A132153 -id:A132153 - cp's OEIS frontend

A175734 Largest n-digit number with 3 divisors.

9, 49, 961, 9409, 97969, 994009, 9840769, 99460729, 999002449, 9998200081, 99996985729, 999966000289, 9999995824729, 99999820000081, 999997874844049, 9999997800000121, 99999977853408361, 999999874000003969, 9999999828172082689, 99999999340000001089

Offset: 1

Jaroslav Krizek, Aug 24 2010

Because every number with exactly three divisors must be the square of a prime, the last digit of each term in this sequence is either 1 or 9. - Harvey P. Dale, Aug 18 2011 [with thanks to Ant King and Christopher Hunt Gribble]

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000

Cf. A131581, A132153, A136582, A175733.

A175734 := func; // Jason Kimberley, Aug 18 2011

Join[{9},Table[NextPrime[Sqrt[10^n-1],-1]^2,{n,2,20}]](* Harvey P. Dale, Aug 18 2011 *)

A175734(n)=precprime(sqrtint(10^n))^2  \\ M. F. Hasler, Aug 18 2011

a(n) = A132153(n)^2.

a(14)-a(20) from Harvey P. Dale, Aug 18 2011

A131581 The next prime greater than the square root of 10^n.

Original entry on oeis.org

2, 5, 11, 37, 101, 317, 1009, 3163, 10007, 31627, 100003, 316241, 1000003, 3162283, 10000019, 31622777, 100000007, 316227767, 1000000007, 3162277669, 10000000019, 31622776621, 100000000003, 316227766069, 1000000000039

Offset: 0

Robert G. Wilson v, Jan 12 2007

The difference between a(n) and floor(sqrt(10^n)): 1, 2, 1, 6, 1, 1, 9, 1, 7, 5, 3, 14, 3, 6, 19, 1, 7, 1, 7, 9, ....

Values for which the difference between a(n) and floor(sqrt(10^n)) equals one: 0, 2, 4, 5, 7, 15, 17, 25, 145, 1079, ..., (1350). Only even terms are 0, 2 & 4; all the rest must be odd.

Cf. A003617, A132153, A136582, A175733, A175734.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Table[ NextPrim@ Floor@ Sqrt[10^n], {n, 0, 25}]
Table[NextPrime[Sqrt[10^n]],{n,0,30}] (* Harvey P. Dale, Aug 15 2017 *)

a(n) = sqrt(A175733(n+1)). [From Jaroslav Krizek, Aug 24 2010]

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).

A175733 a(n) is the smallest n-digit number with 3 divisors.

Original entry on oeis.org

4, 25, 121, 1369, 10201, 100489, 1018081, 10004569, 100140049, 1000267129, 10000600009, 100008370081, 1000006000009, 10000033772089, 100000380000361, 1000000025191729, 10000001400000049, 100000000621806289, 1000000014000000049, 10000000055856073561

Offset: 1

Jaroslav Krizek, Aug 24 2010

Cf. A131581, A132153, A136582, A175734.

Table[Prime[1 + PrimePi[Sqrt[10^n]]]^2, {n, 0, 25}] (* T. D. Noe, Aug 19 2011 *)

from math import sqrt
from sympy import nextprime
def a(n): return nextprime(int(sqrt(10**n+1)))**2
print([a(n) for n in range(20)]) # Michael S. Branicky, Apr 25 2021

a(n) = A131581(n-1)^2.

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

A133225 Largest prime <= 2^((n+1)/2).

Original entry on oeis.org

2, 2, 3, 5, 7, 11, 13, 19, 31, 43, 61, 89, 127, 181, 251, 359, 509, 719, 1021, 1447, 2039, 2887, 4093, 5791, 8191, 11579, 16381, 23167, 32749, 46337, 65521, 92681, 131071, 185363, 262139, 370723, 524287, 741431, 1048573, 1482907, 2097143, 2965819

Offset: 1

Anthony C Robin, Jan 03 2008

nonn

If one is trying to decide whether an n+1 digit binary number is prime, this is the largest prime for which one needs to test divisibility. For example a six digit number like 110101 must be below 64, so only primes up to 7 are needed to test divisibility. Compare with sequence A132153.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A132153.

seq(prevprime(floor(2^((n+1)*1/2))+1),n=1..40); # Emeric Deutsch
A017910 := proc(n) floor(2^(n/2)) ; end: A007917 := proc(n) prevprime(n+1) ; end: A133225 := proc(n) A007917(A017910(n+1)) ; end: seq(A133225(n),n=1..60) ; # R. J. Mathar

PrevPrim[n_] := Block[{k = n}, While[ !PrimeQ@k, k-- ]; k]; f[n_] := PrevPrim@ Floor@ Sqrt[2^(n + 1)]; Array[f, 42] (* Robert G. Wilson v *)
Table[Prime[PrimePi[2^((n + 1)/2)]], {n, 1, 50}] (* Stefan Steinerberger *)
lp[n_]:=Module[{c=2^((n+1)/2)},If[PrimeQ[c],c,NextPrime[c,-1]]]; Array[lp,50] (* Harvey P. Dale, Aug 25 2013 *)

a(n) = A007917[A017910(n+1)]. - R. J. Mathar

More terms from Stefan Steinerberger, R. J. Mathar, Robert G. Wilson v and Emeric Deutsch, Jan 06 2008

cp's OEIS Frontend

A175734 Largest n-digit number with 3 divisors.

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A131581 The next prime greater than the square root of 10^n.

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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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A175733 a(n) is the smallest n-digit number with 3 divisors.

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

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A133225 Largest prime <= 2^((n+1)/2).

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