A018248 -id:A018248 - 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

A331549 15-adic integer x = ...8978C2E9CE8570624D4BDA86 satisfying x^2 = x.

Original entry on oeis.org

6, 8, 10, 13, 11, 4, 13, 4, 2, 6, 0, 7, 5, 8, 14, 12, 9, 14, 2, 12, 8, 7, 9, 8, 5, 6, 7, 14, 10, 12, 4, 13, 12, 5, 6, 14, 3, 1, 8, 3, 8, 8, 2, 9, 12, 5, 14, 13, 9, 13, 4, 5, 3, 6, 6, 0, 14, 2, 8, 14, 13, 13, 2, 0, 12, 1, 9, 1, 0, 5, 4, 2, 0, 5, 8, 8, 14, 2, 2, 7

Offset: 0

Patrick A. Thomas, Jan 20 2020

The base-15 version of A018248. A, B, C, D, and E are the standard representations of the hexadecimal digits 10, 11, 12, 13, and 14, respectively.

			x^2 = ...8978C2E9CE8570624D4BDA86 = x.

Cf. A018248.

\\ See A331548, with initial b=6 instead of b=3.

A331549_vec(n)=Vecrev(digits(lift(chinese(Mod(0,3^n),Mod(1,5^n))),15)) \\ or simpler but slower: Vecrev(digits(lift(Mod(3^4,15^n)^5^(n-1)),15)) \\ M. F. Hasler, Jan 26 2020

x = 15-adic lim_{n->infinity} 3^(4*(5^n)).

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.

A381460 Smallest n-th perfect power greater than 1 satisfying A373387(a(n)) = n.

Original entry on oeis.org

2, 49, 15625, 625, 7737809375, 735091890625, 1253790880222890625, 6634204312890625, 47312447868976594992787109375, 72624607478879073313928212890625, 471781339858152691906169456697218212890625, 1344888824246298437178134918212890625

Offset: 1

Marco Ripà, Feb 24 2025

This sequence has infinitely many terms since a (trivial) upper bound for a(n) is given by (10^(n - v_{10}(n)) + 1)^n, where v_{10} corresponds to the number of trailing 0's of n (if any), and each of these terms is mandatorily a perfect power of degree n or a multiple of n (e.g., a(3) = 25^3 = 5^6).
All the a(n) are generated by the 13 nontrivial 10-adic solutions of the fundamental equation y^5 = y: ...480163574218751 (see A063006), ...263499879186432 (see A120817), ...996418333704193 (see A290375), ...476581907922943 (see A290373), ...259918212890624 (see A091664), ...259918212890625 (see A018247), ...740081787109375 (see A091663), ...740081787109376 (see A018248), ...003581666295807 (see A290372), ...523418092077057 (see A290374), ...736500120813568 (see A120818), ...519836425781249 (see A091661), and ...999999999999999.
The bases of a(11), a(12), ..., a(50) have been provided by Max Alekseyev on February 14, 2025 (see "Closed form for the general term of 2, 49, 15625, 625, ..." in Links).
It is conjectured that if n > 2 is given, then a(n) is generated by {5^2^k}_oo or -{5^2^k}_oo (this probabilistic argument is based on the study of a(n) up to n = 50 since a(50) is a 762 digit number generated by {5^2^k}_oo = ...6259918212890625).

			a(3) = 15625 since 15625 = 25^3 and 15625 is the smallest perfect cube whose constant congruence speed equals 3.

Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43—61.
Marco Ripà, Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration, ResearchGate, 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.
Marco Ripà, On the relation between perfect powers and tetration frozen digits, Journal of AppliedMath, 2024, 2(5), 1771.
Math Overflow, Closed form for the general term of 2, 49, 15625, 625, ....
Wikipedia, Tetration.

Cf. A018247, A018248, A063006, A091661, A091663, A091664, A120817, A120818, A290372, A290373, A290374, A290375, A317905, A373387, A379243.

A378421 Positive integers in A376446 sorted according to their appearance in that sequence.

Original entry on oeis.org

8, 64, 2486, 5, 4268, 8426, 2, 4, 4862, 46, 82, 6248, 6842, 8624, 2684, 28, 6, 9, 7139, 3179, 19, 1397, 1793, 91, 3971, 7931, 9713, 9317

Offset: 2

Marco Ripà, Nov 25 2024

Since a(28) = A376446(700001) = 9317, the present sequence has at least 28 terms.
If we merge A376446(n) and A377124(n*10), taking A376446(n) if and only if n is not a multiple of 10 and A376446(n*10) otherwise, we should get the sequence: 8, 64, 2486, 5, 4268, 8426, 2, 1, 4, 4862, 46, 82, 6248, 6, 6842, 8624, 2684, 28, 9, 7139, 3179, 19, 1397, 1793, 91, 3971, 7931, 9713, 9317 (which the author conjectures to be complete, as the present one).
Moreover, by construction, each term of this sequence is necessarily a circular permutation of the digits of one term of A376842 (e.g., a(4) = 2486 since A376842(4) = 6248).

			a(2) = 64 since A376446(2) = 64 (which is different from A376446(1) = 8).

Marco Ripà, Graham's number stable digits: an exact solution, arXiv:2411.00015 [math.GM], 2024.
Marco Ripà, Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration, ResearchGate, 2024. See pp. 9, 14, 27.
Wikipedia, Graham's Number.
Wikipedia, Tetration.

Cf. A317905, A370211, A372490, A373387, A376446, A376842, A377124.

Cf. A000007, A018247, A018248, A063006, A091661, A091663, A091664, A120817, A120818, A290372, A290373, A290374, A290375.

a(1) = 8, a(2) = 64, ..., a(28) = 9317 (and a(28) is the last term of the present sequence - conjectured).

cp's OEIS Frontend

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

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A331549 15-adic integer x = ...8978C2E9CE8570624D4BDA86 satisfying x^2 = x.

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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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A381460 Smallest n-th perfect power greater than 1 satisfying A373387(a(n)) = n.

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A378421 Positive integers in A376446 sorted according to their appearance in that sequence.

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