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

%I A067270 #33 Jul 26 2022 01:35:09
%S A067270 0,1,5,25,625,9376,90625,890625,7109376,12890625,212890625,1787109376,
%T A067270 81787109376,59918212890625,259918212890625,3740081787109376,
%U A067270 56259918212890625,256259918212890625,7743740081787109376
%N A067270 Numbers m such that m-th triangular number (A000217) ends in m.
%C A067270 Thanks to David W. Wilson for the proof that this sequence is a proper subset of A003226.
%C A067270 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
%C A067270 This sequence is the intersection of A093534 and A301912. - _Robert Dawson_, Aug 01 2018
%H A067270 Chai Wah Wu, <a href="/A067270/b067270.txt">Table of n, a(n) for n = 1..888</a>
%H A067270 Robert Dawson, <a href="https://www.emis.de/journals/JIS/VOL21/Dawson/dawson6.html">On Some Sequences Related to Sums of Powers</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
%e A067270 The 5th triangular = 15 ends in 5, hence 5 is a term of the sequence.
%t A067270 (* a5=A018247 less the commas; a6=A018248 less the commas; *)
%t A067270 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}]]
%o A067270 (Python)
%o A067270 from itertools import count, islice
%o A067270 from sympy.ntheory.modular import crt
%o A067270 def A067270_gen(): # generator of terms
%o A067270     a = 0
%o A067270     yield from (0,1)
%o A067270     for n in count(0):
%o A067270         if (b := int(min(crt(m:=(1<<(n+1),5**n),(0,1))[0], crt(m,(1,0))[0]))) > a:
%o A067270             yield b
%o A067270             a = b
%o A067270 A067270_list = list(islice(A067270_gen(),15)) # _Chai Wah Wu_, Jul 25 2022
%Y A067270 Proper subset of A003226. Cf. A007185, A018247, A016090, A018248.
%Y A067270 Intersection of A093534 and A301912.
%K A067270 base,nonn
%O A067270 1,3
%A A067270 _Joseph L. Pe_, Feb 21 2002
%E A067270 Edited and extended by _Robert G. Wilson v_, Nov 20 2002
%E A067270 0 prepended by _David A. Corneth_, Aug 02 2018