A348491 Positive numbers whose square starts and ends with exactly one 9.

3, 97, 303, 307, 313, 953, 957, 963, 967, 973, 977, 983, 987, 993, 3003, 3007, 3013, 3017, 3023, 3027, 3033, 3037, 3043, 3047, 3053, 3057, 3063, 3067, 3073, 3077, 3083, 3087, 3093, 3097, 3103, 3107, 3113, 3117, 3123, 3127, 3133, 3137, 3143, 9487, 9493, 9497, 9503, 9507, 9513, 9517

Offset

1,1

Comments

When a square ends with 9, it ends with only one 9.

From Marius A. Burtea, Nov 02 2021 : (Start)

The sequence is infinite because the numbers 303, 3003, 30003, ..., 3*(10^k + 1), k >= 2, are terms with squares 91809, 9018009, 900180009, 90001800009, ... 9*(10^(2*k) + 2*10^k + 1), k >= 2.

Numbers 97, 967, 9667, 96667, 966667, ..., (29*10^n + 1) / 3, k >= 1, are terms and have no digits 0, because their squares are 9409, 935089, 93450889, 9344508889, 934445088889, ...

Also 963, 9663, 96663, 966663, 9666663, 96666663, ... (29*10^k - 11) / 3, k >= 2, are terms and have no digits 0, because their squares are 927369, 93373569, 9343735569, 934437355569, 93444373555569, 9344443735555569, ... (End)

Links

Table of n, a(n) for n=1..50.

Example

97^2 = 9409, hence 97 is a term.
997^2 = 994009, hence  997 is not a term.

Mathematica

Join[{3}, Select[Range[10, 10^4], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 9 && d[[2]] != 9 &]] (* Amiram Eldar, Nov 02 2021 *)

Prog

(Magma) [3] cat [n:n in [4..9600]|Intseq(n*n)[1] eq 9 and Intseq(n*n)[#Intseq(n*n)] eq 9]; // Marius A. Burtea, Nov 02 2021
(Python)
from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("9")) == len(s.lstrip("9")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [3, 7]))
  return [k for k in r if ok(k)]
print(aupto(9517)) # Michael S. Branicky, Nov 02 2021
(PARI) isok(k) = my(d=digits(sqr(k))); (d[1]==9) && (d[#d]==9) && if (#d>2, (d[2]!=9) && (d[#d-1]!=9), 1); \\ Michel Marcus, Nov 03 2021
(PARI) list(lim)=my(v=List([3])); for(d=2, 2*#digits(lim\=1), my(s=sqrtint(9*10^(d-1)-1)+1); s+=[3, 2, 1, 0, 3, 2, 1, 0, 5, 4][s%10+1]; forstep(n=s, min(sqrtint(10^d-10^(d-2)-1), lim), if(s%10==3, [4, 6], [6, 4]), listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Nov 03 2021

Crossrefs

Subsequence of A305719, A063226, and A045863.

Cf. A017377, A045863, A273374 (squares ending with 9).

Similar to: A348487 (k=1), A348488 (k=4), A348489 (k=5), A348490 (k=6), this sequence (k=9).

Sequence in context: A209554 A320513 A320517 * A243155 A201843 A278202

Adjacent sequences: A348488 A348489 A348490 * A348492 A348493 A348494

Keyword

nonn,base,easy

Author

Bernard Schott, Nov 02 2021

Status

approved