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A065391 Numbers m such that A062401(m) = phi(sigma(m)) is increasing to a record value, i.e., A062401(m) represents a new peak, so that A062401(m) > A062401(k) for all k < m.

%I A065391 #32 Mar 22 2025 04:41:48
%S A065391 1,2,4,8,9,16,32,36,64,100,144,256,324,400,576,900,1296,1600,2304,
%T A065391 2916,3600,5184,8100,9216,11664,14400,20736,22500,25600,30276,32400,
%U A065391 41616,46656,57600,69696,72900,82944,90000,104976,115600,121104,129600
%N A065391 Numbers m such that A062401(m) = phi(sigma(m)) is increasing to a record value, i.e., A062401(m) represents a new peak, so that A062401(m) > A062401(k) for all k < m.
%C A065391 The terms > 2 are exact powers and except for 2, 8 and 32 all the terms seem to be squares.
%C A065391 Indices of records in A062401. - _Michael De Vlieger_, Dec 06 2018
%H A065391 Amiram Eldar, <a href="/A065391/b065391.txt">Table of n, a(n) for n = 1..249</a> (terms below 10^11; terms 1..100 from Harry J. Smith)
%e A065391 Initial segment of A062401: {1, 2, 2, 6, 2, 4, 4, 8, 12, 6, 4, 12, 6, 8, 8, 30, 6, ...}. The peak values (those exceeding all previous ones) are 1, 2, 6, 8, 12, 30, reached at positions 1, 2, 4, 8, 9, 16, respectively.
%t A065391 a = 0; s = 0; Do[s = EulerPhi[DivisorSigma[1, n]]; If[s > a, a = s; Print[n]], {n, 1, 10^6}]
%t A065391 (* Second program: *)
%t A065391 With[{s = Array[EulerPhi@ DivisorSigma[1, #] &, 2*10^5]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* _Michael De Vlieger_, Dec 06 2018 *)
%t A065391 DeleteDuplicates[Table[{n,EulerPhi[DivisorSigma[1,n]]},{n,150000}],GreaterEqual[ #1[[2]],#2[[2]]]&] [[;;,1]] (* _Harvey P. Dale_, May 12 2023 *)
%o A065391 (PARI) { n=r=0; for (m=1, 10^9, x=eulerphi(sigma(m)); if (x > r, r=x; write("b065391.txt", n++, " ", m); if (n==100, return)) ) } \\ _Harry J. Smith_, Oct 18 2009
%Y A065391 Cf. A000010, A000203, A062401, A062402, A065392.
%K A065391 nonn
%O A065391 1,2
%A A065391 _Labos Elemer_, Nov 05 2001