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Lexicographically earliest sequence of positive numbers in which no set of consecutive terms sums to a prime.
6

%I #26 Jan 21 2023 19:43:28

%S 1,8,1,15,9,1,14,6,30,6,9,15,6,4,8,12,10,14,6,12,8,10,12,18,12,6,6,6,

%T 24,6,6,8,1,9,6,10,8,12,6,14,10,6,4,8,12,10,20,6,18,6,6,4,8,12,6,4,12,

%U 8,10,8,6,6,18,6,6,20,10,12,8,4,6,12,12,6,12,6,12

%N Lexicographically earliest sequence of positive numbers in which no set of consecutive terms sums to a prime.

%C Terms >= 30 seem to be very rare. Up to a(450000), 30 appears only 7 times: at n = 9, 288, 2507, 15902, 54405, 242728, 425707.

%C For n <= 450000, the largest term is 32; it appears at n = 335308 and 370687.

%H Alois P. Heinz, <a href="/A332941/b332941.txt">Table of n, a(n) for n = 1..10000</a>

%p s:= proc(i, j) option remember; `if`(i>j, 0, a(j)+s(i, j-1)) end:

%p a:= proc(n) option remember; local k; for k while

%p ormap(isprime, [k+s(i, n-1)$i=1..n]) do od; k

%p end:

%p seq(a(n), n=1..100); # _Alois P. Heinz_, Mar 23 2020

%t s[i_, j_] := s[i, j] = If[i > j, 0, a[j] + s[i, j-1]];

%t a[n_] := a[n] = Module[{k}, For[k = 1, AnyTrue[k+Table[s[i, n-1], {i, 1, n}], PrimeQ], k++]; k];

%t Array[a, 100] (* _Jean-François Alcover_, Nov 17 2020, after _Alois P. Heinz_ *)

%o (Python)

%o def A(ee):

%o a=[1]

%o print(1)

%o n=1

%o while n<=ee:

%o i=1

%o while i>0:

%o ii=i

%o iz=c=0

%o while iz<=len(a):

%o c=0

%o if ii>2:

%o for j in range(2, int((ii)**0.5+1.5)):

%o if ii%j==0:

%o c=1

%o break

%o if c==0 and ii>1:

%o break

%o else:

%o iz += 1

%o ii=ii+a[n-iz]

%o if c==1:

%o n += 1

%o a.append(i)

%o print(i)

%o break

%o if i<4:

%o i=4

%o else:

%o i += 1

%o return a

%Y Cf. A254337, A084833.

%K nonn

%O 1,2

%A _S. Brunner_, Mar 03 2020