A016090 -id:A016090 - cp's OEIS frontend

A067270 Numbers m such that m-th triangular number (A000217) ends in m.

0, 1, 5, 25, 625, 9376, 90625, 890625, 7109376, 12890625, 212890625, 1787109376, 81787109376, 59918212890625, 259918212890625, 3740081787109376, 56259918212890625, 256259918212890625, 7743740081787109376

Offset: 1

Joseph L. Pe, Feb 21 2002

Thanks to David W. Wilson for the proof that this sequence is a proper subset of A003226.
Also, numbers m such that the m-th k-gonal number ends in m for k == 1, 3, 5, or 9 (mod 10). - Robert Dawson, Jul 09 2018
This sequence is the intersection of A093534 and A301912. - Robert Dawson, Aug 01 2018

			The 5th triangular = 15 ends in 5, hence 5 is a term of the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..888
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.

Proper subset of A003226. Cf. A007185, A018247, A016090, A018248.

Intersection of A093534 and A301912.

(* a5=A018247 less the commas; a6=A018248 less the commas; *)
b5 = FromDigits[ Reverse[ IntegerDigits[a5]]]; b6 = FromDigits[ Reverse[ IntegerDigits[a6]]]; f[0] = 1; f[n_] := Block[{c5 = Mod[b5, 10^n], c6 = Mod[b6, 10^n]}, If[ Mod[c5(c5 + 1)/2, 10^n] == c5, c5, c6]]; Union[ Table[ f[n], {n, 0, 20}]]

from itertools import count, islice
from sympy.ntheory.modular import crt
def A067270_gen(): # generator of terms
    a = 0
    yield from (0,1)
    for n in count(0):
        if (b := int(min(crt(m:=(1<<(n+1),5**n),(0,1))[0], crt(m,(1,0))[0]))) > a:
            yield b
            a = b
A067270_list = list(islice(A067270_gen(),15)) # Chai Wah Wu, Jul 25 2022

Edited and extended by Robert G. Wilson v, Nov 20 2002

0 prepended by David A. Corneth, Aug 02 2018

A181607 Numbers n with k digits such that n^2 == 1 (mod 10^k).

Original entry on oeis.org

1, 9, 49, 51, 99, 249, 251, 499, 501, 749, 751, 999, 1249, 3751, 4999, 5001, 6249, 8751, 9999, 18751, 31249, 49999, 50001, 68751, 81249, 99999, 218751, 281249, 499999, 500001, 718751, 781249, 999999, 4218751, 4999999, 5000001, 5781249, 9218751

Offset: 1

Robert G. Wilson v, Nov 01 2010

Least term of n digits: 1, 49, 249, 1249, 18751, 218751, 4218751, ..., .
If n of k digits is present then 10^k-n is present.
The union of A002283, A198971, A199685, A224473, A224474, A224475, and A224476 (except that this sequence omits 0, 4, and 6). - Eric M. Schmidt, Jan 26 2016

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

Cf. A007185, A016090, A033819.

Table[ Select[ Range[10^(k - 1), 10^k - 1], Mod[ #^2, 10^k] == 1 &], {k, 7}] // Flatten

A076308 Product of 10-adic numbers defined in A018247 and A018248.

Original entry on oeis.org

3, 5, 4, 5, 3, 8, 5, 2, 5, 8, 5, 1, 5, 9, 4, 9, 8, 9, 4, 8, 7, 3, 8, 2, 9, 2, 2, 2, 2, 7, 0, 6, 4, 4, 7, 2, 3, 5, 0, 9, 7, 2, 0, 1, 7, 7, 7, 5, 4, 7, 8, 6, 9, 0, 1, 8, 9, 7, 3, 3, 8, 5, 1, 4, 1, 2, 4, 7, 0, 5, 8, 0, 6, 3, 6, 2, 4, 1, 6, 4, 6, 8, 3, 6, 8, 5, 1, 1, 1, 9, 3, 6, 3, 7, 0, 4, 0, 3, 9, 0, 0, 2, 5, 4, 0

Offset: 1

Robert G. Wilson v, Oct 05 2002

Cf. A018247, A018248, A007185, A016090.

(* execute the programming in both A018247 & A018248 *) IntegerDigits[ FromDigits[ Reverse[a]]*FromDigits[ Reverse[b]]]

A216102 10's-complement of A216103.

Original entry on oeis.org

9, 3, 2, 7, 6, 0, 6, 8, 7, 7, 5, 9, 5, 9, 9, 5, 3, 3, 5, 6, 4, 7, 5, 2, 4, 3, 0, 8, 2, 3, 5, 7, 1, 1, 7, 5, 7, 5, 3, 4, 4, 8, 8, 0, 0, 9, 9, 4, 5, 2, 8, 3, 4, 6, 6, 8, 0, 9, 9, 6, 1, 4, 6, 3, 6, 1, 4, 4, 8, 5, 5, 6, 1, 9, 1, 0, 5, 9, 4, 8, 7, 1, 2, 2, 9, 5, 4, 7, 4, 7, 2, 6, 8, 8, 1, 4

Offset: 1

V. Raman, Sep 01 2012

Cf. A216103, A007185, A016090, A091663, A018248, A091664, A018247.

a(n) = 10 - [{3^(5*10^(n+1)) mod 10^(2n+3) - 1}/(10^(2n+2)) mod 10].

Edited by N. J. A. Sloane, Sep 02 2012

A216103 [3^(5*10^(n+1)) mod 10^(2n+3) - 1]/(10^(2n+2)) mod 10.

Original entry on oeis.org

1, 6, 7, 2, 3, 9, 3, 1, 2, 2, 4, 0, 4, 0, 0, 4, 6, 6, 4, 3, 5, 2, 4, 7, 5, 6, 9, 1, 7, 6, 4, 2, 8, 8, 2, 4, 2, 4, 6, 5, 5, 1, 1, 9, 9, 0, 0, 5, 4, 7, 1, 6, 5, 3, 3, 1, 9, 0, 0, 3, 8, 5, 3, 6, 3, 8, 5, 5, 1, 4, 4, 3, 8, 0, 8, 9, 4, 0, 5, 1, 2, 8, 7, 7, 0, 4, 5, 2, 5, 2, 7, 3, 1, 1, 8, 5

Offset: 1

V. Raman, Sep 01 2012

3^500 ends in 7610001
3^5000 ends in 276100001
3^50000 ends in 32761000001
3^500000 ends in 9327610000001
The last digits before the zeros are converging to the present sequence.

Cf. A007185, A016090, A091663, A018248, A091664, A018247.

A216102 gives the 10's-complement.

A216104 10's complement of A216105.

Original entry on oeis.org

7, 9, 5, 3, 4, 2, 2, 0, 8, 3, 2, 2, 9, 8, 4, 5, 2, 5, 5, 3, 0, 0, 4, 5, 3, 2, 0, 8, 2, 1, 4, 7, 8, 2, 3, 8, 6, 9, 1, 2, 0, 1, 5, 6, 0, 2, 7, 4, 0, 5, 2, 6, 6, 1, 6, 5, 4, 8, 4, 3, 9, 7, 1, 2, 2, 8, 3, 7, 9, 4, 1, 1, 4, 8, 3, 6, 6, 2, 6, 2, 0, 8, 2, 4, 3, 1, 9, 6, 8, 7, 3, 7, 6, 2, 8, 6, 6, 3, 8, 4, 4, 9, 0, 0, 2, 0, 5, 9, 3, 0, 6, 1

Offset: 1

V. Raman, Sep 02 2012

nonn

7^500 ends in 64030001
7^5000 ends in 5640300001
7^50000 ends in 756403000001
7^500000 ends in 77564030000001
The last digits before the zeros tend to converge to A216105.

Cf. A007185, A016090, A091663, A018248, A091664, A018247, A216105.

A216105 [7^(5*10^(n+1)) mod 10^(2n+3) - 1]/(10^(2n+2)) mod 10.

Original entry on oeis.org

3, 0, 4, 6, 5, 7, 7, 9, 1, 6, 7, 7, 0, 1, 5, 4, 7, 4, 4, 6, 9, 9, 5, 4, 6, 7, 9, 1, 7, 8, 5, 2, 1, 7, 6, 1, 3, 0, 8, 7, 9, 8, 4, 3, 9, 7, 2, 5, 9, 4, 7, 3, 3, 8, 3, 4, 5, 1, 5, 6, 0, 2, 8, 7, 7, 1, 6, 2, 0, 5, 8, 8, 5, 1, 6, 3, 3, 7, 3, 7, 9, 1, 7, 5, 6, 8, 0, 3, 1, 2, 6, 2, 3, 7, 1, 3, 3, 6, 1, 5, 5, 0, 9, 9, 7, 9, 4, 0, 6, 9, 3, 8

Offset: 1

V. Raman, Sep 02 2012

7^500 ends in 64030001
7^5000 ends in 5640300001
7^50000 ends in 756403000001
7^500000 ends in 77564030000001
The last digits before the zeros tend to converge to the present sequence.

Cf. A007185, A016090, A091663, A018248, A091664, A018247, A216104.

A216159 10's-complement of A216161.

Original entry on oeis.org

7, 7, 7, 5, 6, 4, 2, 6, 8, 2, 9, 5, 5, 8, 7, 3, 6, 5, 8, 2, 2, 8, 9, 0, 3, 1, 5, 6, 3, 8, 1, 6, 8, 5, 3, 4, 0, 8, 0, 1, 7, 7, 8, 5, 6, 4, 4, 4, 6, 6, 0, 0, 9, 0, 8, 5, 1, 8, 4, 1, 7, 3, 0, 3, 2, 3, 3, 6, 3, 0, 0, 7, 0, 7, 5, 7, 5, 0, 6, 6, 8, 7, 1, 5, 0, 3, 2, 4, 4, 8, 4, 1, 0, 3, 2, 8, 8, 7, 3, 3, 9, 0, 5, 5, 7, 0

Offset: 1

V. Raman, Sep 02 2012

nonn

11^50 ends in 23001
11^500 ends in 2230001
11^5000 ends in 422300001
11^50000 ends in 34223000001
The last digits before the zeros tend to converge.

Cf. A216161, A007185, A016090, A091663, A018248, A091664, A018247.

a(n) = 10 - [{11^(5*10^n) mod 10^(2n+2) - 1}/(10^(2n+1)) mod 10].

A216161 [11^(5*10^n) mod 10^(2n+2) - 1]/(10^(2n+1)) mod 10.

Original entry on oeis.org

3, 2, 2, 4, 3, 5, 7, 3, 1, 7, 0, 4, 4, 1, 2, 6, 3, 4, 1, 7, 7, 1, 0, 9, 6, 8, 4, 3, 6, 1, 8, 3, 1, 4, 6, 5, 9, 1, 9, 8, 2, 2, 1, 4, 3, 5, 5, 5, 3, 3, 9, 9, 0, 9, 1, 4, 8, 1, 5, 8, 2, 6, 9, 6, 7, 6, 6, 3, 6, 9, 9, 2, 9, 2, 4, 2, 4, 9, 3, 3, 1, 2, 8, 4, 9, 6, 7, 5, 5, 1, 5, 8, 9, 6, 7, 1, 1, 2, 6, 6, 0, 9, 4, 4, 2, 9

Offset: 1

V. Raman, Sep 02 2012

11^50 ends in 23001
11^500 ends in 2230001
11^5000 ends in 422300001
11^50000 ends in 34223000001
The last digits before the zeros converge to this sequence.

Cf. A007185, A016090, A091663, A018248, A091664, A018247.

A216159 gives 10's-complement.

A094190 Least n-digit automorphic number.

Original entry on oeis.org

0, 25, 376, 9376, 90625, 109376, 2890625, 12890625, 212890625, 1787109376, 18212890625, 918212890625, 9918212890625, 40081787109376, 259918212890625, 3740081787109376, 43740081787109376, 256259918212890625, 2256259918212890625

Offset: 1

Lekraj Beedassy, May 25 2004

Cf. A003226, A003226, A007185, A016090.

f[n_] := If[n == 1, 0, Block[{a5 = PowerMod[5, 2^n, 10^n], a6 = PowerMod[16, 5^n, 10^n]}, If[a5 < 10^(n - 1), a5 = 10^(n + 1)]; If[a6 < 10^(n - 1), a6 = 10^(n + 1)]; Min[a5, a6]]]; Array[f, 19] (* Robert G. Wilson v, Aug 27 2006 *)

a(n) = Min( 5^(2^n) (mod 10^n), 16^(5^n) (mod 10^n) ). - Robert G. Wilson v, Aug 27 2006

Corrected and extended by Robert G. Wilson v, Aug 27 2006

cp's OEIS Frontend

A067270 Numbers m such that m-th triangular number (A000217) ends in m.

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A181607 Numbers n with k digits such that n^2 == 1 (mod 10^k).

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A076308 Product of 10-adic numbers defined in A018247 and A018248.

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A216102 10's-complement of A216103.

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A216103 [3^(5*10^(n+1)) mod 10^(2n+3) - 1]/(10^(2n+2)) mod 10.

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A216104 10's complement of A216105.

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A216105 [7^(5*10^(n+1)) mod 10^(2n+3) - 1]/(10^(2n+2)) mod 10.

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A216159 10's-complement of A216161.

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A216161 [11^(5*10^n) mod 10^(2n+2) - 1]/(10^(2n+1)) mod 10.

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A094190 Least n-digit automorphic number.

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