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A113117 a(1) = 2; for n>1, a(n) is the smallest integer > a(n-1) such that all primes <= a(n-1) divide at least one integer k for a(n-1) < k <= a(n).

%I A113117 #23 Feb 25 2024 10:49:16
%S A113117 2,4,6,10,15,26,46,86,166,326,634,1262,2518,5006,10006,19946,39874,
%T A113117 79738,159398,318778,637502,1274998,2549978,5099902,10199786,20399534,
%U A113117 40799062,81598082,163196134,326392258,652784498,1305568942,2611137838,5222275634,10444551254
%N A113117 a(1) = 2; for n>1, a(n) is the smallest integer > a(n-1) such that all primes <= a(n-1) divide at least one integer k for a(n-1) < k <= a(n).
%C A113117 It appears that A113118 and this sequence agree except for the 5th term.
%e A113117 The primes <= a(5) = 15 are 2, 3, 5, 7, 11 and 13. So a(6) is the smallest integer >15 such that each prime <= 15 divides at least one integer between 16 and a(6). Setting a(6) to 26 = 2*13 satisfies the conditions. (2 divides 16. 3 divides 18. 5 divides 20. 7 divides 21. 11 divides 22. And 13 divides 26.)
%o A113117 (PARI) {m=21; print1(a=2, ", "); for(n=2, m, nxt=a+1; forprime(p=1, a, k=a+1; while(k%p>0, k++); nxt=max(nxt, k)); print1(a=nxt, ", "))} \\ _Klaus Brockhaus_, Jan 07 2006
%o A113117 (Python)
%o A113117 from sage.all import primes
%o A113117 n, a = 1, 2
%o A113117 data = ""
%o A113117 while True:
%o A113117     print(n, a)
%o A113117     data += str(a) + ", "
%o A113117     if len(data) > 262: print(data[:-2]); break
%o A113117     n += 1
%o A113117     a = max((a//p) * p + p for p in list(primes(a+1)))
%o A113117 # _Lucas A. Brown_, Feb 22 2024
%Y A113117 Cf. A113118.
%K A113117 nonn
%O A113117 1,1
%A A113117 _Leroy Quet_, Jan 03 2006
%E A113117 a(8)-a(21) from _Klaus Brockhaus_, Jan 07 2006
%E A113117 a(22)-a(35) from _Lucas A. Brown_, Feb 22 2024