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The minimal value of A001414(i) where prime(n) < i < prime(n+1).
1

%I #20 Dec 17 2016 12:10:17

%S 4,5,6,7,8,8,9,9,10,10,11,12,11,11,11,12,12,14,12,13,12,14,13,14,22,

%T 15,13,15,14,14,14,28,14,15,20,14,19,16,17,15,16,15,18,19,16,15,16,26,

%U 21,21,16,15,16,22,20,16,21,18,52,16,17,22,22,18,16,18,21

%N The minimal value of A001414(i) where prime(n) < i < prime(n+1).

%H Lei Zhou, <a href="/A239440/b239440.txt">Table of n, a(n) for n = 2..10001</a>

%e At n = 2, the second and third prime numbers are 3 and 5. 4 is the only number between them. 4=2^2, 2*2=4. So a(1) = 4;

%e ...

%e At n = 11, the 11th and 12th prime numbers are 31 and 37. Testing from 32 to 36:

%e 32 = 2^5, sum of prime factors = 2*5 = 10;

%e 33 = 3*11, sum of prime factors = 3+11 = 14;

%e 34 = 2*17, sum of prime factors = 2+17 = 19;

%e 35 = 5*7, sum of prime factors = 5+7 = 12;

%e 36 = 2^2*3^2, sum of prime factors = 2*2+3*2 = 10;

%e The smallest sum of prime factors of these numbers is 10. So a(11) = 10.

%t Table[p1 = Prime[n]; p2 = Prime[n + 1]; a = p2; Do[f = FactorInteger[i]; l = Length[f]; sum = 0; Do[sum = sum + f[[j, 1]]*f[[j, 2]], {j, 1, l}]; If[sum < a, a = sum], {i, p1 + 1, p2 - 1}]; a, {n, 2, 68}]

%Y Cf. A000040, A001414, A239439.

%K nonn,easy

%O 2,1

%A _Lei Zhou_, Mar 18 2014