A271384 Least k with precisely n partitions k = x + y satisfying phi(k) = phi(x) + phi(y), where phi(k) is the Euler totient function of k.
3, 14, 20, 28, 44, 92, 112, 224, 266, 260, 404, 380, 476, 552, 558, 696, 860, 984, 846, 1062, 1388, 1128, 1278, 1752, 1494, 1422, 2034, 1926, 1704, 1992, 2358, 2466, 2712, 2424, 2718, 3222, 3006, 3258, 4924, 3288, 3582, 4296, 3798, 4008, 4518, 5688, 5094, 5352
Offset: 1
Examples
phi(28) = phi(6) + phi(22) = phi(8) + phi(20) = phi(12) + phi(16) = phi(14) + phi(14) = 12 and 28 is the least number with 4 partitions of two numbers with this property: therefore a(4) = 28; phi(112) = phi(14) + phi(98) = phi(24) + phi(88) = phi(30) + phi(82) = phi(32) + phi(80) = phi(36) + phi(76) = phi(48) + phi(64) = phi(56) + phi(56) = 48 and 112 is the least number with 7 partitions of two numbers with this property: therefore a(7) = 112.
Links
- Paolo P. Lava, Table of n, a(n) for n = 1..50
Programs
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Maple
with(numtheory): P:=proc(q) local a,h,k,n; for h from 1 to q do for n from 2*h to q do a:=0; for k from 1 to trunc(n/2) do if phi(n)=phi(k)+phi(n-k) then a:=a+1; fi; od; if a=h then print(n); break; fi; od; od; end: P(10^9);
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Mathematica
Table[SelectFirst[Range[10 + 5 n^2], Function[k, With[{e = EulerPhi@ k}, Count[Transpose@ {Range[k - 1, Ceiling[k/2], -1], Range@ Floor[k/2]}, x_ /; Total@ EulerPhi@ x == e] == n]]], {n, 25}] (* Michael De Vlieger, Apr 06 2016, Version 10 *)