A016090 -id:A016090 - cp's OEIS frontend

A259986 This sequence and A259987 are base-6 analogs of A007185 and A016090, written in base 10.

3, 9, 81, 81, 6561, 29889, 76545, 636417, 3995649, 24151041, 326481921, 689278977, 11573190657, 76876660737, 155240824833, 2035980763137, 4857090670593, 55637069004801, 157197025673217, 1375916505694209, 19656708706009089, 129341461907898369

Offset: 1

N. J. A. Sloane, Jul 13 2015

See Schut (1991) for precise definition.

Ignoring repetitions, the subsequence of A237583 of terms ending in 3 in base 6. - Eric M. Schmidt, Jul 18 2015

C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A007185, A016090, A201821, A237583, A259986-A259991.

def a(n) : return crt(1, 0, 2^n, 3^n) # Eric M. Schmidt, Jul 18 2015

More terms from Eric M. Schmidt, Jul 18 2015

A259991 This sequence and A259990 are base-14 analogs of A007185 and A016090, written in base 10.

Original entry on oeis.org

8, 148, 344, 36016, 151264, 1764736, 46941952, 1101076992, 15858967552, 139825248256, 2453862488064, 14602557997056, 127990382747648, 921705156001792, 56481739283791872, 523186025957228544, 15768859390622826496, 198716939766610001920, 3186868919241067200512

Offset: 1

N. J. A. Sloane, Jul 13 2015

See Schut (1991) for precise definition.

Ignoring repetitions, the subsequence of A201919 of terms ending in 8 in base 14. - Eric M. Schmidt, Jul 18 2015

C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.

Eric M. Schmidt, Table of n, a(n) for n = 1..500

Cf. A007185, A016090, A201919, A259986-A259991.

def a(n) : return crt(0, 1, 2^n, 7^n) # Eric M. Schmidt, Jul 18 2015

More terms from Eric M. Schmidt, Jul 18 2015

A214881 2-adic valuation of A016090.

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 8, 8, 9, 13, 14, 14, 14, 14, 15, 17, 17, 18, 21, 21, 22, 23, 26, 26, 26, 26, 27, 28, 30, 30, 31, 32, 34, 34, 36, 36, 38, 38, 39, 42, 41, 48, 43, 44, 49, 46, 47, 49, 52, 51, 51, 52, 53, 56, 56, 56, 63, 58, 59, 60, 61, 65, 64, 64, 66, 66, 67

Offset: 1

Eric M. Schmidt, Jul 29 2012

a(n) >= n.

It appears that the density of n such that a(n) = n is 1/2.

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
Index entries for sequences related to automorphic numbers

Cf. A016090, A214880, A214882, A214883.

A214881 := function(max) local result, i; result := [6]; for i in [2..max] do Add(result, (2*result[i-1] - result[i-1]^2) mod 10^i); od; result := List(result, x->Valuation(x, 2)); return result; end;

A214883 A016090(n)/2^n.

Original entry on oeis.org

3, 19, 47, 586, 293, 1709, 55542, 340271, 1537323, 1745224, 39935112, 19967556, 9983778, 2446398139, 22585503757, 57069119066, 333710340783, 2837143256329, 14770012057852, 7385006028926, 289794797936338, 621734557171294, 2695053069601272, 1347526534800636

Offset: 1

Eric M. Schmidt, Jul 31 2012

Conjecture: For any m coprime to 5 and for any k, the density of n such that a(n) == k (mod m) is 1/m.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Index entries for sequences related to automorphic numbers

Cf. A214880, A214881, A214882.

A214883 := function(max) local result, i; result := [6]; for i in [2..max] do Add(result, (2*result[i-1] - result[i-1]^2) mod 10^i); od; result := List([1..max], n->result[n]/2^n); return result; end;

A259987 This sequence and A259986 are base 6 analogs of A007185 and A016090, written in base 10.

Original entry on oeis.org

4, 28, 136, 1216, 1216, 16768, 203392, 1043200, 6082048, 36315136, 36315136, 1487503360, 1487503360, 1487503360, 314944159744, 785129144320, 12069568774144, 45922887663616, 452162714337280, 2280241934368768, 2280241934368768, 2280241934368768, 2280241934368768

Offset: 1

N. J. A. Sloane, Jul 13 2015

See Schut (1991) for precise definition.

Ignoring repetitions, the subsequence of A237583 of terms ending in 4 in base 6. - Eric M. Schmidt, Jul 18 2015

C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A007185, A016090, A201821, A237583, A259986-A259991.

def a(n) : return crt(0, 1, 2^n, 3^n) # Eric M. Schmidt, Jul 18 2015

More terms from Eric M. Schmidt, Jul 18 2015

A259988 This sequence and A259989 are base-6 analogs of A007185 and A016090, written in base 6.

Original entry on oeis.org

3, 13, 213, 213, 50213, 350213, 1350213, 21350213, 221350213, 2221350213, 52221350213, 152221350213, 5152221350213, 55152221350213, 155152221350213, 4155152221350213, 14155152221350213, 314155152221350213, 1314155152221350213, 21314155152221350213

Offset: 1

N. J. A. Sloane, Jul 13 2015

See Schut (1991) for precise definition.

Ignoring repetitions, the subsequence of A201821 of terms ending in 3. - Eric M. Schmidt, Jul 18 2015

C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A007185, A016090, A201821, A237583, A259986-A259991.

def a(n) : return Integer(crt(1, 0, 2^n, 3^n).str(6)) # Eric M. Schmidt, Jul 18 2015

More terms from Eric M. Schmidt, Jul 18 2015

A259989 This sequence and A259988 are base-6 analogs of A007185 and A016090, written in base 6.

Original entry on oeis.org

4, 44, 344, 5344, 5344, 205344, 4205344, 34205344, 334205344, 3334205344, 3334205344, 403334205344, 403334205344, 403334205344, 400403334205344, 1400403334205344, 41400403334205344, 241400403334205344, 4241400403334205344, 34241400403334205344

Offset: 1

N. J. A. Sloane, Jul 13 2015

See Schut (1991) for precise definition.

Ignoring repetitions, the subsequence of A201821 of terms ending in 4. - Eric M. Schmidt, Jul 18 2015

C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A007185, A016090, A201821, A237583, A259986-A259991.

def a(n) : return Integer(crt(0, 1, 2^n, 3^n).str(6)) # Eric M. Schmidt, Jul 18 2015

Corrected and extended by Eric M. Schmidt, Jul 18 2015

A259990 This sequence and A259991 are base-14 analogs of A007185 and A016090, written in base 10.

Original entry on oeis.org

7, 49, 2401, 2401, 386561, 5764801, 58471553, 374712065, 4802079233, 149429406721, 1595702681601, 42091354378241, 665724390506497, 10190301669556225, 99086356274020353, 1654767311852142593, 14722487338708369409, 228161914444026740737, 2789435039707847196673

Offset: 1

N. J. A. Sloane, Jul 13 2015

See Schut (1991) for precise definition.

Ignoring repetitions, the subsequence of A201919 of terms ending in 7 in base 14. - Eric M. Schmidt, Jul 18 2015

C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.

Eric M. Schmidt, Table of n, a(n) for n = 1..500

Cf. A007185, A016090, A201919, A259986-A259991.

def a(n) : return crt(1, 0, 2^n, 7^n) # Eric M. Schmidt, Jul 18 2015

More terms from Eric M. Schmidt, Jul 18 2015

A003226 Automorphic numbers: m^2 ends with m.

Original entry on oeis.org

0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376, 59918212890625, 259918212890625, 740081787109376

Offset: 1

N. J. A. Sloane

Also called curious numbers.
For entries after the second, two successive terms sum up to a total having the form 10^n + 1. - Lekraj Beedassy, Apr 29 2005 [This comment is clearly wrong as stated. The sums of two consecutive terms are 1, 6, 11, 31, 101, 452, 1001, 10001, 100001, 200001, 1000001, 3781250, .... - T. D. Noe, Nov 14 2010]
If a d-digit number n is in the sequence, then so is 10^d+1-n. However, the same number can be 10^d+1-n for different n in the sequence (e.g., 10^3+1-376 = 10^4+1-9376 = 625), which spoils Beedassy's comment. - Robert Israel, Jun 19 2015
Substring of both its square and its cube not congruent to 0 (mod 10). See A029943. - Robert G. Wilson v, Jul 16 2005
a(n)^k ends with a(n) for k > 0; see also A029943. - Reinhard Zumkeller, Nov 26 2011
Apart from initial term, a subsequence of A046831. - M. F. Hasler, Dec 05 2012
This is also the sequence of numbers such that the n-th m-gonal number ends in n for any m == 0,4,8,16 (mod 20). - Robert Dawson, Jul 09 2018
Apart from 6, a subsequence of A301912. - Robert Dawson, Aug 01 2018

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 76, p. 26, Ellipses, Paris 2008.
V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170-174.
Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 253-254.
B. A. Naik, 'Automorphic numbers' in 'Science Today'(subsequently renamed '2001') May 1982 pp. 59, Times of India, Mumbai.
Ya. I. Perelman, Algebra can be fun, pp. 97-98.
Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Hoboken, 2005, p. 64.
C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Chai Wah Wu, Table of n, a(n) for n = 1..1786 (terms n = 1..200 from Vincenzo Librandi)
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2.3 (1969), 170-174. (Annotated scanned copy)
V. deGuerre & R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1.3 (1968), 173-179
M. Beeler, R. W. Gosper, and R. Schroeppel, HAKMEM. MIT AI Memo 239, Feb 29 1972 (Item 59 provides a 40-digit member of this sequence).
Shyam Sunder Gupta, Elegance of Squares, Cubes, and Higher Powers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 2, 29-81.
W. Schneider, Automorphic Numbers
C. P. Schut, Idempotents, Report AM-R9101, Centre for Mathematics and Computer Science, Amsterdam, 1991. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Automorphic Number
Wikipedia, Automorphic number
Xiaolong Ron Yu, Curious Numbers, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823.
Index entries for sequences related to automorphic numbers

Cf. A008851, A018247, A018248, A018834, A033819, A035383, A046831, A052228.

import Data.List (isSuffixOf)
a003226 n = a003226_list !! (n-1)
a003226_list = filter (\x -> show x `isSuffixOf` show (x^2)) a008851_list
-- Reinhard Zumkeller, Jul 27 2011

[n: n in [0..10^7] | Intseq(n^2)[1..#Intseq(n)] eq Intseq(n)]; // Vincenzo Librandi, Jul 03 2015

V:= proc(m) option remember;
  select(t -> t^2 - t mod 10^m = 0, map(s -> seq(10^(m-1)*j+s, j=0..9), V(m-1)))
end proc:
V(0):= {0,1}:
V(1):= {5,6}:
sort(map(op,[V(0),seq(V(i) minus V(i-1),i=1..50)])); # Robert Israel, Jun 19 2015

f[k_] := (r = Reduce[0 < 10^k < n < 10^(k + 1) && n^2 == m*10^(k + 1) + n, {n, m}, Integers]; If[Head[r] === And, n /. ToRules[r], n /. {ToRules[r]}]); Flatten[ Join[{0, 1}, Table[f[k], {k, 0, 13}]]] (* Jean-François Alcover, Dec 01 2011 *)
Union@ Join[{1}, Array[PowerMod[5, 2^#, 10^#] &, 16, 0], Array[PowerMod[16, 5^#, 10^#] &, 16, 0]] (* Robert G. Wilson v, Jul 23 2018 *)

is_A003226(n)={n<2 || 10^valuation(n^2-n,10)>n} \\ M. F. Hasler, Dec 05 2012

A003226(n)={ n<3 & return(n-1); my(i=10,j=10,b=5,c=6,a=b); for( k=4,n, while(b<=a, b=b^2%i*=10); while(c<=a, c=(2-c)*c%j*=10); a=min(b,c)); a } \\ M. F. Hasler, Dec 06 2012

from itertools import count, islice
from sympy.ntheory.modular import crt
def A003226_gen(): # generator of terms
    a = 0
    yield from (0,1)
    for n in count(0):
        b = sorted((int(crt(m:=(1< a:
            yield from b
            a = b[1]
        elif b[1] > a:
            yield b[1]
            a = b[1]
A003226_list = list(islice(A003226_gen(),15)) # Chai Wah Wu, Jul 25 2022

def automorphic(maxdigits, pow, base=10) :
    morphs = [[0]]
    for i in range(maxdigits):
        T=[d*base^i+x for x in morphs[-1] for d in range(base)]
        morphs.append([x for x in T if x^pow % base^(i+1) == x])
    res = list(set(sum(morphs, []))); res.sort()
    return res
# call with pow=2 for this sequence, Eric M. Schmidt, Feb 09 2014

Equals {0, 1} union A007185 union A016090.

More terms from Michel ten Voorde, Apr 11 2001
Edited by David W. Wilson, Sep 26 2002
Incorrect statement removed from title by Robert Dawson, Jul 09 2018

A007185 Automorphic numbers ending in digit 5: a(n) = 5^(2^n) mod 10^n.

Original entry on oeis.org

5, 25, 625, 625, 90625, 890625, 2890625, 12890625, 212890625, 8212890625, 18212890625, 918212890625, 9918212890625, 59918212890625, 259918212890625, 6259918212890625, 56259918212890625, 256259918212890625, 2256259918212890625, 92256259918212890625

Offset: 1

N. J. A. Sloane

Conjecture: For any m coprime to 10 and for any k, the density of n such that a(n) == k (mod m) is 1/m. - Eric M. Schmidt, Aug 01 2012

a(n) is the unique positive integer less than 10^n such that a(n) is divisible by 5^n and a(n) - 1 is divisible by 2^n. - Eric M. Schmidt, Aug 18 2012

			625 is in the sequence because 625^2 = 390625, which ends in 625.
90625 is in the sequence because 90625^2 = 8212890625, which ends in 90625.
90635 is not in the sequence because 90635^2 = 8214703225, which does not end in 90635.

V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170-174.
Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 253-4.
Ya. I. Perelman, Algebra can be fun, pp. 97-98.
C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
C. P. Schut, Idempotents, Report AM-R9101, Centre for Mathematics and Computer Science, Amsterdam, 1991. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Automorphic Number
Xiaolong Ron Yu, Curious Numbers, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823.
Index entries for sequences related to automorphic numbers

A018247 gives the associated 10-adic number.

A003226 = {0, 1} union (this sequence) union A016090.

[Modexp(5, 2^n, 10^n): n in [1..30]]; // Vincenzo Librandi, Jun 11 2016

a:= n-> 5&^(2^n) mod 10^n: seq(a(n), n=1..25);  # Alois P. Heinz, Mar 11 2018

Table[PowerMod[5, 2^n, 10^n], {n, 25}] (* Vincenzo Librandi, Jun 11 2016 *)

A007185(n)=lift(Mod(5,10^n)^2^n)  \\ M. F. Hasler, Dec 05 2012

[crt(1, 0, 2^n, 5^n) for n in range(1, 1001)] # Eric M. Schmidt, Aug 18 2012

a(n) = 5^(2^n) mod 10^n.
a(n)^2 == a(n) (mod 10^n), that is, a(n) is an idempotent in Z[10^n].
a(n+1) = a(n)^2 mod 10^(n+1). - Eric M. Schmidt, Jul 28 2012
a(2n) = (3*a(n)^2 - 2*a(n)^3) mod 10^(2n). - Sylvie Gaudel, Mar 10 2018

More terms from David W. Wilson
Edited by David W. Wilson, Sep 26 2002
Further edited by N. J. A. Sloane, Jul 21 2010
Comment moved to name by Alonso del Arte, Mar 10 2018

cp's OEIS Frontend

A259986 This sequence and A259987 are base-6 analogs of A007185 and A016090, written in base 10.

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A259991 This sequence and A259990 are base-14 analogs of A007185 and A016090, written in base 10.

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A214881 2-adic valuation of A016090.

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GAP

A214883 A016090(n)/2^n.

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GAP

A259987 This sequence and A259986 are base 6 analogs of A007185 and A016090, written in base 10.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Sage

Extensions

A259988 This sequence and A259989 are base-6 analogs of A007185 and A016090, written in base 6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Sage

Extensions

A259989 This sequence and A259988 are base-6 analogs of A007185 and A016090, written in base 6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Sage

Extensions

A259990 This sequence and A259991 are base-14 analogs of A007185 and A016090, written in base 10.

Views

Author

Keywords

Comments

References