A174824 -id:A174824 - cp's OEIS frontend

A185075 The number of distinct residues modulo n of {i^i: i=1,2...}.

1, 2, 3, 3, 5, 5, 7, 6, 7, 8, 11, 8, 13, 11, 15, 10, 17, 11, 19, 13, 17, 17, 23, 15, 21, 20, 19, 18, 29, 21, 31, 18, 33, 26, 35, 18, 37, 29, 31, 24, 41, 23, 43, 28, 35, 35, 47, 27, 43, 32, 51, 33, 53, 29, 47, 33, 45, 44, 59, 36, 61, 47, 45, 34, 65, 45, 67

Offset: 1

José María Grau Ribas, Jan 23 2012

nonn

David W. Wilson, Table of n, a(n) for n = 1..10099

Cf. A000312, A174824.

res[mod_] := Length[Union[Table[PowerMod[i,i,mod], {i, 1, mod + LCM[mod*CarmichaelLambda[mod]]}]]]; Table[res[n], {n, 100}]

period(n) = lcm(n, znstar(n)[2]); \\ A174824
a(n) = {v = []; for (i=1, period(n), v = Set(concat(v, Mod(i, n)^i));); #v;} \\ Michel Marcus, Mar 18 2016

A204693 a(n) = n^n (mod 7).

Original entry on oeis.org

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

Offset: 0

José María Grau Ribas, Jan 18 2012

For n>0, periodic with period 42 = A174824(7): repeat[1, 4, 6, 4, 3, 1, 0, 1, 1, 4, 2, 1, 6, 0, 1, 2, 5, 1, 5, 1, 0, 1, 4, 1, 4, 4, 6, 0, 1, 1, 3, 2, 6, 1, 0, 1, 2, 2, 1, 2, 6, 0].

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A204671, A174824.

Mathematica
```
Table[PowerMod[n,n,7], {n,0,100}]
```

a(n)=lift(Mod(n,7)^n) \\ Charles R Greathouse IV, Jan 23 2012

a(n) = a(n-42), n>42. - R. J. Mathar, Sep 25 2014

G.f.: ( -1 -4*x -2*x^37 -6*x^2 -4*x^22 -4*x^24 -4*x^25 -6*x^26 -x^28 -x^29 -3*x^30 -6*x^40 -2*x^31 -6*x^32 -x^33 -x^35 -x^17 -5*x^18 -x^19 -x^21 -x^23-x^38 -2*x^39 -2*x^36 -4*x^3 -3*x^4 -x^5 -x^7 -x^8 -4*x^9 -2*x^10 -x^11 -6*x^12 -x^14 -2*x^15 -5*x^16 ) / ( (x-1) *(1+x^6+x^5+x^4+x^3+x^2+x) *(1+x+x^2) *(1-x+x^3-x^4+x^6-x^8+x^9-x^11+x^12) *(1+x) *(1-x+x^2-x^3+x^4-x^5+x^6) *(1-x+x^2)*(1+x-x^3-x^4+x^6-x^8-x^9+x^11+x^12) ). - R. J. Mathar, Sep 25 2014

A204694 a(n) = n^n (mod 8).

Original entry on oeis.org

1, 1, 4, 3, 0, 5, 0, 7, 0, 1, 0, 3, 0, 5, 0, 7, 0, 1, 0, 3, 0, 5, 0, 7, 0, 1, 0, 3, 0, 5, 0, 7, 0, 1, 0, 3, 0, 5, 0, 7, 0, 1, 0, 3, 0, 5, 0, 7, 0, 1, 0, 3, 0, 5, 0, 7, 0, 1, 0, 3, 0, 5, 0, 7, 0, 1, 0, 3, 0, 5, 0, 7, 0, 1, 0, 3, 0, 5, 0, 7, 0, 1, 0, 3, 0, 5, 0

Offset: 0

José María Grau Ribas, Jan 18 2012

Eventually periodic with period 8 = A174824(8): a(1)=1, a(2)=4 and repeat [3, 0, 5, 0, 7, 0, 1, 0].

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A000312, A174824.

Table[PowerMod[n,n,8], {n,0,100}]
Join[{1, 1, 4},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{3, 0, 5, 0, 7, 0, 1, 0},84]] (* Ray Chandler, Aug 25 2015 *)
PadRight[{1,1,4},120,{0,1,0,3,0,5,0,7}] (* Harvey P. Dale, Aug 06 2018 *)

a(n)=lift(Mod(n,8)^n) \\ Charles R Greathouse IV, Jan 23 2012

G.f.: (4*x^10 + x^8 - 7*x^7 - 5*x^5 - 3*x^3 - 4*x^2 - x - 1)/(x^8 - 1). - Chai Wah Wu, Jun 04 2016

a(n) = A000312(n) mod 8. - Michel Marcus, Jun 04 2016

A204695 a(n) = n^n (mod 9).

Original entry on oeis.org

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

Offset: 0

José María Grau Ribas, Jan 18 2012

For n>0, periodic with period 18 = A174824(9): repeat [1, 4, 0, 4, 2, 0, 7, 1, 0, 1, 5, 0, 4, 7, 0, 7, 8, 0].

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A000312, A174824.

Table[PowerMod[n,n,9], {n,0,100}]
Join[{1},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 4, 0, 4, 2, 0, 7, 1, 0, 1, 5, 0, 4, 7, 0, 7, 8, 0},86]] (* Ray Chandler, Aug 27 2015 *)

a(n)=lift(Mod(n,9)^n) \\ Charles R Greathouse IV, Jan 23 2012

G.f.: (x^18 - 8*x^17 - 7*x^16 - 7*x^14 - 4*x^13 - 5*x^11 - x^10 - x^8 - 7*x^7 - 2*x^5 - 4*x^4 - 4*x^2 - x - 1)/(x^18 - 1). - Chai Wah Wu, Jun 04 2016

a(n) = A000312(n) mod 9. - Michel Marcus, Jun 04 2016

A260031 Final nonzero digit of n^n in base 12.

Original entry on oeis.org

1, 4, 3, 4, 5, 3, 7, 4, 9, 4, 11, 1, 1, 4, 3, 4, 5, 3, 7, 4, 9, 4, 11, 4, 1, 4, 3, 4, 5, 3, 7, 4, 9, 4, 11, 9, 1, 4, 3, 4, 5, 3, 7, 4, 9, 4, 11, 4, 1, 4, 3, 4, 5, 3, 7, 4, 9, 4, 11, 1, 1, 4, 3, 4, 5, 3, 7, 4, 9, 4, 11, 9, 1, 4, 3, 4, 5, 3, 7, 4, 9, 4, 11, 1

Offset: 1

N. J. A. Sloane, Jul 19 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000
José María Grau and Antonio M. Oller-Marcén, On the last digit and the last non-zero digit of n^n in base b, arXiv preprint arXiv:1203.4066 [math.NT], 2012.

Cf. A174824, A204819, A230024.

Cf. A000312, A010881.

import Math.NumberTheory.Moduli (powerMod)
a260031 n = if x > 0 then x else f $ div (n ^ n) 12
          where x = powerMod n n 12
                f z = if m == 0 then f z' else m
                      where (z', m) = divMod z 12
-- Reinhard Zumkeller, Jul 19 2015

from gmpy2 import mpz, digits
def A260031(n):
    s = digits(mpz(n)**mpz(n),12)
    t = s[-1]
    while t == '0':
        s = s[:-1]
        t = s[-1]
    return int(t,12) # Chai Wah Wu, Jul 19 2015

More terms from Chai Wah Wu, Jul 19 2015

A327570 a(n) = n*phi(n)^2, phi = A000010.

Original entry on oeis.org

1, 2, 12, 16, 80, 24, 252, 128, 324, 160, 1100, 192, 1872, 504, 960, 1024, 4352, 648, 6156, 1280, 3024, 2200, 11132, 1536, 10000, 3744, 8748, 4032, 22736, 1920, 27900, 8192, 13200, 8704, 20160, 5184, 47952, 12312, 22464, 10240, 65600, 6048, 75852, 17600, 25920, 22264, 99452, 12288

Offset: 1

Jianing Song, Sep 17 2019

a(n) is the order of the group consisting of all upper-triangular (or equivalently, lower-triangular) matrices in GL(2, Z_n). That is to say, a(n) = |G_n|, where G_n = {{{a, b}, {0, d}} : gcd(a, n) = gcd(d, n) = 1}. The group G_n is well-defined because the product of two upper-triangular matrices is again an upper-triangular matrix. For example,{{a, b}, {0, d}} * {{x, y}, {0, z}} = {{a*x, a*y+b*z}, {0, d*z}}.
The exponent of G_n (i.e., the least positive integer k such that x^k = e for all x in G_n) is A174824(n). (Note that {{1, 1}, {0, 1}} is an element with order n and there exists some r such that {{r, 0}, {0, r}} is an element with order psi(n), psi = A002322. It is easy to show that x^lcm(n, psi(n)) = Id = {{1, 0}, {0, 1}} for all x in G_n.)
If only upper-triangular matrices in SL(2, Z_n) are wanted, we get a group of order n*phi(n) = A002618(n) and exponent A174824(n).

			G_3 = {{{1, 0}, {0, 1}}, {{1, 1}, {0, 1}}, {{1, 2}, {0, 1}}, {{1, 0}, {0, 2}}, {{1, 1}, {0, 2}}, {{1, 2}, {0, 2}}, {{2, 0}, {0, 1}}, {{2, 1}, {0, 1}}, {{2, 2}, {0, 1}}, {{2, 0}, {0, 2}}, {{2, 1}, {0, 2}}, {{2, 2}, {0, 2}}} with order 12, so a(3) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000252, A002618, A174824, A011379, A065464.

Table[n * EulerPhi[n]^2, {n, 1, 100}] (* Amiram Eldar, Sep 19 2020 *)

PARI
```
a(n) = n*eulerphi(n)^2
```

Multiplicative with a(p^e) = (p-1)^2*p^(3e-2).
a(n) = A000010(n)*A002618(n).
a(p) = A011379(p-1) for p prime. - Peter Luschny, Sep 17 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/((p-1)^3 * (p^2 + p + 1))) = 1.7394747912949637836019917301710010334604379331855033150372654868327481539... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^4, where c = (1/4) * Product_{p prime} (1 - (2*p-1)/p^3) = A065464 / 4 = 0.1070623764... . - Amiram Eldar, Nov 05 2022

A309428 Irregular triangle read by rows: T(n,k) is the multiplicative order of {{A038566(n,k), 1}, {0, 1}} modulo n, n >= 1, 1 <= k <= A000010(n).

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 4, 4, 2, 6, 2, 7, 3, 6, 3, 6, 2, 8, 4, 8, 2, 9, 6, 9, 6, 9, 2, 10, 4, 4, 2, 11, 10, 5, 5, 5, 10, 10, 10, 5, 2, 12, 4, 6, 2, 13, 12, 3, 6, 4, 12, 12, 4, 3, 6, 12, 2, 14, 6, 6, 6, 6, 2, 15, 4, 6, 12, 4, 10, 12, 2, 16, 8, 16, 4, 16, 8, 16, 2, 17, 8, 16, 4, 16, 16, 16, 8

Offset: 1

Jianing Song, Sep 18 2019

Let M = {{r, 1}, {0, 1}}, then M^e = {{r^e, 1 + r + r^2 + ... + r^(e-1)}, {0, 1}}. As a result, for gcd(r, n) = 1, the multiplicative order of {{r, 1}, {0, 1}} modulo n is n if r == 1 (mod n) and ord(r,n*(r-1)) otherwise, where ord(r,t) is the multiplicative order of r modulo t.

			Table starts
  1,
  2,
  3, 2,
  4, 2,
  5, 4, 4, 2,
  6, 2,
  7, 3, 6, 3, 6, 2,
  8, 4, 8, 2,
  9, 6, 9, 6, 9, 2,
  10, 4, 4, 2,
  11, 10, 5, 5, 5, 10, 10, 10, 5, 2,
  12, 4, 6, 2,
  13, 12, 3, 6, 4, 12, 12, 4, 3, 6, 12, 2,
  14, 6, 6, 6, 6, 2,
  15, 4, 6, 12, 4, 10, 12, 2,
  16, 8, 16, 4, 16, 8, 16, 2,
  17, 8, 16, 4, 16, 16, 16, 8, 8, 16, 16, 16, 4, 16, 8, 2,
  18, 6, 18, 6, 18, 2,
  19, 18, 18, 9, 9, 9, 3, 6, 9, 18, 3, 6, 18, 18, 18, 9, 9, 2,
  20, 4, 4, 4, 10, 4, 4, 2,
  ...
For n = 14 and k = 4, let M = {{A038566(n,k), 1}, {0, 1}} = {{9, 1}, {0, 1}}, then:
- M^2 mod 14 = {{11, 10}, {0, 1}};
- M^3 mod 14 = {{1, 7}, {0, 1}};
- M^4 mod 14 = {{9, 8}, {0, 1}};
- M^5 mod 14 = {{11, 3}, {0, 1}};
- M^6 mod 14 = {{1, 0}, {0, 1}}.
So T(14,4) = d(14,9) = 6.

Cf. A000010, A038566, A174824.

row(n) = my(v=vector(n,i,i),u=vector(eulerphi(n),i,n)); v=select(i->gcd(n,i)==1,v); for(i=2, #v, u[i]=znorder(Mod(v[i], n*(v[i]-1)))); u

For gcd(n,r) = 1, 1 <= r <= n, let d(n,r) be the multiplicative order of {{r, 1}, {0, 1}}, then T(n,k) = d(n,A038566(k)).
(a) If p is an odd prime, then d(p^e,r) = p^e if r == 1 (mod p), ord(r,p^e) otherwise;
(b) d(2^e,r) = 2^(e+1-v2(r+1)), where v2(t) is the 2-adic valuation of t;
(c) For gcd(m,n) = 1, d(m*n,r) = lcm(d(m,r mod m),d(n,r mod n)).
The LCM of the n-th row is A174824(n).

A342144 Numbers m with integer solution to x^x == (x+1)^(x+1) (mod m) with x > 0.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 127, 129, 131, 133, 137, 139, 141, 143, 145, 147, 149, 151, 155, 157, 159, 161, 163, 167

Offset: 1

Owen C. Keith, Mar 01 2021

nonn

Some values of m have multiple solutions.
For example, for m = 49, 25^25 == 26^26 (mod 49) and 37^37 == 38^38 (mod 49).
All terms are odd.
First differs from A334420 at a(70) which is 167 for this sequence and 165 for A334420.
First differs from A056911 at a(21) which is 49 for this sequence and 51 for A056911.

			3 is a term since 1^1 == 2^2 (mod 3).
5 is a term since 11^11 == 12^12 (mod 5).

Cf. A056911, A084820, A174824, A334420, A338445.

seqQ[n_] := AnyTrue[Range[LCM[n, CarmichaelLambda[n]]+1], PowerMod[#, #, n] == PowerMod[# + 1, # + 1, n] &]; Select[Range[145], seqQ]

A347671 a(n) = n^n mod 100.

Original entry on oeis.org

1, 1, 4, 27, 56, 25, 56, 43, 16, 89, 0, 11, 56, 53, 16, 75, 16, 77, 24, 79, 0, 21, 84, 67, 76, 25, 76, 3, 36, 69, 0, 31, 76, 13, 36, 75, 36, 17, 4, 59, 0, 41, 64, 7, 96, 25, 96, 63, 56, 49, 0, 51, 96, 73, 56, 75, 56, 57, 84, 39, 0, 61, 44, 47, 16, 25, 16, 23

Offset: 0

John Bibby, Sep 10 2021

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A000312 (n^n), A056849 (mod 10), A174824.

Table[PowerMod[n,n,100],{n,0,70}] (* Harvey P. Dale, Aug 13 2023 *)

def a(n): return pow(n, n, 100)
print([a(n) for n in range(101)]) # Michael S. Branicky, Sep 26 2021

For n >= 101, a(n) = a(n-100), i.e., cyclic with period A174824(100) = 100, disregarding a(0). - Michael S. Branicky, Sep 26 2021

cp's OEIS Frontend

A185075 The number of distinct residues modulo n of {i^i: i=1,2...}.

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A204693 a(n) = n^n (mod 7).

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A204694 a(n) = n^n (mod 8).

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A204695 a(n) = n^n (mod 9).

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A260031 Final nonzero digit of n^n in base 12.

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A327570 a(n) = n*phi(n)^2, phi = A000010.

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A309428 Irregular triangle read by rows: T(n,k) is the multiplicative order of {{A038566(n,k), 1}, {0, 1}} modulo n, n >= 1, 1 <= k <= A000010(n).

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A342144 Numbers m with integer solution to x^x == (x+1)^(x+1) (mod m) with x > 0.

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