A065294 -id:A065294 - cp's OEIS frontend

A065389 Numbers m such that sigma(phi(m)) sets a new record, i.e., sigma(phi(m)) > sigma(phi(k)) for all k < m numbers.

1, 3, 5, 7, 11, 13, 17, 19, 25, 29, 31, 37, 43, 53, 61, 73, 97, 109, 127, 143, 151, 157, 181, 211, 241, 313, 331, 337, 397, 403, 421, 527, 541, 601, 631, 661, 757, 779, 899, 1009, 1147, 1201, 1321, 1333, 1517, 1621, 1763, 1801, 2017, 2161, 2341, 2501, 2521

Offset: 1

Labos Elemer, Nov 05 2001

nonn

Numbers m at which A000203(A000010(m)) = A062402(m) reaches a new maximal value.

			A062402 begins {1, 1, 3, 3, 7, 3, 12, 7, 12, 7, 18, 7, 28, 12, 15, 15, 31, 12, 39, 15, 28, 18, 36, 15, 42}. New peaks are reached at positions 3, 5, 7, 11, 13, 17, 19, 25. These peak values are 3, 7, 12, 18, 28, 31, 39, 42, respectively.

Amiram Eldar, Table of n, a(n) for n = 1..958 (terms below 10^11; terms 1..500 from Harry J. Smith)

Cf. A000010, A000203, A062402, A065390, A065391, A065392, A065393, A065294.

a=0; s=0; Do[s=DivisorSigma[1, EulerPhi[n]]; If[s>a, a=s; Print[n]], {n, 1, 10000}];
DeleteDuplicates[Table[{n,DivisorSigma[1,EulerPhi[n]]},{n,2600}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Jul 20 2023 *)

{ n=r=0; for (m=1, 10^9, x=sigma(eulerphi(m)); if (x > r, r=x; write("b065389.txt", n++, " ", m); if (n==500, return)) ) } \\ Harry J. Smith, Oct 18 2009

A065293 Number of values of s, 0 <= s <= n-1, such that 2^s mod n = s.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 2, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 0, 3, 0, 0, 1, 1, 2, 0, 1, 2, 1, 0, 2, 0, 2, 0, 1, 1, 1, 1, 0, 2, 1, 0, 0, 0, 2, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 28 2001

nonn

			For n=5 we have (2^0) mod 5 = 1, (2^1) mod 5 = 2, (2^2) mod 5 = 4, (2^3) mod 5 = 3, (2^4) mod 5 = 1. Only for s=3 does (2^s) mod 5=s, so a(5)=1

Michel Marcus, Table of n, a(n) for n = 1..1000

Cf. A065294.

Table[Count[Range[0, n - 1], ?(Mod[2^#, n] == # &)], {n, 105}] (* _Michael De Vlieger, Jun 19 2018 *)
Table[Count[Range[0,n-1],?(PowerMod[2,#,n]==#&)],{n,110}] (* _Harvey P. Dale, Aug 02 2024 *)

a(n) = sum(s=0, n-1, Mod(2, n)^s == s); \\ Michel Marcus, Jun 19 2018

a(1) corrected by Michel Marcus, Jun 20 2018

cp's OEIS Frontend

A065389 Numbers m such that sigma(phi(m)) sets a new record, i.e., sigma(phi(m)) > sigma(phi(k)) for all k < m numbers.

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