) = 3.
%e A060646 a(75)=18 because 75-18+1=58 < 61=prime(18), but 75-17+1=59=prime(17).
%t A060646 Table[j=0; While[j++; n-j+1 >= Prime[j]]; j, {n, 1, 76}] (* _Jean-François Alcover_, Aug 30 2011 *)
%o A060646 (Haskell)
%o A060646 import Data.List (findIndex)
%o A060646 import Data.Maybe (fromJust)
%o A060646 a060646 n = (fromJust $ findIndex ((n+1) <) a014688_list) + 1
%o A060646 -- _Reinhard Zumkeller_, Sep 16 2011
%o A060646 (Python)
%o A060646 from sympy import nextprime
%o A060646 from itertools import count, islice
%o A060646 def agen(): # generator of terms
%o A060646     n, pj = 1, 2
%o A060646     for j in count(1):
%o A060646         while n - j + 1 < pj: yield j; n += 1
%o A060646         pj = nextprime(pj)
%o A060646 print(list(islice(agen(), 76))) # _Michael S. Branicky_, Aug 09 2022
%Y A060646 Cf. A014688.
%K A060646 easy,nonn,nice
%O A060646 1,2
%A A060646 _Frank Ellermann_, Apr 17 2001