A343740 a(n) is the digit position of the first appearance of the last digit to appear in sqrt(n) (or -1 if n is a square).

-1, 19, 23, -1, 37, 39, 45, 36, -1, 27, 17, 25, 15, 36, 19, -1, 20, 36, 25, 37, 28, 13, 27, 52, -1, 39, 17, 38, 27, 26, 17, 23, 24, 37, 19, -1, 25, 26, 26, 41, 58, 57, 25, 12, 25, 22, 24, 19, -1, 33, 48, 23, 41, 49, 23, 32, 32, 23, 30, 19, 17, 31, 27, -1, 24

Offset

1,2

Comments

A343739(n) is the last digit to appear in the decimal expansion of sqrt(n) (or -1 if n is a square), so a(n) is the digit position of the first appearance of the digit A343739(n) in sqrt(n).

(The first digit of sqrt(n) is counted as digit position 1; the decimal point is disregarded.)

Links

Table of n, a(n) for n=1..65.

Formula

a(100^q*n) = a(n), q > 0. - Bernard Schott, Jul 29 2021

Example

a(2)=19 because A343739(2)=8 and the first appearance of an 8 in sqrt(2) = 1.414213562373095048... is at the 19th digit;
a(24)=52 because A343739(24)=0 and the first appearance of a 0 in sqrt(24) = 4.898979485566356196394568149411782783931894961313340... is at the 52nd digit.

Mathematica

Table[If[IntegerQ@ Sqrt@ n, -1, Function[s, Max@ Array[FirstPosition[s, #][[1]] &, 10, 0]]@ RealDigits[Sqrt[n], 10, 120][[1]]], {n, 65}] (* Michael De Vlieger, Jul 06 2021 *)

Prog

(Python 3.8+)
from math import isqrt
def A343740(n):
    i = isqrt(n)
    if i**2 == n:
        return -1
    m, dset, pos = n, set(), 0
    for d in (int(s) for s in str(i)):
        dset.add(d)
        pos += 1
        if len(dset) == 10:
            return pos
    while len(dset) < 10:
        m *= 100
        d = isqrt(m) % 10
        dset.add(d)
        pos += 1
    return pos # Chai Wah Wu, Jul 07 2021

Crossrefs

Cf. A023961, A343739, A343741, A343742.

Sequence in context: A303299 A072305 A095222 * A317254 A226687 A070299

Adjacent sequences: A343737 A343738 A343739 * A343741 A343742 A343743

Keyword

sign,base

Author

Jon E. Schoenfield, Jul 05 2021

Status

approved