A288845 -id:A288845 - cp's OEIS frontend

A049302 Numbers k such that k is a substring of 4^k.

6, 10, 17, 25, 36, 42, 50, 59, 60, 61, 72, 73, 78, 79, 81, 84, 86, 87, 89, 92, 93, 95, 96, 160, 200, 212, 222, 225, 227, 239, 260, 261, 269, 290, 291, 296, 300, 301, 304, 311, 313, 315, 324, 326, 327, 330, 336, 344, 345, 348, 350, 355, 362, 372, 378, 379, 381

Offset: 1

David W. Wilson

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

Cf. A000302, A032740, A049301, A049303, A049304, A049305, A049306, A049307.

Includes A288845.

A320930 Numbers k such that 4^k starts with k.

Original entry on oeis.org

10, 17, 556, 1771, 4695, 38537, 56969, 345797, 141419115, 1788191728

Offset: 1

Robert Israel, Oct 24 2018

			4^10 = 1048576 starts with 10, so 10 is in the sequence.

Cf. A000302, A100129, A288845.

filter:= proc(n) local t, b;
  t:= 4^n;
b:= ilog10(t) - ilog10(n);
floor(t/10^b) = n
end proc:
select(filter, [$1..10^5]);

def afind(limit, startk=1):
    k, pow4 = startk, 4**startk
    for k in range(startk, limit+1):
        if str(pow4).startswith(str(k)):
            print(k, end=", ")
        pow4 *= 4
afind(10**4) # Michael S. Branicky, Oct 17 2021

a(8)-a(10) from Giovanni Resta, Oct 25 2018

A306686 Values of n such that 9^n ends in n, or expomorphic numbers relative to "base" 9.

Original entry on oeis.org

9, 89, 289, 5289, 45289, 745289, 2745289, 92745289, 392745289, 7392745289, 97392745289, 597392745289, 7597392745289, 87597392745289, 8087597392745289, 48087597392745289, 748087597392745289, 10748087597392745289, 610748087597392745289, 5610748087597392745289

Offset: 1

Bernard Schott, Mar 05 2019

Definition: For positive integers b (as base) and n, the positive integer (allowing initial zeros) k(n) is expomorphic relative to base b (here 9) if k(n) has exactly n decimal digits and if b^k(n) == k(n) (mod 10^n) or, equivalently, b^k(n) ends in k(n). [See Crux Mathematicorum link.]
For sequences in the OEIS, no term is allowed to begin with a digit 0 (except for the 1-digit number 0 itself). However, in the problem as defined in the Crux Mathematicorum article, leading 0 digits are allowed; under that definition a(n) = k(n) until the first k(n) which begins with digit 0. When k(n) begins with 0, then, a(n) is the next term of the sequence k(n) which doesn't begin with digit 0.
Conjecture: if k(n) is expomorphic relative to "base" b, then, the next one in the sequence, k(n+1), consists of the last n+1 digits of b^k(n).
a(n) is the backward concatenation of A133619(0) through A133619(n-1). So, a(1) = 9, a(2) = 89 and so on, with recognition of the former comments about the OEIS and terms beginning with 0. - Davis Smith, Mar 07 2019

			9^9 = 387420489 ends in 9, so 9 is a term; 9^89 = .....289 ends in 89, so 89 is another term.

Davis Smith, Table of n, a(n) for n = 1..944
Charles W. Trigg, Problem 559, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.
Emil Vaughan, Problem 226.8 - 999 nines, M500 Magazine of the Open University, number 226, February 2009, page 21; and Tony Forbes, Solution 226.8 - 999 nines, M500 Magazine of the Open University, number 232, February 2010, pages 8-9, calculating a(9) = 392745289.

Cf. A064541 (base 2), A183613 (base 3), A288845 (base 4), A306570 (base 5), A290788 (base 6), A321970 (base 7), A289138 (smallest expomorphic number in base n).
Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers).
Cf. A133619 (leading digits).

tetrmod(b, n, m)=my(t=b); for(i=1, n, if(i>1, t=lift(Mod(b,m)^t), t)); t
for(n=1, 21,if(tetrmod(9,n,10^n)!=tetrmod(9,n-1,10^(n-1)),print1(tetrmod(9,n,10^(n-1)),", "))) \\ Davis Smith, Mar 09 2019

a(8)-a(20) from Davis Smith, Mar 07 2019

A289138 a(n) = smallest expomorphic number in base n: least integer k such that n^k ends in k, or 0 if no such k exists.

Original entry on oeis.org

1, 36, 7, 6, 5, 6, 3, 56, 9, 0, 1, 16, 7, 6, 5, 6, 3, 76, 9, 0, 1, 96, 7, 6, 5, 6, 3, 96, 9, 0, 1, 76, 7, 6, 5, 6, 3, 16, 9, 0, 1, 56, 7, 6, 5, 6, 3, 36, 9, 0, 1, 36, 7, 6, 5, 6, 3, 56, 9, 0, 1, 16, 7, 6, 5, 6, 3, 76, 9, 0, 1, 96, 7, 6, 5, 6, 3, 96, 9, 0, 1, 76, 7, 6, 5, 6, 3, 16, 9, 0, 1, 56, 7, 6, 5, 6, 3, 36, 9, 0

Offset: 1

Bernard Schott and Robert G. Wilson v, Jun 28 2017

Definition: For positive integers b (the base) and n, the positive integer (allowing initial zeros) a(n) is expomorphic relative to base b if a(n) has exactly n decimal digits and if b^a(n) == a(n) (mod 10^n) or, equivalently, b^a(n) ends in a(n). [See Crux Mathematicorum link.]
The only twelve values a(n) can take are 0, 1, 3, 5, 6, 7, 9, 16, 36, 56, 76 and 96;
and the percentages of the time these occur are 10, 10, 10, 10, 20, 10, 10, 4, 4, 4, 4 and 4, respectively.
The bases, n, for which k is:
n == 0 (mod 10)
n == 1 (mod 10)
n == 7 (mod 10)
n == 5 (mod 10)
n == 4 or 6 (mod 10)
n == 3 (mod 10)
n == 9 (mod 10)
n == +/- 12 (mod 50)
n == +/-  2 (mod 50)
n == +/-  8 (mod 50)
n == +/- 18 (mod 50)
n == +/- 22 (mod 50).
Periodicity is 50.

			a(4) is 6 since 4^6 = 4096 which ends in 6.

Charles W. Trigg, Problem 559, Crux Mathematicorum, page 192, Vol. 7, Jun. 1981.
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, 1).

Cf. A288845.

f:= proc(n) local k;
   if n mod 10 = 0 then return 0 fi;
   for k from 1 do if n^k - k mod 10^(1+ilog10(k)) = 0 then return k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Jul 07 2017

f[n_] := If[ Mod[n, 10] > 0, Block[{k = 1}, While[ PowerMod[n, k, 10^IntegerLength[k]] != k, k++]; k], 0]; Array[f, 88]

def a(n):
    if n%10==0: return 0
    k=1
    while pow(n, k, 10**len(str(k)))!=k: k+=1
    return k
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 29 2017

A306570 Values of n such that 5^n ends in n, or expomorphic numbers relative to "base" 5.

Original entry on oeis.org

5, 25, 125, 3125, 203125, 8203125, 408203125, 8408203125, 18408203125, 618408203125, 2618408203125, 52618408203125, 152618408203125, 3152618408203125, 93152618408203125, 493152618408203125, 7493152618408203125, 17493152618408203125, 117493152618408203125, 7117493152618408203125, 87117493152618408203125

Offset: 1

Bernard Schott, Feb 24 2019

Definition: For positive integers b (as base) and n, the positive integer (allowing initial zeros) k(n) is expomorphic relative to base b (here 5) if k(n) has exactly n decimal digits and if b^k(n) == k(n) (mod 10^n) or, equivalently, b^k(n) ends in k(n). [See Crux Mathematicorum link.]
For sequences in the OEIS, no term is allowed to begin with a digit 0 (except for the 1-digit number 0 itself). However, in the problem as defined in the Crux Mathematicorum article, leading 0 digits are allowed; under that definition a(n) = k(n) until the first k(n) which begins with digit 0. When k(n) begins with 0, then, a(n) is the next term of the sequence k(n) which doesn't begin with digit 0.
Under that definition, the term after a(4) = 3125 is not "03125" but a(5) = 203125. [Comments from Jon E. Schoenfield in A288845 and discussion with Rémy Sigrist]
Conjecture: if k(n) is expomorphic relative to "base" b, then, the next one in the sequence, k(n+1), consists of the last n+1 digits of b^k(n).
a(n) is the backward concatenation of A133615(0) through A133615(n-1). So, a(1) is 5, a(2) is 25, and so on, with recognition of the comments about the OEIS and terms beginning with 0 (for example, when n = 5, A133615(n-1) = 0, so the next nonzero digit is concatenated as well, reducing the amount subtracted from n by 1). - Davis Smith Mar 07 2019

			5^5 = 25 ends in 5, so 5 is a term; 5^25 = ...125 ends in 25, so 25 is another term.

Davis Smith, Table of n, a(n) for n = 1..894
Charles W. Trigg, Problem 559, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.

Cf. A064541 (base 2), A183613 (base 3), A288845 (base 4), A290788 (base 6), A321970 (base 7), A306686 (base 9), A289138 (smallest expomorphic number in base n).
Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers).
Cf. A133615 (leading digits).

is(n) = my(t=#digits(n)); lift(Mod(5, 10^t)^n)==n
for(n=1, oo, my(x=n*5); if(lift(Mod(5, 10)^x)==x%10, if(is(x), print1(x, ", ")))) \\ Felix Fröhlich, Feb 24 2019

tetrmod(b,n,m)=my(t=b); for(i=1, n, if(i>1, t=lift(Mod(b,m)^t), t)); t
for(n=1, 21,if(tetrmod(5,n,10^n)!=tetrmod(5,n-1,10^(n-1)),print1(tetrmod(5,n,10^(n-1)),", "))) \\ Davis Smith, Mar 09 2019

a(5)-a(7) from Felix Fröhlich, Feb 24 2019
a(8) from Michel Marcus, Mar 02 2019
a(9)-a(21) from Davis Smith, Mar 07 2019

A290788 Values of n such that 6^n ends in n, or expomorphic numbers in "base" 6.

Original entry on oeis.org

6, 56, 656, 8656, 38656, 238656, 7238656, 47238656, 447238656, 7447238656, 27447238656, 227447238656, 3227447238656

Offset: 1

Bernard Schott, Aug 10 2017

Definition: For positive integers b (as base) and n, the positive integer (allowing initial 0's) a(n) is expomorphic relative to base b (here 6) if a(n) has exactly n decimal digits and if b^a(n) == a(n) (mod 10^n) or, equivalently, b^a(n) ends in a(n). [See Crux Mathematicorum link.]

			6^6 = 46656 ends in 6, so 6 is a term.
6^56 = ...656 ends in 56, so 56 is another term.

Charles W. Trigg, Problem 559, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.

Cf. A064541 (base 2), A183613 (base 3), A288845 (base 4), A289138, A306570 (base 5), A306686 (base 9).

Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers).

Select[Range[10^6], PowerMod[6, #, 10^(1 + Floor@ Log10[#])] == # &] (* Michael De Vlieger, Apr 13 2021 *)

is(n)=my(m=10^#digits(n)); Mod(6,m)^n==n \\ Charles R Greathouse IV, Aug 10 2017

a(6)-a(9) from Charles R Greathouse IV, Aug 10 2017

a(10)-a(13) from Chai Wah Wu, Apr 13 2021

A351410 Numbers m such that the decimal representation of 8^m ends in m.

Original entry on oeis.org

56, 856, 5856, 25856, 225856, 5225856, 95225856, 895225856, 6895225856, 16895225856, 416895225856, 5416895225856, 35416895225856, 7035416895225856, 77035416895225856, 577035416895225856, 1577035416895225856, 21577035416895225856, 521577035416895225856, 1521577035416895225856, 81521577035416895225856

Offset: 1

Bernard Schott, Feb 10 2022

The Crux Mathematicorum link calls these numbers "expomorphic" relative to "base" b, with here b = 8.
Under that definition, the term after a(13) = 35416895225856 is not "035416895225856" or "35416895225856" but a(14) = 7035416895225856.
Conjecture: if k(n) is "expomorphic" relative to "base" b, then the next one in the sequence, k(n+1), consists of the last n+1 digits of b^k(n).
This conjecture is true. See A133618. - David A. Corneth, Feb 10 2022

			8^56 = 374144419156711147060143317175368453031918731001856, so 56 is a term.
8^856 = ...5856 ends in 856, so 856 is another term.

Charles W. Trigg, Problem 559, Crux Mathematicorum, pp. 192-194, Vol. 7, Jun. 1981.

Cf. A064541, A183613, A288845, A306570, A290788, A321970, this sequence, A306686, A289138.
Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers).
Cf. A133618 (leading digits).

a(7)-a(8) from Michel Marcus, Feb 10 2022

More terms from David A. Corneth, Feb 10 2022

cp's OEIS Frontend

A049302 Numbers k such that k is a substring of 4^k.

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A320930 Numbers k such that 4^k starts with k.

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A306686 Values of n such that 9^n ends in n, or expomorphic numbers relative to "base" 9.

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A289138 a(n) = smallest expomorphic number in base n: least integer k such that n^k ends in k, or 0 if no such k exists.

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A306570 Values of n such that 5^n ends in n, or expomorphic numbers relative to "base" 5.

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PARI

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A290788 Values of n such that 6^n ends in n, or expomorphic numbers in "base" 6.

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Mathematica

PARI

Extensions

A351410 Numbers m such that the decimal representation of 8^m ends in m.

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