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

1,3

Comments

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

Links

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.

Example

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

Mathematica

(* 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}]]

Prog

(Python)
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

Crossrefs

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

Intersection of A093534 and A301912.

Sequence in context: A007185 A175852 A030995 * A215118 A218150 A176594

Adjacent sequences: A067267 A067268 A067269 * A067271 A067272 A067273

Keyword

base,nonn

Author

Joseph L. Pe, Feb 21 2002

Extensions

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

0 prepended by David A. Corneth, Aug 02 2018

Status

approved