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The smallest number k that can be partitioned in n ways as the sum of two numbers from A003601.
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%I #5 Feb 20 2024 10:29:24

%S 1,2,6,12,18,20,26,28,38,34,42,48,44,50,52,58,62,60,68,80,74,76,88,84,

%T 92,129,98,110,100,114,108,122,134,116,144,132,171,138,140,146,148,

%U 160,174,162,182,156,170,172,184,198,200,180,206,188,257,253,196,204

%N The smallest number k that can be partitioned in n ways as the sum of two numbers from A003601.

%e a(0) = 1 because 1 cannot be written as the sum of two terms in A003601.

%e 2 = 1 + 1 = A003601(1) + A003601(1), so a(1) = 2.

%e The numbers 3 and 5 cannot be written as the sum of two terms in A003601, and 4 = 1 + 3 = A003601(1) + A003601(2) is the only writing with terms in A003601. The number 6 can be written 6 = 1 + 5 = A003601(1) + A003601(3) and 6 = 3 + 3 = A003601(2) + A003601(2), so a(2) = 6.

%o (Magma) f:=func<n|DivisorSigma(1,n) mod DivisorSigma(0,n) eq 0>; b:=[k: k in [1..1000] |f(n)]; a:=[]; for n in [0..60] do k:=1; while #RestrictedPartitions(k,2,Set(b)) ne n do k:=k+1; end while; Append(~a,k); end for; a;

%Y Cf. A000005, A000203, A003601.

%K nonn

%O 0,2

%A _Marius A. Burtea_, Jan 25 2024