A348490 Positive numbers whose square starts and ends with exactly one 6.

26, 246, 254, 256, 264, 776, 784, 786, 794, 796, 804, 806, 824, 826, 834, 836, 2454, 2456, 2464, 2466, 2474, 2476, 2484, 2486, 2494, 2496, 2504, 2506, 2514, 2516, 2524, 2526, 2534, 2536, 2544, 2546, 2554, 2556, 2564, 2566, 2594, 2596, 2604, 2606, 2614, 2616, 2624, 2626, 2634, 2636, 2644, 7746

Offset

1,1

Comments

When a square ends with 6, it ends with only one 6.

From Marius A. Burtea, Oct 30 2021 : (Start)

The sequence is infinite because the numbers 806, 8006, 80006, ..., 8*10^k + 6, k >= 2, are terms with squares 649636, 64096036, 6400960036, 640009600036, ..., 64*10^(2*k) + 96*10^k + 36, k >= 2.

Numbers 796, 7996, 79996, 799996, 7999996, 79999996, ..., 10^k*8 - 4, k >= 2, are terms and have no digits 0, because their squares are 633616, 63936016, 6399360016, 639993600016, 63999936000016, 6399999360000016, ....

Also 794, 7994, 79994, 799994, ..., (8*10^k - 6), k >= 2, are terms and have no digits 0, because their squares are 630436, 63904036, 6399040036, 639990400036, 63999904000036, 6399999040000036, ... (End)

Links

Table of n, a(n) for n=1..52.

Example

26^2 = 676, hence 26 is a term.
814^2 = 662596, hence 814 is not a term.

Mathematica

Select[Range[10, 7750], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 6 && d[[2]] != 6 &] (* Amiram Eldar, Oct 30 2021 *)

Prog

(Python)
from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("6")) == len(s.lstrip("6")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [4, 6]))
  return [k for k in r if ok(k)]
print(aupto(2644)) # Michael S. Branicky, Oct 29 2021
(PARI) isok(k) = my(d=digits(sqr(k))); (d[1]==6) && (d[#d]==6) && if (#d>2, (d[2]!=6) && (d[#d-1]!=6), 1); \\ Michel Marcus, Oct 30 2021
(Magma) [n:n in [4..7500]|Intseq(n*n)[1] eq 6 and Intseq(n*n)[#Intseq(n*n)] eq 6 and Intseq(n*n)[-1+#Intseq(n*n)] ne 6 ]; // Marius A. Burtea, Oct 30 2021

Crossrefs

Cf. A045789, A045860, A273373 (squares ending with 6).

Similar to: A348487 (k=1), A348488 (k=4), A348489 (k=5), this sequence (k=6), A348491 (k=9).

Subsequence of A305719.

Sequence in context: A020537 A184461 A088889 * A060105 A108645 A086023

Adjacent sequences: A348487 A348488 A348489 * A348491 A348492 A348493

Keyword

nonn,base

Author

Bernard Schott, Oct 29 2021

Status

approved