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A386871 Least k such that A056100(k) = n or -1 if no such k exists.

%I A386871 #31 Aug 11 2025 10:21:21
%S A386871 6,124357252657,4,33,8,145,9,37063859,16,51,26,1441,15,2353,34,69,20,
%T A386871 1011377103546119,27,7201,25,87,115,9911837,56,385,58,45,62,86125529,
%U A386871 57,30721,35,123,74,295,90,15686608811,82,141,86,70561,49,77857739,94,159,329,34884199
%N A386871 Least k such that A056100(k) = n or -1 if no such k exists.
%C A386871 Wanted terms: a(18), a(38), a(72), a(80).
%C A386871 a(38) = 15686608811 (found by _Tomas Rokicki_), a(80) = 4222433393407, and a(18), a(72) > 6*10^12. - _Giovanni Resta_, Aug 09 2025
%C A386871 a(72) > a(18). - _Martin Ehrenstein_, Aug 10 2025
%H A386871 Michel Marcus, <a href="https://groups.google.com/g/seqfan/c/0VzC7nuSFE0">Extension to A056100</a>, Seqfan, Aug 05 2025.
%H A386871 Neil Sloane, <a href="https://mailman.xmission.com/hyperkitty/list/math-fun@mailman.xmission.com/thread/ZV7MW32PAC32AG6JMYJ3LR4FDE3KK2VR/">Smallest inverse to f(n) = phi(n)*sigma(n) + 1 mod n ?</a>, Math Fun, Aug 05 2025.
%t A386871 a[n_]:=Module[{k=0},Until[Mod[DivisorSigma[1, k]*EulerPhi[k] + 1, k]==n,k++];k] (* _James C. McMahon_, Aug 06 2025 *)
%Y A386871 Cf. A056100, A386856 (odd bisection), A015706.
%K A386871 nonn,hard
%O A386871 1,1
%A A386871 _Michel Marcus_, Aug 06 2025
%E A386871 a(2) from _Tomas Rokicki_, Aug 05 2025
%E A386871 a(18) from _Martin Ehrenstein_, Aug 10 2025
%E A386871 a(19)-a(37) from _Hugo Pfoertner_, Aug 11 2025
%E A386871 a(38) from _Tomas Rokicki_, Aug 09 2025
%E A386871 More terms from _Hugo Pfoertner_, Aug 11 2025