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%I #29 Feb 22 2020 20:54:24
%S 1,2,2,2,2,3,2,2,2,3,2,3,3,3,2,2,2,3,2,3,3,5,2,3,3,3,3,3,3,3,2,2,2,3,
%T 2,3,3,3,2,3,3,3,3,3,5,5,2,3,3,3,3,5,3,3,3,3,3,3,3,5,3,3,2,2,2,3,2,5,
%U 3,7,2,3,3,3,3,5,3,5,2,3,3,3,3,3,3,3,3
%N a(n) is the minimum prime (or 1) needed to write integer n into the form n = a + b such that all prime factors of a and b are smaller or equal to a(n).
%C Any integer n that is greater than 1 can be written into the sum of two other positive integers, such that n = a + b. There are IntegerPart[n / 2] ways to do this assuming a <= b. For each of the ways, we can have a set of prime factor of a and b, defined as sa = FactorSet[a] and sb = FactorSet[b], quoting the function in the Mathematica program. Then we can define a union set s=Union[sa, sb], which is a list of prime factors that can factor either a or b. In this way we obtain IntegerPart[n / 2] of possible set s. Define p_i is the largest prime number in each of set s_i, i = 1,2...IntegerPart[n / 2], a(n) = the smallest s_i.
%C Though 2 = 1 + 1 and 1 is not a prime number, a(2) can still be defined as 1.
%C The Mathematica program generates up to term 88.
%C The first occurrence of a(n)=k forms sequence A000229. - _Lei Zhou_, Feb 06 2014
%H T. D. Noe, <a href="/A196941/b196941.txt">Table of n, a(n) for n = 2..10000</a>
%e n = 3, 3 = 1 + 2, the largest prime factor of 1 and 2 is 2, so a[3] = 2;
%e n = 4, 4 = 2 + 2, the largest prime factor of 2 and 2 is 2, so a[4] = 2;
%e [in 4 = 1 + 3, the largest prime factor of 1 and 3 is 3, which is larger than a[4] = 2]
%e ...
%e n = 23, 23 = 3 + 20 = 3 + 2^2 * 5, the largest prime factor of 3 and 20 is 5, so a[23] = 5;
%t FactorSet[seed_] := Module[{fset2, a, l, i}, a = FactorInteger[seed]; l = Length[a]; fset2 = {}; Do[fset2 = Union[fset2, {a[[i]][[1]]}], {i, 1, l}]; fset2]; Table[min = n; Do[r = n - k; s = Union[FactorSet[k], FactorSet[r]]; If[a = s[[Length[s]]]; a < min, min = a], {k, 1, IntegerPart[n/2]}]; min, {n, 2, 88}]
%t LPF[n_] := FactorInteger[n][[-1,1]]; Table[Min[Table[Max[{LPF[i], LPF[n-i]}], {i, Floor[n/2]}]], {n, 2, 100}] (* _T. D. Noe_, Oct 07 2011 *)
%Y Cf. A173786 (n for which a(n)=2), A196526, A000229.
%K nonn,easy
%O 2,2
%A _Lei Zhou_, Oct 07 2011
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