A082576 -id:A082576 - cp's OEIS frontend

A338128 a(n) is the least k > 1 such that the base n representation of k^k ends with that of k.

3, 4, 3, 6, 3, 4, 3, 4, 5, 12, 3, 5, 4, 5, 5, 9, 4, 7, 5, 4, 11, 12, 3, 6, 5, 10, 4, 17, 5, 16, 9, 11, 9, 13, 4, 10, 7, 5, 5, 9, 4, 7, 11, 9, 12, 24, 5, 8, 6, 9, 5, 54, 10, 11, 7, 7, 17, 60, 5, 13, 16, 4, 9, 5, 11, 37, 9, 12, 13, 36, 7, 37, 10, 6, 7, 16, 5, 27

Offset: 2

Rémy Sigrist, Oct 11 2020

			a(10) = A082576(2) = 5.

Rémy Sigrist, Table of n, a(n) for n = 2..10000
Index entries for sequences related to automorphic numbers

Cf. A065296, A082576.

a(n) = for (k=2, oo, if (Mod(k, n^#digits(k,n))^k==k, return (k)))

a(n) <= n+1 with equality iff n belongs to A065296.

A371048 Numbers formed by the rightmost decimal digits of n that are the same as those n^n, where -1 indicates that n <> n^n (mod 10).

Original entry on oeis.org

0, 1, -1, -1, -1, 5, 6, -1, -1, 9, 0, 11, -1, 3, -1, 5, 16, 7, -1, 9, 0, 21, -1, -1, -1, 25, 6, -1, -1, 9, 0, 31, -1, 3, -1, 5, 36, 7, -1, 9, 0, 41, -1, -1, -1, 5, 6, -1, -1, 49, 0, 51, -1, 3, -1, 5, 56, 57, -1, 9, 0, 61, -1, -1, 5, 6, -1, -1, 9, 0, 71, -1, 3

Offset: 0

Marco Ripà, Mar 10 2024

The common digits might include leading 0's (such as at n = 51 or n = 57) and they are discarded (in particular, a(0) = 0 indicates that the corresponding zero digit term results in a 0 integer entry).
Assuming that c > 0 is an integer not a multiple of 10, then a(c*10^k) = 0 for every positive integer k, since (c*10^k) and (c*10^k)^(c*10^k) have in common only their rightmost k digits.
a(n) is equal to -1 if and only if n == 2,4,8 (mod 10) or n == 3,7 (mod 20).
A082576 is a subsequence of the present one.

			For n = 51, 51^51 = 1219211305094648479473193481872927834667576992593770717189298225284399541977208231315051 and 51^51 == 5051 (mod 10^4), so there might be three common final digits by including a leading 0 that should instead be disregarded. Consequently, a(51) = 51.

Jorge Jiménez Urroz and José Luis Andrés Yebra, On the Equation a^x == x (mod b^n), Journal of Integer Sequences, Article 09.8.8, 2009.
Marco Ripà, Congruence speed of tetration bases ending with 0, arXiv:2402.07929 [math.NT], 2024.

Cf. A000312, A082576, A317905, A349425, A369624, A369826, A370211.

If n <> 2,4,8 (mod 10) or n <> 3,7 (mod 20), then a(n) = n (mod 10^k), where k is such that n == n^n (mod 10^k) and n <> n^n (mod 10^(k+1)), whereas a(n) = -1 otherwise.

A371074 Number of the rightmost decimal digits of n that are the same as those of n^n.

Original entry on oeis.org

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

Offset: 0

Marco Ripà, Mar 10 2024

The common digits might include leading 0's (such as at n = 51 or n = 57) and they are included in the total.
Let c be a positive integer and assume that k is a positive integer that is not a multiple of 10. If n = k*10^c, then a(n) = c which is all the rightmost 0's of n.
For every n >= 0, a(n) is the congruence speed of n at height 1 by Definitions 1.1 and 1.3 of the paper entitled "Number of stable digits of any integer tetration" (see Links).

			a(0) = 0 since 0^0 = 1 so that 0 and 0^0 have no digits in common.
For n = 51, a(n) = 3 since 51^51 == 5051 (mod 10^4).

Jorge Jiménez Urroz and José Luis Andrés Yebra, On the Equation a^x == x (mod b^n), Journal of Integer Sequences, Article 09.8.8, 2009.
Marco Ripà, Congruence speed of tetration bases ending with 0, arXiv:2402.07929 [math.NT], 2024.
Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441-457.
Wikipedia, Tetration.

Cf. A000312, A082576, A317905, A349425, A369826, A369624, A370211.

For any n >= 2, a(n) is such that n == n^n (mod 10^(a(n))) and n <> n^n (mod 10^(a(n)+1)).

A373206 Numbers m such that m^m == m (mod 10^(len(m) + 2)), where len(m) is the number of digits of m (A055642).

Original entry on oeis.org

1, 751, 1001, 2001, 2751, 3001, 4001, 5001, 5376, 6001, 6751, 7001, 8001, 9001, 10001, 18751, 20001, 30001, 40001, 50001, 58751, 60001, 69376, 70001, 80001, 90001, 98751, 100001, 110001, 120001, 130001, 138751, 140001, 150001, 160001, 170001, 180001, 190001

Offset: 1

Marco Ripà, May 28 2024

By definition, this sequence is a subsequence of A082576 and also a subsequence of A373205.
For each integer r >= 3 the sequence contains 10^r + 1.
All terms > 9001 end in 0001 (e.g., 10001), 0625 (e.g., 390625), 1249 (e.g., 781249), 8751 (e.g., 18751), 9376 (e.g., 69376), and possibly in 4193, 7057, or 9375.

			751 is a term since 751 is a 3-digit number and 751^751 == 500751 (mod 10^6) and thus 751^751 == 751 (mod 10^(3 + 2)).

Harvey P. Dale, Table of n, a(n) for n = 1..351 (All terms up to 20 million)
Index entries for sequences related to automorphic numbers

Cf. A055642, A082576, A373205.

Select[Range[200000],PowerMod[#,#,10^(IntegerLength[#]+2)]==#&] (* Harvey P. Dale, Dec 05 2024 *)

for (len_m = 1, 5, for (m = 10^(len_m - 1), 10^len_m - 1, if (m == Mod(m, 10^(len_m + 1))^m, print1(m, ", "))));

from itertools import count
def A373206_gen(): # generator of terms
    yield from (1, 751, 1001, 2001, 2751, 3001, 4001, 5001, 5376, 6001, 6751, 7001, 8001, 9001)
    for i in count(10000,10000):
        for j in (1,625,1249,4193,7057,8751,9375,9376):
            m = i+j
            if pow(m,m,100*10**(len(str(m)))) == m:
                yield m
A373206_list = list(islice(A373206_gen(),20)) # Chai Wah Wu, Jun 02 2024

For all terms m, m^m == m (mod 10^(floor(log_10(m)) + 3)).

A338129 Positive numbers k such that the binary representation of k^k ends with that of k.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 25, 31, 33, 41, 49, 57, 63, 65, 81, 97, 113, 127, 129, 145, 161, 177, 193, 209, 225, 241, 255, 257, 289, 321, 353, 385, 417, 449, 481, 511, 513, 545, 577, 609, 641, 673, 705, 737, 769, 801, 833, 865, 897, 929, 961, 993, 1023, 1025

Offset: 1

Rémy Sigrist, Oct 11 2020

This sequence is infinite as it contains the positive terms of A000225.
All terms are odd.
Run lengths in first differences appear to be regular and suggest a simple procedure to generate the sequence.

			The binary representation of 3^3 ("11011") ends with that of 3 ("11"), so 3 is a term.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Binary plot of the terms < 2^16
Index entries for sequences related to automorphic numbers

Cf. A000225, A082576, A338128, A338130.

Select[Range[1200],Take[IntegerDigits[#^#,2],-IntegerLength[ #,2]] == IntegerDigits[ #,2]&] (* Harvey P. Dale, Jan 12 2022 *)

is(n, base=2) = Mod(n, base^#digits(n, base))^n==n

A338130 Positive numbers k such that the ternary representation of k^k ends with that of k.

Original entry on oeis.org

1, 4, 7, 10, 19, 28, 37, 46, 55, 64, 73, 82, 109, 136, 163, 190, 217, 244, 271, 298, 325, 352, 379, 406, 433, 460, 487, 514, 541, 568, 595, 622, 649, 676, 703, 730, 811, 892, 973, 1054, 1135, 1216, 1297, 1378, 1459, 1540, 1621, 1702, 1783, 1864, 1945, 2026

Offset: 1

Rémy Sigrist, Oct 11 2020

All terms are of the form 3*m + 1 for some m >= 0.
The first differences appear to contain only powers of 3 and to be weakly increasing.
Run lengths in first differences appear to be regular and suggest a simple procedure to generate the sequence.

			The ternary representation of 4^4 ("100111") ends with that of 4 ("11"), so 4 is a term.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Ternary plot of the terms < 3^10 (blue squares correspond to 1's and red squares to 2's)
Index entries for sequences related to automorphic numbers

Cf. A082576, A338128, A338129.

Select[Range[2100],Take[IntegerDigits[#^#,3],-IntegerLength[#,3]] == IntegerDigits[ #,3]&] (* Harvey P. Dale, Feb 13 2022 *)

is(n, base=3) = Mod(n, base^#digits(n, base))^n==n

A373205 Numbers m such that m^m == m (mod 10^(len(m) + 1)), where len(m) is the number of digits of m (A055642).

Original entry on oeis.org

1, 51, 57, 101, 151, 176, 201, 301, 351, 401, 501, 551, 576, 601, 625, 701, 751, 801, 901, 951, 976, 1001, 1376, 2001, 2057, 2751, 3001, 4001, 4193, 4751, 5001, 5376, 6001, 6249, 6751, 7001, 8001, 8751, 9001, 9375, 9376, 10001, 10751, 11001, 12001, 13001

Offset: 1

Marco Ripà, May 27 2024

By definition, the present sequence is a subsequence of A082576.
For each integer r >= 2 this sequence contains 10^r + 1.
All terms > 1 end in 01, 25, 49, 51, 57, 75, 76, or 93.

			51 is a term since 51 is a 2-digit number and 51^51 == 5051 (mod 10^4) and thus 51^51 == 51 (mod 10^(2 + 1)).

Index entries for sequences related to automorphic numbers

Cf. A055642, A082576, A373206.

for (len_m = 1, 5, for (m = 10^(len_m - 1), 10^len_m - 1, if (m == Mod(m, 10^(len_m + 1))^m, print1(m, ", "))))

from itertools import count
def A373205_gen(): # generator of terms
    for i in count(0,100):
        for j in (1, 25, 49, 51, 57, 75, 76, 93):
            m = i+j
            if pow(m,m,10*10**(len(str(m)))) == m:
                yield m
A373205_list = list(islice(A373205_gen(),20)) # Chai Wah Wu, Jun 02 2024

A358348 Numbers k such that k == k^k (mod 9).

Original entry on oeis.org

1, 4, 7, 9, 10, 13, 16, 17, 18, 19, 22, 25, 27, 28, 31, 34, 35, 36, 37, 40, 43, 45, 46, 49, 52, 53, 54, 55, 58, 61, 63, 64, 67, 70, 71, 72, 73, 76, 79, 81, 82, 85, 88, 89, 90, 91, 94, 97, 99, 100, 103, 106, 107, 108, 109, 112, 115, 117, 118, 121, 124, 125, 126

Offset: 1

Ivan Stoykov, Nov 11 2022

Each multiple of 9 is in the sequence. Additionally, the squares are also present.

			4 is a term since 4^4 = 256 == 4 (mod 9).

M. Fujiwara and Y. Ogawa, Introduction to Truly Beautiful Mathematics. Tokyo: Chikuma Shobo, 2005.

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

Cf. A007953, A010888, A082576, A189510.

A358348 := proc(n)
    2*(n+1)-op(modp(n,9)+1,[2,3,2,1,1,2,1,0,1]) ;
end proc:
seq(A358348(n),n=1..50) ; # R. J. Mathar, Mar 29 2023

Select[Range[130], MemberQ[{0, 1, 4, 7, 9, 10, 13, 16, 17}, Mod[#, 18]] &] (* Amiram Eldar, Nov 12 2022 *)

isok(k) = k == Mod(k,9)^k; \\ Michel Marcus, Nov 22 2022

def A358348(n):
    return ((0, 1, 4, 7, 9, 10, 13, 16, 17)[m := n % 9]
         + (n - m << 1))  # Chai Wah Wu, Feb 09 2023

G.f.: x*(x+1)*(x^7+3*x^5+x^3+x^2+2*x+1)/((1-x)^2*(1+x^3+x^6)*(1+x+x^2)). - Alois P. Heinz, Feb 08 2023

a(n) = 2*(n+1) - b(n) where b(n>=0) = 2,3,2,1,1,2,1,0,1,2,3,2,... has period 9. - Kevin Ryde, Mar 26 2023

A373386 Smallest integer m > 1 such that m == m^m (mod 10^(len(m) + n)), where len(m) is the number of digits of m.

Original entry on oeis.org

5, 51, 751, 10001, 100001, 1000001, 10000001, 100000001, 1000000001

Offset: 0

Marco Ripà, Jun 02 2024

By definition, this sequence is a subsequence of A082576.

It is not known if a(n) = 10^(n + 1) + 1 holds for all n >= 3.

			a(2) = 751 since m = 751 is the smallest integer satisfying m == m^m (mod 10^(len(m) + 2)), given the fact that 751 is a 3-digit number and 751^751 == 500751 (mod 10^6) and thus 751^751 == 751 (mod 10^(3 + 2)).

Index entries for sequences related to automorphic numbers

Cf. A000533, A055642, A082576, A373205, A373206.

a(n) = my(im); for (len_m = 1, oo, if (len_m==1, im=2, im=10^(len_m - 1)); for (m = im, 10^len_m - 1, if (m == Mod(m, 10^(len_m + n))^m, return(m)))); \\ Michel Marcus, Jun 03 2024

a(7)-a(8) from Michel Marcus, Jun 03 2024

cp's OEIS Frontend

A338128 a(n) is the least k > 1 such that the base n representation of k^k ends with that of k.

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A371048 Numbers formed by the rightmost decimal digits of n that are the same as those n^n, where -1 indicates that n <> n^n (mod 10).

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A371074 Number of the rightmost decimal digits of n that are the same as those of n^n.

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A373206 Numbers m such that m^m == m (mod 10^(len(m) + 2)), where len(m) is the number of digits of m (A055642).

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A338129 Positive numbers k such that the binary representation of k^k ends with that of k.

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A338130 Positive numbers k such that the ternary representation of k^k ends with that of k.

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A373205 Numbers m such that m^m == m (mod 10^(len(m) + 1)), where len(m) is the number of digits of m (A055642).

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A358348 Numbers k such that k == k^k (mod 9).

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