A065295 -id:A065295 - cp's OEIS frontend

A065296 Values of n such that A065295(n) = 1.

2, 3, 5, 11, 53, 59, 83, 107, 179, 227, 269, 293, 317, 347, 389, 467, 557, 563, 587, 797, 1019, 1109, 1187, 1259, 1283, 1307, 1523, 1579, 1619, 1733, 1907, 2027, 2099, 2459, 2477, 2579, 2693, 2819, 2909, 2957, 2963, 3203, 3413, 3467, 3779, 3803, 3947, 4139

Offset: 1

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

nonn

For all these the only value of s such that s^s mod n = s is s=1; all the values appear to be primes.

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A065295.

isok(k) = { for (s=2, k - 1, if (Mod(s, k)^s == s, return(0))); k > 1 } \\ Harry J. Smith, Oct 15 2009

Corrected by T. D. Noe, Nov 01 2006

A382752 Numbers k such that A000005(k) = A065295(k).

Original entry on oeis.org

6, 7, 8, 9, 10, 13, 19, 23, 29, 32, 37, 47, 54, 71, 109, 149, 167, 173, 223, 229, 263, 283, 359, 383, 479, 503, 509, 653, 659, 719, 739, 773, 839, 863, 887, 971, 983, 1229, 1319, 1367, 1439, 1487, 1493, 1637, 1699, 1823, 1949, 1997, 2039, 2063, 2207, 2309, 2411, 2447

Offset: 1

Juri-Stepan Gerasimov, Jun 02 2025

nonn

			6 is a term because the number of divisors of 6 is equal to 4 (1, 2, 3 and 6) and 1^1 = 1 (mod 6), 2^2 = 4 (mod 6), 3^3 = 27 == 3 (mod 6), 4^4 = 256 == 4 (mod 6), 5^5 + 3125 == 5 (mod 6).

Cf. A000005, A065295.

[k: k in [2..2500] | #Divisors(k) eq #[s: s in [0..k-1] | s^s mod k eq s]];

isok(k) = numdiv(k) == sum(s=1, k-1, Mod(s, k)^s == s); \\ Michel Marcus, Jun 03 2025

A384854 The number of divisors d of n such that (-d)^d == d (mod n).

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 5, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1

Offset: 1

Juri-Stepan Gerasimov, Jun 10 2025

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A032741, A065295, A384237, A384781.

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

a:= n-> add(`if`((-d)&^d-d mod n=0, 1, 0), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 10 2025

a(n) = sumdiv(n, d, Mod(-d, n)^d == d); \\ Michel Marcus, Jun 11 2025

a(n) = 1 + number of proper divisors h of n such that (-h)^h = h (mod n).

A384781 Number of values of s, 0 < s <= n - 1, such that (-s)^s == s (mod n).

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 1, 0, 1, 4, 0, 1, 1, 3, 3, 0, 0, 4, 0, 1, 2, 3, 1, 1, 3, 6, 1, 3, 1, 6, 1, 0, 3, 2, 2, 3, 3, 3, 2, 1, 1, 6, 0, 3, 5, 3, 1, 1, 3, 8, 2, 2, 2, 4, 3, 2, 1, 5, 0, 3, 3, 3, 7, 0, 5, 6, 0, 1, 3, 8, 1, 3, 3, 8, 5, 3, 4, 6, 1, 1, 4, 3, 0, 5, 2, 4, 6, 2, 4, 10, 5, 2, 3, 3, 2, 1, 4, 8, 5, 5

Offset: 1

Juri-Stepan Gerasimov, Jun 09 2025

nonn

Cf. A065295, A151821, A373901 (k such that a(k) = 0), A382752, A384854.

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

a:= n-> add(`if`((-s)&^s-s mod n=0, 1, 0), s=1..n-1):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 09 2025

a[n_]:=Length[Select[Range[n-1],PowerMod[-#,#,n]==# &]]; Array[a,100] (* Stefano Spezia, Jun 11 2025 *)

a(n) = sum(s=1, n-1, Mod(-s, n)^s == s); \\ Michel Marcus, Jun 11 2025

A385392 The number of divisors d of n such that -(d^d) == d (mod n).

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1

Offset: 1

Juri-Stepan Gerasimov, Jun 27 2025

Cf. A032741, A065295, A384237, A384781, A384854, A385103.

[1+#[d: d in [1..n-1] | n mod d eq 0 and Modexp(d, d, n) eq (n-d)]: n in [1..100]]; // Juri-Stepan Gerasimov, Jun 28 2025

a:= n-> add(`if`(d&^d+d mod n=0, 1, 0), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 27 2025

a[n_] := DivisorSum[n, 1 &, PowerMod[#, #, n] == n-# &]; Array[a, 100] (* Amiram Eldar, Jun 27 2025 *)

a(n) = sumdiv(n, d, -Mod(d, n)^d == d); \\ Michel Marcus, Jun 27 2025

A385540 Number of values of nonnegative s < n such that s^s == (-s)^s == s (mod n).

Original entry on oeis.org

1, 1, 0, 0, 0, 2, 1, 0, 1, 2, 0, 1, 0, 3, 2, 0, 0, 4, 0, 1, 2, 3, 1, 1, 2, 4, 1, 3, 0, 4, 1, 0, 3, 2, 1, 3, 1, 3, 2, 1, 1, 6, 0, 3, 4, 3, 1, 1, 3, 6, 2, 2, 0, 4, 3, 2, 1, 3, 0, 3, 1, 3, 7, 0, 3, 6, 0, 1, 3, 6, 1, 3, 1, 4, 5, 3, 4, 6, 1, 1, 4, 3, 0, 5, 0, 4, 4, 2, 1, 8, 4, 2, 3, 3, 2, 1, 0, 8, 5, 5

Offset: 1

Juri-Stepan Gerasimov, Jul 02 2025

nonn

Cf. A065295, A384781, A385541.

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

a[n_] := Count[Range[0, n-1], ?(PowerMod[#, #, n] == PowerMod[-#, #, n] == # &)]; Array[a, 100] (* _Amiram Eldar, Jul 03 2025 *)

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

A233518 Primes p such that x^x == x (mod p) for some number x with 1 < x < p.

Original entry on oeis.org

7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 263, 271, 277, 281, 283, 307, 311, 313, 331, 337

Offset: 1

T. D. Noe, Feb 18 2014

nonn

Complement of A065296.

T. D. Noe, Table of n, a(n) for n = 1..1000
Pär Kurlberg, Florian Luca, and Igor Shparlinski, On the fixed points of the map x -> x^x modulo a prime, arX1v 1402.4464

Cf. A065295, A065296, A233519, A233520.

fQ[p_] := Min[Table[Mod[PowerMod[x, x, p] - x, p], {x, 2, p - 1}]] == 0; Select[Prime[Range[2, 100]], fQ[#] &]

A233519 The number of times x^x == x (mod prime(n)) for x in 0 < x < prime(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 2, 3, 2, 5, 3, 2, 1, 1, 4, 3, 2, 5, 4, 1, 4, 5, 3, 4, 1, 2, 5, 5, 3, 5, 4, 2, 5, 7, 3, 2, 2, 1, 5, 3, 9, 3, 7, 6, 2, 1, 2, 6, 3, 8, 5, 5, 2, 1, 8, 3, 10, 2, 1, 8, 6, 9, 1, 9, 12, 1, 5, 5, 2, 4, 6, 6, 2, 1, 4, 5, 9, 4, 4, 3, 6, 4, 5, 6

Offset: 1

T. D. Noe, Feb 19 2014

nonn

This is A065295 restricted to the primes. The plot is significantly different.

T. D. Noe, Table of n, a(n) for n = 1..10000
Pär Kurlberg, Florian Luca, and Igor Shparlinski On the fixed points of the map x -> x^x modulo a prime arX1v 1402.4464

Cf. A065295, A233518, A233520.

f[p_] := Module[{x = Range[p - 1]}, Count[PowerMod[x, x, p] - x, 0]]; Table[f[n], {n, Prime[Range[100]]}]

A233520 The number of distinct values of x^x (mod n) - x for x in 0 < x < n.

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 5, 4, 5, 5, 6, 4, 10, 7, 8, 9, 11, 5, 12, 9, 12, 10, 15, 9, 14, 12, 14, 12, 19, 11, 21, 19, 18, 16, 19, 12, 28, 18, 18, 18, 30, 13, 33, 20, 22, 23, 36, 18, 28, 20, 23, 27, 39, 17, 35, 24, 32, 30, 43, 20, 46, 33, 26, 37, 37, 22, 49, 34, 34, 30

Offset: 1

T. D. Noe, Feb 19 2014

nonn

According to Kurlberg et al. (who quote Crocker and Somer), for primes p, the count is between floor(sqrt((p-1)/2)) and 3p/4 + O(p^(1/2 + o(1))).

Note that the subtraction is not done mod n. - Robert Israel, Dec 17 2014

			For n = 5 the a(5) = 4 values are 1-1=0, 4-2=2, 2-3=-1, 1-4=-3. - _Robert Israel_, Dec 17 2014

T. D. Noe, Table of n, a(n) for n = 1..10000
Roger Crocker, On residues of n^n, Amer. Math. Monthly, 76 (1969), 1028-1029.
Pär Kurlberg, Florian Luca, and Igor Shparlinski, On the fixed points of the map x -> x^x modulo a prime, arXiv:1402.4464 [math.NT], 2014.
Lawrence Somer, The residues of n^n modulo p, The Fibonacci Quart., 19 (1981), 110-117.

Cf. A065295, A233518, A233519, A233521.

f:= n -> nops({seq((x &^ x mod n - x) , x = 1 .. n-1)}):
seq(f(n), n=1..100); # Robert Israel, Dec 17 2014

fs[p_] := Module[{x = Range[p - 1]}, Length[Union[PowerMod[x, x, p] - x]]]; Table[fs[n], {n, 100}]

a(n) = #Set(vector(n-1, j, lift(Mod(j, n)^j) - j)); \\ Michel Marcus, Dec 16 2014

A385103 Number of values of s, 0 < s < n, such that -(s^s) == s (mod n).

Original entry on oeis.org

0, 1, 1, 0, 2, 2, 1, 0, 1, 4, 2, 1, 4, 3, 2, 0, 1, 2, 2, 1, 3, 3, 1, 1, 2, 6, 1, 1, 3, 6, 1, 0, 2, 2, 5, 1, 4, 3, 3, 1, 1, 4, 3, 1, 2, 3, 1, 1, 1, 4, 1, 2, 4, 2, 3, 2, 3, 5, 2, 3, 4, 3, 1, 0, 5, 5, 2, 1, 2, 8, 3, 1, 3, 8, 3, 1, 3, 4, 2, 1, 1, 3, 2, 3, 5, 4, 3, 1, 4, 6, 5, 2, 3, 3, 2, 1, 5, 2, 3, 1

Offset: 1

Juri-Stepan Gerasimov, Jun 17 2025

nonn

Cf. A065295, A151821, A373901, A382752, A384781, A384854.

a:= n-> add(`if`(s&^s+s mod n=0, 1, 0), s=1..n-1):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 20 2025

a(n) = sum(s=1, n-1, -Mod(s, n)^s == s); \\ Michel Marcus, Jun 19 2025

cp's OEIS Frontend

A065296 Values of n such that A065295(n) = 1.

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A382752 Numbers k such that A000005(k) = A065295(k).

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A384854 The number of divisors d of n such that (-d)^d == d (mod n).

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A384781 Number of values of s, 0 < s <= n - 1, such that (-s)^s == s (mod n).

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A385392 The number of divisors d of n such that -(d^d) == d (mod n).

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A385540 Number of values of nonnegative s < n such that s^s == (-s)^s == s (mod n).

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A233518 Primes p such that x^x == x (mod p) for some number x with 1 < x < p.

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A233519 The number of times x^x == x (mod prime(n)) for x in 0 < x < prime(n).

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A233520 The number of distinct values of x^x (mod n) - x for x in 0 < x < n.

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