A065392 -id:A065392 - cp's OEIS frontend

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.

1, 2, 4, 8, 9, 16, 32, 36, 64, 100, 144, 256, 324, 400, 576, 900, 1296, 1600, 2304, 2916, 3600, 5184, 8100, 9216, 11664, 14400, 20736, 22500, 25600, 30276, 32400, 41616, 46656, 57600, 69696, 72900, 82944, 90000, 104976, 115600, 121104, 129600

Offset: 1

Labos Elemer, Nov 05 2001

nonn

The terms > 2 are exact powers and except for 2, 8 and 32 all the terms seem to be squares.

Indices of records in A062401. - Michael De Vlieger, Dec 06 2018

			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.

Amiram Eldar, Table of n, a(n) for n = 1..249 (terms below 10^11; terms 1..100 from Harry J. Smith)

Cf. A000010, A000203, A062401, A062402, A065392.

a = 0; s = 0; Do[s = EulerPhi[DivisorSigma[1, n]]; If[s > a, a = s; Print[n]], {n, 1, 10^6}]
(* Second program: *)
With[{s = Array[EulerPhi@ DivisorSigma[1, #] &, 2*10^5]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Dec 06 2018 *)
DeleteDuplicates[Table[{n,EulerPhi[DivisorSigma[1,n]]},{n,150000}],GreaterEqual[ #1[[2]],#2[[2]]]&] [[;;,1]] (* Harvey P. Dale, May 12 2023 *)

{ 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

A385729 Number of nonnegative values s < n such that (-s) == (-s)^s == s^s (mod n).

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 3, 3, 1, 0, 1, 2, 2, 1, 3, 3, 1, 1, 1, 4, 1, 1, 2, 4, 1, 0, 2, 2, 4, 1, 2, 3, 3, 1, 1, 4, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 2, 3, 1, 0, 3, 5, 2, 1, 2, 6, 3, 1, 1, 4, 3, 1, 3, 4, 2, 1, 1, 3, 2, 3, 3, 4, 1, 1, 1, 4, 4, 2, 3, 3, 2, 1, 1, 2, 3, 1

Offset: 1

Juri-Stepan Gerasimov, Jul 08 2025

nonn

Cf. A065392, A384781, A385540, A385731.

[1] cat [#[s: s in [0..n-1] | Modexp(s, s, n) eq n-s and Modexp(-s, s, n) eq n-s]: n in [2..100]];

a[n_] := Count[Range[0, n-1], ?(PowerMod[#, #, n] == PowerMod[-#, #, n] == Mod[-#, n] &)]; Array[a, 100] (* _Amiram Eldar, Jul 15 2025 *)

a(n) = sum(s=0, n-1, (-s == Mod(s, n)^s) && (-s == Mod(-s, n)^s)); \\ Michel Marcus, Jul 09 2025

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.

Original entry on oeis.org

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

A065390 Peak values reached by A062402 at the sites listed in A065389.

Original entry on oeis.org

1, 3, 7, 12, 18, 28, 31, 39, 42, 56, 72, 91, 96, 98, 168, 195, 252, 280, 312, 360, 372, 392, 546, 576, 744, 840, 864, 992, 1092, 1170, 1344, 1512, 1680, 1860, 1872, 2016, 2240, 2418, 2880, 3224, 3600, 3844, 4320, 4368, 4914, 5082, 5952, 6045, 6552, 7440

Offset: 1

Labos Elemer, Nov 05 2001

nonn

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

Cf. A062402, A000203, A000010, A065389, A065391, A065392, A065393, A065394.

a=0; s=0; Do[s=DivisorSigma[1, EulerPhi[n]]; If[s>a, a=s; Print[s]], {n, 1, 10000}]
(* Second program: *)
Union@ FoldList[Max, Array[DivisorSigma[1, EulerPhi[#]] &, 2200]] (* Michael De Vlieger, Jun 19 2018 *)

{ n=r=0; for (m=1, 10^9, x=sigma(eulerphi(m)); if (x > r, r=x; write("b065390.txt", n++, " ", x); if (n==500, return)) ) } \\ Harry J. Smith, Oct 18 2009

a(n) = A062402(A065389(n)). - Amiram Eldar, Mar 22 2025

cp's OEIS Frontend

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.

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A385729 Number of nonnegative values s < n such that (-s) == (-s)^s == s^s (mod n).

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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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A065390 Peak values reached by A062402 at the sites listed in A065389.

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