A017934 -id:A017934 - cp's OEIS frontend

A132153 Largest prime <= square root of 10^n.

3, 7, 31, 97, 313, 997, 3137, 9973, 31607, 99991, 316223, 999983, 3162277, 9999991, 31622743, 99999989, 316227731, 999999937, 3162277633, 9999999967, 31622776589, 99999999977, 316227766003, 999999999989, 3162277660153, 9999999999971, 31622776601657

Offset: 1

Anthony C Robin, Nov 01 2007

nonn

To check if an (n+1)-digit number is prime, a(n) is the largest prime which one needs to check is not a factor of the (n+1)-th digit number. For example, to check that a general four-digit number is not prime, we need to test its divisibility by all the primes up to and including 97.

Charles R Greathouse IV, Table of n, a(n) for n = 1..2000

Cf. A017934, A131581, A136582, A175733, A175734.

Cf. A000040, A122121, A003618.

Table[NextPrime[Sqrt[10^n],-1],{n,27}] (* James C. McMahon, Mar 04 2025 *)

a(n)=precprime(sqrtint(10^n)) \\ Charles R Greathouse IV, Aug 18 2011

from sympy import prevprime, integer_nthroot
def a(n): return prevprime(integer_nthroot(10**n, 2)[0]+1)
print([a(n) for n in range(1, 28)]) # Michael S. Branicky, Dec 23 2021

a(n) = A000040(A122121(n)). a(2n) = A003618(n). - R. J. Mathar, Nov 06 2007 [Corrected by Jaroslav Krizek, Jul 12 2010]

a(n) = sqrt(A175734(n)). - Jaroslav Krizek, Aug 24 2010

More terms from N. J. A. Sloane, Jan 05 2008

A062940 Number of squares (including 0) with n digits.

Original entry on oeis.org

4, 6, 22, 68, 217, 683, 2163, 6837, 21623, 68377, 216228, 683772, 2162278, 6837722, 21622777, 68377223, 216227767, 683772233, 2162277661, 6837722339, 21622776602, 68377223398, 216227766017, 683772233983, 2162277660169

Offset: 1

Amarnath Murthy, Jul 07 2001

Sum of first 2n terms = 10^n. - Zak Seidov, Aug 05 2006

a(n)/a(n-1) ~ 10^(1/2). For the sequence giving the number of members of the sequence a(k)=k^r with n digits we have a(n)/a(n-1) ~ 10^(1/r). - Ctibor O. Zizka, Mar 09 2008

			a(1)=4 because there are 4 one-digit squares: 0,1,4,9. - _Zak Seidov_, Aug 05 2006
a(2)=6 because there are 6 two-digit squares: 16,25,36,49,64,81. - _Zak Seidov_, Aug 05 2006
22 squares (100=10^2, 121=11^2, ..., 961=31^2) have 3 digits, hence a(3)=22.

Harry J. Smith, Table of n, a(n) for n = 1..200

A variant of A049415. A049415(n) = A017936(n+1) - A017936(n) = A049416(n+1) - A049416(n). Cf. A000290, A062941.

Column k=2 of A216653.

r:= proc(n, k) local b; b:= iroot(n, k); b+`if`(b^k r(10^n, 2) -r(10^(n-1), 2) +`if`(n=1, 1, 0):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 12 2012

je=[4]; for(n=2, 45, je=concat(je, ceil(sqrt(10^n))-ceil(sqrt(10^(n-1))))); je

{ default(realprecision, 200); for (n=1, 200, b=ceil(10^(n/2)); if (n>1, a=b - c, a=4); c=b; write("b062940.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 14 2009

a(n) = ceiling(sqrt(10^n)) - ceiling(sqrt(10^(n-1))), n > 1.

a(n) = A017934(n) - A017934(n-1) - (-1)^n, n >= 2. - R. J. Mathar, Mar 17 2008

Corrected and extended by Dean Hickerson and Jason Earls, Jul 10 2001

Edited by R. J. Mathar, Aug 07 2008

A136582 Sqrt(10)-primes: primes obtained by concatenating the first digits in the decimal expansion of sqrt(10).

Original entry on oeis.org

3, 31, 3162277, 316227766016837933, 316227766016837933199889354443271

Offset: 1

Lekraj Beedassy, Jan 09 2008

n such that floor(sqrt(10^(2*n-1))) is prime (1, 2, 7, 18, 33,  ...) are given in A136583.
This sequence is the list of prime terms in A017934.
The next term has 206 digits. - Harvey P. Dale, Dec 06 2023

Cf. A010467, A017934, A017934, A131581, A132153, A136583, A175733, A175734.

// by Jason Kimberley, Aug 2011
for n in [1..499 by 2] do
  f := Isqrt(10^n);
  if IsPrime(f) then
    printf "%o,", f;
  end if;
end for;

Module[{nn=50,sq10},sq10=RealDigits[Sqrt[10],10,nn][[1]];Select[FromDigits/@Table[Take[sq10,n],{n,nn}],PrimeQ]] (* Harvey P. Dale, Dec 06 2023 *)

a(n) = A017934(2*A136583(n)-1).

A018072 Powers of fourth root of 10 rounded down.

Original entry on oeis.org

1, 1, 3, 5, 10, 17, 31, 56, 100, 177, 316, 562, 1000, 1778, 3162, 5623, 10000, 17782, 31622, 56234, 100000, 177827, 316227, 562341, 1000000, 1778279, 3162277, 5623413, 10000000, 17782794, 31622776, 56234132, 100000000, 177827941, 316227766, 562341325, 1000000000, 1778279410

Offset: 0

N. J. A. Sloane

			a(2) = 3 because 10^(2/4) = 10^(1/2) = sqrt(10) = 3.16228...
a(3) = 5 because 10^(3/4) = 5.62341...
a(4) = 10 because 10^(4/4) = 10^1 = 10.
a(5) = 17 because 10^(5/4) = 17.78279...

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A017934, A011007, A010467.

[Floor(10^(n/4)): n in [0..37]]; // Vincenzo Librandi, May 30 2025

Table[Floor[10^(n/4)], {n, 0, 39}] (* Alonso del Arte, Jan 26 2013 *)
Floor[Surd[10,4]^Range[0,40]] (* Harvey P. Dale, Oct 26 2019 *)

a(n)=sqrtint(sqrtint(10^n)) \\ Charles R Greathouse IV, Jan 27 2013

A few additional terms from Alonso del Arte, Jan 26 2013

A091100 Number of Gaussian primes whose norm is less than 10^n.

Original entry on oeis.org

16, 100, 668, 4928, 38404, 313752, 2658344, 23046512, 203394764, 1820205436, 16472216912, 150431552012, 1384262129028, 12819767598972, 119378281788240, 1116953361826164

Offset: 1

T. D. Noe, Dec 19 2003

nonn

Marc Deléglise, Pierre Dusart, and Xavier-Francois Roblot, Counting primes in residue classes, Math. Comp. 73 (2004), no. 247, 1565-1575
Eric Weisstein's World of Mathematics, Gaussian Prime
Index entries for Gaussian integers and primes

Cf. A091098 (number of primes of the form 4k+1 less than 10^n), A091099 (number of primes of the form 4k+3 less than 10^n), A091101, A091102.
Cf. A091134 (number of Gaussian primes whose modulus is less than 10^n).
Cf. A017934, A295996.

Table[lim2=10^n; lim1=Floor[Sqrt[lim2]]; cnt=0; Do[If[x^2+y^2True], cnt++ ], {x, -lim1, lim1}, {y, -lim1, lim1}]; cnt, {n, 6}]

a(2n) = 8*A091098(2n) + 4*A091099(n) + 4.

a(n) ~ 4 Li(10^n) ~ k/n * 10^n, where k = 4/log(10) = 1.737.... - Charles R Greathouse IV, Oct 24 2012

a(10)-a(16) from Seiichi Manyama using the data in A091098, Dec 03 2017

A136583 n such that floor(sqrt(10^(2*n-1))) is (probably) prime.

Original entry on oeis.org

1, 2, 7, 18, 33, 206, 468, 1061, 6831, 40377

Offset: 1

Lekraj Beedassy, Jan 09 2008

Number of digits of sqrt(10)-primes (A136582).

The n such that A017934(2*n-1) is (probably) prime.

Cf. A010467, A017934, A017934, A131581, A132153, A136582, A175733, A175734.

for n in [1..10^6] do if IsPrime(Isqrt(10^(2*n-1))) then printf "%o, ", n; end if; end for; // Jason Kimberley, Sep 03 2011

rd = RealDigits[Sqrt[10], 10, 10^5][[1]]; Do[ If[ PrimeQ@ FromDigits@ Take[rd, n], Print@n], {n, 10^5}] (* Robert G. Wilson v, Jan 20 2008 *)

a(6) - a(8) from Robert G. Wilson v, Jan 20 2008

Probable terms a(9) and a(10) from Jason Kimberley, Aug 19 and Sep 03 2011

A255616 Table read by antidiagonals, T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 4, 5, 4, 1, 1, 2, 5, 8, 9, 5, 1, 1, 2, 6, 11, 16, 15, 8, 1, 1, 2, 7, 14, 25, 32, 27, 11, 1, 1, 3, 8, 18, 36, 55, 64, 46, 16, 1, 1, 3, 9, 22, 49, 88, 125, 128, 81, 22, 1, 1, 3, 10, 27, 64, 129, 216, 279, 256, 140, 32, 1, 1, 3, 11, 31, 81, 181, 343, 529, 625, 512, 243, 45, 1

Offset: 0

Kival Ngaokrajang, Feb 28 2015

			See table in the links.

G. C. Greubel, Table of n, a(n) for the first 100 antidaigonals, flattened
Kival Ngaokrajang, Example of table T(n,k), n = 0..12, k = 1..10

Rows(1..10): A000012, A000196, A000027, A000093, A000290, A155013, A000578, A155014, A000583, A238170.
Columns(1..10): A000012, A017910, A017913, A000079, A017919, A017922, A017925, A017928, A000244, A017934.
Diagonals: A190343, A066642, A075264.

T[n_, k_] := Floor[Sqrt[k^n]]; Table[T[k, n + 1 - k], {n, 0, 15}, {k, 0, n}] (* G. C. Greubel, Dec 30 2017 *)

{for(i=1,20,for(n=0,i-1,a=floor(sqrt((i-n)^n));print1(a,", ")))}

T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

Terms a(81) onward added by G. C. Greubel, Dec 30 2017

A361612 Decimal expansion of sqrt(10) truncated to n places (after the decimal point).

Original entry on oeis.org

3, 31, 316, 3162, 31622, 316227, 3162277, 31622776, 316227766, 3162277660, 31622776601, 316227766016, 3162277660168, 31622776601683, 316227766016837, 3162277660168379, 31622776601683793, 316227766016837933, 3162277660168379331, 31622776601683793319

Offset: 0

Amit Katz, Mar 17 2023

nonn

Cf. A010467, A011547, A017934.

Floor[Sqrt[10]*10^Range[0, 20]] (* Paolo Xausa, Jul 25 2024 *)

from math import isqrt
def A361612(n): return isqrt(10**((n<<1)|1)) # Chai Wah Wu, Mar 26 2023

a(n) = floor(sqrt(10) * 10^n).
a(n) = A017934(2*n + 1).
a(n-1) mod 10 = A010467(n). - Chai Wah Wu, Mar 26 2023

A069658 a(1) = 1; a(n) = smallest nontrivial n-digit perfect power.

Original entry on oeis.org

1, 16, 121, 1024, 10201, 100489, 1002001, 10004569, 100020001, 1000014129, 10000200001, 100000147984, 1000002000001, 10000002149284, 100000020000001, 1000000025191729, 10000000200000001, 100000000621806289, 1000000002000000001

Offset: 1

Amarnath Murthy, Apr 04 2002

Powers of 10 are not allowed.

Cf. A061432, A017936.

Join[{1},(Floor[Sqrt[10^Range[20]]]+1)^2] (* Harvey P. Dale, Mar 08 2022 *)

a(n) = (floor(sqrt(10)^(n-1))+1)^2 = (A017934(n-1)+1)^2. - Vladeta Jovovic, Jun 30 2002

More terms from Sascha Kurz, Jan 28 2003

cp's OEIS Frontend

A132153 Largest prime <= square root of 10^n.

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A062940 Number of squares (including 0) with n digits.

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A136582 Sqrt(10)-primes: primes obtained by concatenating the first digits in the decimal expansion of sqrt(10).

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Magma

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A018072 Powers of fourth root of 10 rounded down.

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Magma

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A091100 Number of Gaussian primes whose norm is less than 10^n.

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Mathematica

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A136583 n such that floor(sqrt(10^(2*n-1))) is (probably) prime.

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Magma

Mathematica

Extensions

A255616 Table read by antidiagonals, T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

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Mathematica

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