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

Author

Joseph L. Pe, Feb 21 2002

Keywords

• base
• nonn

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

Examples

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

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.

Crossrefs

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

Intersection of A093534 and A301912.

Programs

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}]]

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

Extensions

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

0 prepended by David A. Corneth, Aug 02 2018