A064541 -id:A064541 - cp's OEIS frontend

A064540 Leading digits in A064541.

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

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 08 2001

Can be obtained from A133612 by removing the first term and all zeros. - Max Alekseyev, Aug 22 2013

More terms from David Wasserman, Jul 23 2002

A100129 Numbers k such that 2^k starts with k.

Original entry on oeis.org

6, 10, 1542, 77075, 113939, 1122772, 2455891300, 2830138178, 136387767490, 2111259099790, 3456955336468, 4653248164310, 10393297007134, 321249146279171, 972926121017616, 72780032758751764

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 15 2004

According to van de Lune, Erdős observed that 2^6 = 64 and 2^10 = 1024 were two examples where the decimal expansion of 2^k starts with that of k. At that time no other examples were known. Jan van de Lune computed the first 6 terms in 1978. - Juan Arias-de-Reyna, Feb 12 2016

			2^6 = 64, which begins with 6;
2^10 = 1024, which begins with 10.

J. van de Lune, A note on a problem of Erdős, Department of Pure Mathematics, ZW 87/78, Mathematisch Centrum, Amsterdam 1978, 1-3.

Cf. A064541 (2^k ending with k), A032740 (k a substring of 2^k), A131494.

f[n_] := Floor[ 10^Floor[ Log[10, n]](10^FractionalPart[n*N[ Log[10, 2], 24]])]; Do[ If[ f[n] == n, Print[n]], {n, 125000000}] (* Robert G. Wilson v, Nov 16 2004 *)

# Caveat: fails for large n due to rounding error.
from math import log10 as log
frac = lambda x: x - int(x)
is_a100129 = lambda n: 0 <= frac(n * log(2)) - frac(log(n)) < log(n + 1) - log(n) # David Radcliffe, Jun 02 2019

from itertools import count, islice
def A100129_gen(startvalue=1): # generator of terms
    a = 1<<(m:=max(startvalue,1))
    for n in count(m):
        if (s:=str(n))==str(a)[:len(s)]:
            yield n
        a <<= 1
A100129_list = list(islice(A100129_gen(),4)) # Chai Wah Wu, Apr 10 2023

The sequence contains k if and only if 0 <= {k*log_10(2)} - {log_10(k)} < log_10(k+1) - log_10(k), where {x} denotes the fractional part of x. See the van de Lune article. - David Radcliffe, Jun 02 2019

a(5) and a(6) from Robert G. Wilson v, Nov 16 2004

More terms from Robert Gerbicz, Aug 22 2006

A288845 Values of n such that 4^n ends in n, or expomorphic numbers in base 4.

Original entry on oeis.org

6, 96, 896, 8896, 28896, 728896, 1728896, 11728896, 411728896, 90411728896, 290411728896, 5290411728896, 55290411728896, 555290411728896, 2555290411728896, 302555290411728896, 2302555290411728896, 22302555290411728896, 622302555290411728896, 3622302555290411728896

Offset: 1

Bernard Schott, Jun 18 2017

Definition: For positive integers b (as base) and n, the positive integer (allowing initial zeros) a(n) is expomorphic relative to base b (here 4) 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.]

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, "0411728896" would be included because the last 10 digits of 4^0411728896 are 0411728896, and also 02555290411728896" because the last 17 digits of 4^02555290411728896 are "02555290411728896". However, these are not in the sequence as defined here. - Jon E. Schoenfield

			4^6 = 4096 ends in 6, so 6 is a term; 4^96 = ....896 ends in 96, so 96 is another term.

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

Cf. A064541 (base 2), A183613 (base 3).

Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers), A133614.

isok(n) = 10^valuation(4^n-n, 10)>n; \\ after A003226; Michel Marcus, Jun 18 2017

is(n)=my(m=10^#digits(n)); Mod(4, m)^n==n \\ Charles R Greathouse IV, Aug 10 2017

a(6)-a(9) from Gheorghe Coserea, Jun 21 2017

a(10)-a(11) from Robert G. Wilson v, Jun 24 2017

A109405 a(2) = 36; for n >= 3, a(n) = 2^a(n-1) mod 10^n.

Original entry on oeis.org

36, 736, 8736, 48736, 948736, 2948736, 32948736, 432948736, 3432948736, 53432948736, 353432948736, 5353432948736, 75353432948736, 75353432948736, 5075353432948736, 15075353432948736, 615075353432948736

Offset: 2

N. J. A. Sloane, following email from Max Alekseyev, May 06 2007

nonn

Decimal digits read backwards form A133612.

Related to but different from A064541 and A121319.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Same as A113627 except for the initial term (14). - Max Alekseyev, May 11 2007

a = 36; For[n = 3, n < 25, n++, a = PowerMod[2, a, 10^n]; Print[a]] (* Stefan Steinerberger, May 25 2007 *)
nxt[{n_,a_}]:={n+1,PowerMod[2,a,10^(n+1)]}; NestList[nxt,{2,36},20][[All,2]] (* Harvey P. Dale, Jan 22 2023 *)

More terms from Stefan Steinerberger, May 25 2007

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

A067844 Numbers k such that k and 2^k end with the same digit.

Original entry on oeis.org

14, 16, 34, 36, 54, 56, 74, 76, 94, 96, 114, 116, 134, 136, 154, 156, 174, 176, 194, 196, 214, 216, 234, 236, 254, 256, 274, 276, 294, 296, 314, 316, 334, 336, 354, 356, 374, 376, 394, 396, 414, 416, 434, 436, 454, 456, 474, 476, 494, 496, 514, 516, 534, 536

Offset: 1

Benoit Cloitre, Mar 07 2002

Also numbers k such that k and (2^(2*h+1))^k (for n>=0) end with the same digit. - Bruno Berselli, Dec 13 2018

			2^36 = 68719476736 hence 36 is in the sequence.

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

Cf. A064541.

LinearRecurrence[{1,1,-1},{14,16,34},70] (* Harvey P. Dale, Aug 19 2021 *)

isok(n) = (2^n - n) % 10 == 0; \\ Michel Marcus, Nov 23 2013

From Colin Barker, Dec 01 2012: (Start)
G.f.: 2*x*(2*x^2 + x + 7)/((x - 1)^2*(x + 1)).
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) = 10*n - 4*(-1)^n. (End)

Example corrected by Michel Marcus, Nov 23 2013

A113627 a(n) is the smallest number k such that k and 2^k have the same last n digits. Here k may have fewer than n digits and can be padded with leading zeros (cf. A121319).

Original entry on oeis.org

14, 36, 736, 8736, 48736, 948736, 2948736, 32948736, 432948736, 3432948736, 53432948736, 353432948736, 5353432948736, 75353432948736, 75353432948736, 5075353432948736, 15075353432948736, 615075353432948736, 8615075353432948736, 98615075353432948736, 98615075353432948736

Offset: 1

Jon E. Schoenfield, Apr 23 2007

			2^14 = 16384 and 14 end with the same single digit 4, thus a(1) = 14.

Jon E. Schoenfield, Table of n, a(n) for n = 1..81

See A121319, the main entry for this sequence, for further information.
Same as A109405 except for the initial term (14). - Max Alekseyev, May 11 2007
Cf. A007185, A016090, A003226, A035383, A064540, A064541, A109405, A121319, A206636.

A121319 a(n) is the smallest number k such that k and 2^k have the same last n digits. Here k must have at least n digits (cf. A113627).

Original entry on oeis.org

14, 36, 736, 8736, 48736, 948736, 2948736, 32948736, 432948736, 3432948736, 53432948736, 353432948736, 5353432948736, 75353432948736, 1075353432948736, 5075353432948736, 15075353432948736, 615075353432948736, 8615075353432948736, 98615075353432948736

Offset: 1

Tanya Khovanova, Aug 25 2006

			2^14 = 16384 and 14 end with the same single digit 4, thus a(1) = 14.

Jon E. Schoenfield, Table of n, a(n) for n = 1..80
Jon E. Schoenfield, Excel program

Cf. A007185, A016090, A003226, A035383, A064540, A064541, A109405.

f[n_] := Block[{k = If[n == 1, 2, 10], m = 10^n}, While[ PowerMod[2, k, m] != Mod[k, m], k += 2]; k]; Do[ Print@f@n, {n, 9}] (* Robert G. Wilson v *)

A121319(n) = { local(k,tn); tn=10^n ; forstep(k=2,1000000000,2, if ( k % tn == (2^k) % tn, return(k) ; ) ; ) ; return(0) ; } { for(n = 1,13, print( A121319(n)) ; ) ; } \\ R. J. Mathar, Aug 27 2006

If A109405(n) has n digits, a(n) = A109405(n), otherwise a(n) = A109405(n) + 10^n. - Max Alekseyev, May 05 2007

a(6)-a(9) from Robert G. Wilson v and Jon E. Schoenfield, Aug 26 2006
a(10) from Robert G. Wilson v, Sep 26 2006
a(11)-a(16) from Alexander Adamchuk, Jan 28 2007
a(16) corrected by Max Alekseyev, Apr 12 2007

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

cp's OEIS Frontend

A064540 Leading digits in A064541.

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A100129 Numbers k such that 2^k starts with k.

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A288845 Values of n such that 4^n ends in n, or expomorphic numbers in base 4.

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A109405 a(2) = 36; for n >= 3, a(n) = 2^a(n-1) mod 10^n.

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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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A067844 Numbers k such that k and 2^k end with the same digit.

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A113627 a(n) is the smallest number k such that k and 2^k have the same last n digits. Here k may have fewer than n digits and can be padded with leading zeros (cf. A121319).

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A121319 a(n) is the smallest number k such that k and 2^k have the same last n digits. Here k must have at least n digits (cf. A113627).

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