A071901 -id:A071901 - cp's OEIS frontend

A003076 n-th digit after decimal point of square root of n.

0, 0, 1, 2, 0, 6, 9, 3, 2, 0, 1, 5, 7, 9, 4, 6, 0, 4, 6, 2, 1, 8, 6, 3, 8, 0, 0, 4, 2, 4, 8, 5, 9, 9, 0, 4, 0, 0, 3, 2, 1, 2, 4, 3, 9, 8, 9, 0, 1, 0, 4, 3, 4, 3, 5, 7, 0, 8, 4, 3, 7, 7, 5, 7, 0, 6, 2, 6, 0, 5, 7, 0, 0, 3, 6, 4, 3, 0, 2, 5, 9, 0, 3, 8, 8, 8, 4, 0, 1, 8, 5, 4, 7, 3, 9, 6, 0, 7, 0, 2, 0, 6, 6, 5, 8

Offset: 0

N. J. A. Sloane

Regarded as a decimal fraction, 0.01206932015794604621863800... is likely to be an irrational number.

			sqrt(0) = 0.0, so a(0) = 0 with the convention that the 0th digit after the decimal point is the digit before the decimal point,
sqrt(1) = 1.0, where the first digit after the decimal point is a(1) = 0,
sqrt(2) = 1.4142135..., where the 2nd decimal digit is a(2) = 1,
sqrt(3) = 1.7320508..., where the 3rd decimal digit is a(3) = 2,
sqrt(4) = 2.0000000..., where the 4th decimal digit is a(4) = 0,
sqrt(5) = 2.2360679..., where the 5th decimal digit is a(5) = 6,
sqrt(6) = 2.4494897..., where the 6th decimal digit is a(6) = 9, etc.
From _M. F. Hasler_, Jun 22 2024: (Start)
For the frequency of the respective digits among the first 10^k terms, we have:
  k :   0's    1's   2's    3's   4's   5's   6's   7's   8's   9's
----+---------------------------------------------------------------
  1 :     4,     1,    2,     1,    0,    0,    1,    0,    0,    1;
  2 :    22,     6,    9,    11,   12,    7,    8,    8,    9,    8;
  3 :   126,   106,  105,    94,   95,   90,   86,   96,   92,  110;
  4 :  1097,  1026, 1037,  1031,  984,  979, 1000,  956,  922,  968;
  5 : 10320, 10053, 9926, 10122, 9855, 9985, 9934, 9857, 9855, 10093. (End)

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A071901.

Join[{0}, Array[ Function[ n, RealDigits[ N[ Sqrt[ n ], n+6 ] ]// (#[ [ 1, #[ [ 2 ] ]+n ] ])& ], 110 ]]
Table[ Floor[ Mod[10^n*Sqrt[n], 10]], {n, 0, 104}] (* Robert G. Wilson v, Jun 17 2002 *)

A003076(n)=sqrtint(n*100^n)%10;
apply(A003076, [0..99]) \\ M. F. Hasler, Jun 22 2024

From M. F. Hasler, Jun 22 2024: (Start)
a(n) = A000196(n*100^n) % 10, where n % 10 = A010879(n) is the final digit of n.
a(n) = 0 for all n in A000290 (but not only those). This explains that the value 0 is slightly more frequent than the other values. (End)

Extension and program from Olivier Gérard, Oct 15 1997

A071989 a(n) = n-th decimal digit of the fractional part of the square root of the n-th nonsquare number (A000037).

Original entry on oeis.org

4, 3, 6, 4, 5, 7, 6, 9, 5, 4, 7, 7, 6, 8, 3, 3, 0, 4, 5, 9, 8, 3, 3, 0, 8, 9, 6, 2, 4, 1, 0, 4, 4, 0, 6, 7, 9, 5, 1, 7, 4, 1, 3, 5, 7, 5, 7, 7, 4, 8, 8, 9, 5, 0, 5, 0, 6, 5, 1, 7, 3, 3, 9, 9, 7, 7, 6, 1, 4, 9, 9, 2, 7, 8, 5, 8, 4, 9, 4, 5, 4, 2, 8, 0, 2, 1, 7, 7, 4, 7, 4, 8, 1, 8, 4, 5, 7, 5, 8, 0, 0, 0, 1, 4, 3

Offset: 1

Robert G. Wilson v, Jun 17 2002

Regarded as a decimal fraction, 0.43645769547768330... is likely to be an irrational number.

			Sqrt(2)=1.4142135... -> the 1st decimal digit is 4,
sqrt(3)=1.7320508... -> the 2nd decimal digit is 3,
sqrt(5)=2.2360679... -> the 3rd decimal digit is 6,
sqrt(6)=2.4494897... -> the 4th decimal digit is 4, etc.

Martin Aigner & Günter M. Ziegler, Proofs from THE BOOK, Second Edition, Springer-Verlag, Berlin Heidelberg NY, Section of Analysis, Chptr 15, "Sets, function, and the continuum hypothesis", 2000, pp. 87-98.
Georg Cantor, Über eine Eigenschaft des Inbegriffes aller reellen Zahlen ("On the Characteristic Property of All Real Numbers").
Timothy Gowers, Editor, with June Barrow-Green & Imre Leader, Assc. Editors, The Princeton Companion to Mathematics, Princeton Un. Press, Princeton & Oxford, 2008, pp. 171 & 779.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §7.5 Transfinite Numbers, pp. 257-262.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Richard Lipton, Gödel's Lost Letter and P=NP.
Luke Mastin, 19th Century Mathematics - Cantor.
Tom Schaffter, Cantor's Diagonal Argument: Proof and Paradox.
Eric Weisstein's World of Mathematics, Cantor Diagonal Method.
Wikipedia, Cantor's diagonal argument.

Cf. A000037, A071901, A216251.

q[n_] := (m = Floor[n + Sqrt[n + Sqrt[n]]]; Floor[ Mod[ 10^n*Sqrt[m], 10]]); Table[ q[n], {n, 1, 105}]

from math import isqrt
def A071989(n): return isqrt(10**(n<<1)*(n+(k:=isqrt(n))+int(n>=k*(k+1)+1)))%10 # Chai Wah Wu, Jul 20 2024

a(n) = floor(sqrt(A000037(n))*10^n) mod 10. - Jason Yuen, Aug 20 2024

A111422 a(n) = n-th decimal digit of the fractional part of the cube root of the n-th prime.

Original entry on oeis.org

2, 4, 9, 9, 8, 4, 5, 4, 9, 6, 9, 5, 7, 2, 4, 0, 4, 5, 0, 0, 6, 3, 7, 8, 4, 6, 7, 9, 3, 6, 7, 7, 8, 2, 5, 9, 0, 6, 1, 8, 8, 8, 3, 9, 1, 6, 6, 9, 9, 9, 4, 4, 3, 7, 7, 2, 4, 4, 7, 6, 7, 1, 8, 4, 6, 6, 9, 0, 6, 5, 7, 9, 8, 9, 7, 5, 2, 4, 5, 1, 7, 0, 9, 4, 7, 0, 6, 3, 1, 7, 3, 9, 3, 7, 0, 9, 4, 0, 9, 7, 0, 9, 7, 2, 0

Offset: 2

Cino Hilliard, Nov 13 2005

			The 2nd prime is 3. 3^(1/3) = 1.442249..., The 2nd entry after the decimal point is 4 the 2nd entry in the table.

John D. Barrow, The Infinite Book, Pantheon Book New York 2005, pp. 69-76.

Cf. A071901.

a[n_] := Block[{rd = RealDigits[(Prime@n)^(1/3), 10, 111]}, rd[[1, n + rd[[2]]]]];
Array[a, 105] (* Robert G. Wilson v, Nov 17 2005 *)

a(n) = localprec(n+1); floor(frac(sqrtn(prime(n), 3))*10^n) % 10; \\ Michel Marcus, Feb 22 2024

More terms from Robert G. Wilson v, Nov 17 2005

A111421 a(n) = n-th decimal digit + 1 of the fractional part formed by the square root of the n-th prime.

Original entry on oeis.org

5, 4, 7, 8, 3, 2, 7, 5, 4, 2, 4, 9, 9, 1, 5, 3, 8, 5, 0, 4, 2, 5, 3, 1, 5, 2, 9, 7, 5, 0, 9, 9, 2, 5, 4, 5, 1, 9, 5, 2, 1, 3, 9, 7, 4, 3, 4, 8, 5, 8, 7, 7, 3, 6, 1, 2, 3, 4, 2, 4, 8, 5, 5, 8, 8, 5, 4, 7, 0, 7, 2, 3, 2, 0, 9, 0, 5, 3, 0, 0, 4, 6, 7, 0, 1, 5, 0, 4, 9, 7, 0, 7, 4, 7, 5, 3, 7, 4, 6, 0, 4, 8, 9, 0, 7

Offset: 2

Cino Hilliard, Nov 12 2005

Also a Cantor diagonal proving the irrational numbers are a non-denumerable infinite set. Also A071901(n)+ 1.

			The 2nd prime is 3. Sqrt(3) = 1.7320508..., The 2nd entry after the decimal point is 3 and 3+1=4, the 2nd entry in the table.

John D. Barrow, The Infinite Book, Pantheon Book New York 2005, pp. 69-76.

Cf. A071901.

f[n_] := Block[{rd = RealDigits[ Sqrt@Prime@n, 10, 111]}, Mod[rd[[1, n + rd[[2]]]] + 1, 10]]; Array[f, 105] (* Robert G. Wilson v, Nov 17 2005 *)

a(n) = localprec(n+1); (floor(frac(sqrt(prime(n)))*10^n)+1) % 10; \\ Michel Marcus, Feb 22 2024

More terms from Robert G. Wilson v, Nov 17 2005

A111423 a(n) = n-th decimal digit of the fractional part formed by the 4th root of the n-th prime.

Original entry on oeis.org

1, 1, 5, 5, 6, 8, 1, 2, 3, 1, 7, 9, 2, 1, 5, 5, 0, 3, 4, 3, 4, 7, 5, 0, 1, 3, 4, 4, 5, 0, 8, 1, 1, 0, 3, 1, 3, 6, 6, 0, 9, 0, 9, 4, 8, 6, 4, 0, 8, 1, 0, 1, 7, 7, 2, 9, 9, 6, 9, 5, 6, 0, 6, 1, 1, 1, 1, 4, 4, 0, 6, 9, 9, 4, 6, 6, 3, 9, 2, 4, 8, 9, 3, 5, 2, 5, 0, 7, 8, 4, 8, 8, 6, 2, 8, 6, 0, 6, 9, 0, 6, 1, 8, 6, 5

Offset: 1

Cino Hilliard, Nov 13 2005

Cf. A071901, A111422.

a[n_] := Block[{rd = RealDigits[Sqrt@Sqrt@Prime@n, 10, 111]}, rd[[1, n + rd[[2]] ]]];
Array[a, 105] (* Robert G. Wilson v, Nov 17 2005 *)

a(n) = localprec(n+1); floor(frac(sqrtn(prime(n), 4))*10^n) % 10; \\ Michel Marcus, Feb 22 2024

More terms from Robert G. Wilson v, Nov 17 2005

A178903 n-th decimal digit of the fractional part of the square root of the n-th semiprime.

Original entry on oeis.org

0, 4, 0, 2, 5, 3, 6, 5, 0, 5, 3, 5, 6, 7, 8, 1, 0, 7, 7, 3, 1, 0, 1, 2, 5, 0, 6, 7, 9, 0, 2, 6, 3, 2, 7, 6, 0, 2, 5, 0, 7, 6, 7, 5, 2, 6, 7, 7, 8, 0, 9, 7, 7, 0, 7, 0, 3, 1, 2, 7, 5, 4, 0, 5, 6, 7, 3, 7, 5, 1, 1, 8, 9, 5, 1, 7, 1, 9, 7, 9, 1, 9, 2, 9, 4, 0, 2, 1, 5, 0, 4, 8, 8, 8, 5, 8, 7, 5, 6, 5, 1, 5, 9, 0, 4

Offset: 1

Jonathan Vos Post, Jun 22 2010

This is to semiprimes A001358 as A071901 is to prime A000040. Regarded as a decimal fraction, 0.0402536505356781... is likely to be an irrational number.

			semiprime(1) = 4, sqrt(4) = 2.000, first digit of fractional part is 0, so a(1) = 0.
semiprime(2) = 6, sqrt(6) = 2.449, 2nd digit of fractional part is 4, so a(2) = 4.
semiprime(3) = 9, sqrt(9) = 3.000, 3rd digit of fractional part is 0, so a(3) = 0.
semiprime(4) = 10, sqrt(10) = 3.162277, 4th digit of fractional part is 2, so a(4) = 2.
semiprime(5) = 14, sqrt(14) = 3.741657, 5th digit of fractional part is 5, so a(5) = 5.
semiprime(6) = 15, sqrt(15) = 3.8729833, 6th digit of fractional part is 3, so a(6) = 3 semiprime(7) = 21, sqrt(21) = 4.58257569, 7th digit of fractional part is 6, so a(6) = 6.

Cf. A000040, A001358, A071901.

SemiPrimePi[n_] := Sum[ PrimePi[n/Prime@i] - i + 1, {i, PrimePi@ Sqrt@n}]; SemiPrime[n_] := Block[{e = Floor[ Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[ SemiPrimePi@a < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; f[n_] := Mod[ Floor@ N[10^n*Sqrt@ SemiPrime@n, n + 10], 10]; Array[f, 111] (* Robert G. Wilson v, Jul 31 2010 *)

a(16) onwards from Robert G. Wilson v, Jul 31 2010

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A003076 n-th digit after decimal point of square root of n.

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A071989 a(n) = n-th decimal digit of the fractional part of the square root of the n-th nonsquare number (A000037).

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A111422 a(n) = n-th decimal digit of the fractional part of the cube root of the n-th prime.

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A111421 a(n) = n-th decimal digit + 1 of the fractional part formed by the square root of the n-th prime.

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A111423 a(n) = n-th decimal digit of the fractional part formed by the 4th root of the n-th prime.

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A178903 n-th decimal digit of the fractional part of the square root of the n-th semiprime.

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