A091113 -id:A091113 - cp's OEIS frontend

A014076 Odd nonprimes.

1, 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99, 105, 111, 115, 117, 119, 121, 123, 125, 129, 133, 135, 141, 143, 145, 147, 153, 155, 159, 161, 165, 169, 171, 175, 177, 183, 185, 187, 189, 195, 201, 203, 205, 207

Offset: 1

Warut Roonguthai

Same as A071904 except for the initial term 1 (which is not composite).
Numbers n such that product of first n odd numbers divided by sum of the first n odd numbers is an integer : 1*3*5*...*(2*n - 1) / (1 + 3 + 5 + ... + (2*n - 1)) = c. - Ctibor O. Zizka, Jun 26 2010
Conjecture: There exist infinitely many pairs [a(n), a(n)+6] such that a(n)/3 and (a(n)+6)/3 are twin primes. - Eric Desbiaux, Sep 25 2014.
Odd numbers 2*n + 1 such that (2*n)!/(2*n + 1) is an integer. Odd terms of A056653. - Peter Bala, Jan 24 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A002808, A005408; first differences: A067970, A196274; A047846.

Cf. A056653.

a014076 n = a014076_list !! (n-1)
a014076_list = filter ((== 0) . a010051) a005408_list
-- Reinhard Zumkeller, Sep 30 2011

remove(isprime, [seq(i,i=1..1000,2)]); # Robert Israel, May 25 2016
for n from 0 to 120 do
if irem(factorial(2*n), 2*n+1) = 0 then print(2*n+1) end if;
end do: # Peter Bala, Jan 24 2017

Select[Range@210, !PrimeQ@ # && OddQ@ # &] (* Robert G. Wilson v, Sep 22 2008 *)
Select[Range[1, 199, 2], PrimeOmega[#] != 1 &] (* Alonso del Arte, Nov 19 2012 *)

is(n)=n%2 && !isprime(n) \\ Charles R Greathouse IV, Nov 24 2012

from sympy import primepi
def A014076(n):
    if n == 1: return 1
    m, k = n-1, primepi(n) + n - 1 + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    return m # Chai Wah Wu, Jul 31 2024

A000035(a(n))*(1 - A010051(a(n))) = 1. - Reinhard Zumkeller, Sep 30 2011
a(n) ~ 2n. - Charles R Greathouse IV, Jul 02 2013
(a(n+2)-1)/2 - pi(a(n+2)-1) = n. - Anthony Browne, May 25 2016. Proof from Robert Israel: This follows by induction on n. If f(n) = (a(n+2)-1)/2 - pi(a(n+2)-1), one can show f(n+1) - f(n) = 1 (there are three cases to consider, depending on primeness of a(n+2) + 2 and a(n+2) + 4).
Union of A091113 and A091236. - R. J. Mathar, Oct 02 2018

A016105 Blum integers: numbers of the form p * q where p and q are distinct primes congruent to 3 (mod 4).

Original entry on oeis.org

21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, 501, 517, 537, 553, 573, 581, 589, 597, 633, 649, 669, 681, 713, 717, 721, 737, 749, 753, 781, 789

Offset: 1

Robert G. Wilson v

Subsequence of A084109. - Ralf Stephan and David W. Wilson, Apr 17 2005
Subsequence of A046388. - Altug Alkan, Dec 10 2015
Subsequence of A339817. No common terms with A339870. - Antti Karttunen, Dec 26 2020
Named after the Venezuelan-American computer scientist Manuel Blum (b. 1938). - Amiram Eldar, Jun 06 2021
First introduced by Blum, Blum, & Shub for the generation of pseudorandom numbers and later applied (by Manuel Blum and other authors) to zero-knowledge proofs. - Charles R Greathouse IV, Sep 26 2024

Lenore Blum, Manuel Blum, and Mike Shub. A simple unpredictable pseudorandom number generator, SIAM Journal on computing 15:2 (1986), pp. 364-383.

Antti Karttunen, Table of n, a(n) for n = 1..26828 (all terms < 2^19; first 1000 terms from T. D. Noe)
Joe Hurd, Blum Integers, Talk at the Trinity College, Jan 20 1997.
Wikipedia, Blum integer.

Cf. A002145, A006881, A046388, A339870.
Intersection of A005117 and A107978.
Also, subsequence of the following sequences: A046388, A084109, A091113, A167181, A339817.

import Data.Set (singleton, fromList, deleteFindMin, union)
a016105 n = a016105_list !! (n-1)
a016105_list = f [3,7] (drop 2 a002145_list) 21 (singleton 21) where
   f qs (p:p':ps) t s
     | m < t     = m : f qs (p:p':ps) t s'
     | otherwise = m : f (p:qs) (p':ps) t' (s' `union` (fromList pqs))
     where (m,s') = deleteFindMin s
           t' = head $ dropWhile (> 3*p') pqs
           pqs = map (p *) qs
-- Reinhard Zumkeller, Sep 23 2011

N:= 10000: # to get all terms <= N
Primes:= select(isprime, [seq(i,i=3..N/3,4)]):
S:=select(`<=`,{seq(seq(Primes[i]*Primes[j],i=1..j-1),j=2..nops(Primes))},N):
sort(convert(S,list)); # Robert Israel, Dec 11 2015

With[{upto = 820}, Select[Union[Times@@@Subsets[ Select[Prime[Range[ PrimePi[ NextPrime[upto/3]]]], Mod[#, 4] == 3 &], {2}]], # <= upto &]] (* Harvey P. Dale, Aug 19 2011 *)
Select[4Range[5, 197] + 1, PrimeNu[#] == 2 && MoebiusMu[#] == 1 && Mod[FactorInteger[#][[1, 1]], 4] != 1 &] (* Alonso del Arte, Nov 18 2015 *)

list(lim)=my(P=List(),v=List(),t,p); forprimestep(p=3,lim\3,4, listput(P,p)); for(i=2,#P, p=P[i]; for(j=1,i-1, t=p*P[j]; if(t>lim, break); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jul 01 2016, updated Sep 26 2024

isA016105(n) = (2==omega(n)&&2==bigomega(n)&&1==(n%4)&&3==((factor(n)[1,1])%4)); \\ Antti Karttunen, Dec 26 2020

use ntheory ":all"; forcomposites { say if ($ % 4) == 1 && is_square_free($) && scalar(factor($)) == 2 && !scalar(grep { ($ % 4) != 3 } factor($)); } 10000; # _Dana Jacobsen, Dec 10 2015

from sympy import factorint
def ok(n):
    fn = factorint(n)
    return len(fn) == sum(fn.values()) == 2 and all(f%4 == 3 for f in fn)
print([k for k in range(790) if ok(k)]) # Michael S. Branicky, Dec 20 2021

a(n) = A195758(n) * A195759(n). - Reinhard Zumkeller, Sep 23 2011

a(n) ~ 4n log n/log log n. - Charles R Greathouse IV, Sep 17 2022

More terms from Erich Friedman

A091236 Nonprimes of form 4k+3.

Original entry on oeis.org

15, 27, 35, 39, 51, 55, 63, 75, 87, 91, 95, 99, 111, 115, 119, 123, 135, 143, 147, 155, 159, 171, 175, 183, 187, 195, 203, 207, 215, 219, 231, 235, 243, 247, 255, 259, 267, 275, 279, 287, 291, 295, 299, 303, 315, 319, 323, 327, 335, 339, 343, 351, 355, 363

Offset: 1

Labos Elemer, Feb 24 2004

If we define f(n) to be the number of primes (counted with multiplicity) of the form 4k + 3 that divide n, then with this sequence f(a(n)) is always odd. For example, 95 is divisible by 17 and 99 is divisible by 3 (twice) and 11. - Alonso del Arte, Jan 13 2016
Complement of A002145 with respect to A004767. - Michel Marcus, Jan 17 2016
With the Jan 05 2004 Jovovic comment on A078703: The number of 1 and -1 (mod 4) divisors of a(n) are identical. Proof: each number 3 (mod 4) is trivially not a sum of two squares. The number of solutions of n as a sum of two squares is r_2(n) = 4*(d_1(n) - d_3(n)), where d_k(n) is the number of k (mod 4) divisors of n. See e.g., Grosswald, pp. 15-16 for the proof of Jacobi. - Wolfdieter Lang, Jul 29 2016

			27 = 4 * 6 + 3 = 3^3.
35 = 4 * 8 + 3 = 5 * 7.
a(8) = 75 with 2*A078703(19) = 6 divisors [1, 3, 5, 15, 25, 75], which are 1, -1, 1, -1, 1, -1 (mod 4). - _Wolfdieter Lang_, Jul 29 2016

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002145, A004767, A078703.

Cf. A091113-A092256.

A091236 := proc(n)
    option remember  ;
    local a;
    if n = 1 then
        15 ;
    else
        for a from procname(n-1)+1 do
            if not isprime(a) and modp(a,4) = 3 then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(A091236(n),n=1..20) ; # R. J. Mathar, Jul 20 2025

Select[Range[1000], !PrimeQ[#] && IntegerQ[(# - 3)/4] &] (* Harvey P. Dale, Aug 16 2013 *)
Select[4Range[100] - 1, Not[PrimeQ[#]] &] (* Alonso del Arte, Jan 13 2016 *)

lista(nn) = for(n=1, nn, if(!isprime(k=4*n+3), print1(k, ", "))); \\ Altug Alkan, Jan 17 2016

A339817 Squarefree numbers k > 1 for which the 2-adic valuation of phi(k) does not exceed the 2-adic valuation of k-1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 33, 37, 41, 43, 47, 53, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 89, 93, 97, 101, 103, 107, 109, 113, 127, 129, 131, 133, 137, 139, 141, 145, 149, 151, 157, 161, 163, 167, 173, 177, 179, 181, 191, 193, 197, 199, 201, 209, 211, 213, 217, 223, 227, 229, 233, 237, 239, 241

Offset: 1

Antti Karttunen, Dec 19 2020

nonn

After 2, terms of A003961(A019565(A339816(i))) [or equally, of A019565(2*A339816(i))], for i = 1.., sorted into ascending order.
Natural numbers n that satisfy equation y * phi(n) = n - 1 (with y an integer) all occur in this sequence. Lehmer conjectured that there are no solutions such that n is composite (and thus y > 1).
From Antti Karttunen, Dec 22-26 2020: (Start)
Composite terms in this sequence are all of the form 4u+1 (A016813, A091113).
Generally, if any term k > 2 here has x prime divisors (which are all odd and distinct, i.e., A001221(k) = A001222(k) = x), then k is of the form 2^x * u + 1 (where u maybe even or odd), because each prime divisor of k contributes at least one instance of 2 in phi(k). Specifically, each prime factor of the form 4u+3 (A002145) contributes one instance of 2 (+1 to the 2-adic valuation), while primes of the form 4u+1 (A002144) contribute at least +2 to the 2-adic valuation. There must be an even number of 4u+3 primes, as otherwise the product would be of the form 4u+3. On the other hand, although all the terms of A016105 occur here, none of them occurs in A339870.
If the only terms this sequence shares with A339879 are the primes (A000040), then Lehmer's conjecture certainly holds. Similarly if the sequences A339818 and A339869 do not have any common terms.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..21695
D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
Wikipedia, Lehmer's totient problem.

Cf. A000010, A002144, A002145, A007814, A016105, A016813, A091113, A339816.
Cf. A000040, A016105, A339818, A339819 (subsequences).
Cf. also A339879, A339907, A339869, A339870, A339880.

Select[Range[2, 250], SquareFreeQ[#] && IntegerExponent[EulerPhi[#], 2] <= IntegerExponent[# - 1, 2] &] (* Amiram Eldar, Feb 17 2021 *)

isA339817(n) = ((n>1)&&issquarefree(n)&&(valuation(eulerphi(n),2)<=valuation(n-1,2)));

A091300 Nonprimes of the form 6k + 1.

Original entry on oeis.org

1, 25, 49, 55, 85, 91, 115, 121, 133, 145, 169, 175, 187, 205, 217, 235, 247, 253, 259, 265, 289, 295, 301, 319, 325, 343, 355, 361, 385, 391, 403, 415, 427, 445, 451, 469, 475, 481, 493, 505, 511, 517, 529, 535, 553, 559, 565, 583, 589, 595, 625, 637, 649

Offset: 1

Labos Elemer, Feb 24 2004

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section D1.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A091113, A092256.

Cf. A002476. Subsequence of A016921.

Filtered(List([0..110],k->6*k+1),n->not IsPrime(n)); # Muniru A Asiru, Mar 12 2019

for k from 0 to 100 do if(not isprime(6*k+1))then printf("%d, ",6*k+1); fi: od: # Nathaniel Johnston, May 18 2011

Do[If[ !PrimeQ[n]&&Equal[Mod[n, 6], 1 ], Print[n]], {n, 1, 1000}]
DeleteCases[6*Range[0,150]+1,?PrimeQ] (* _Harvey P. Dale, Jun 23 2014 *)

[n for n in (1..650) if ((n-1)/6).is_integer() and not is_prime(n)] # Stefano Spezia, Oct 05 2024

A339870 Composite numbers k of the form 4u+1 for which the odd part of phi(k) divides k-1.

Original entry on oeis.org

85, 561, 1105, 1261, 1285, 2465, 4369, 6601, 8245, 8481, 9061, 9605, 10585, 16405, 16705, 17733, 18721, 19669, 21845, 23001, 28645, 30889, 38165, 42121, 43165, 46657, 54741, 56797, 57205, 62745, 65365, 74593, 78013, 83665, 88561, 91001, 106141, 117181, 124645, 126701, 134521, 136981, 141661, 162401, 171205, 176437

Offset: 1

Antti Karttunen, Dec 22 2020

nonn

From Antti Karttunen, Dec 26 2020: (Start)
Equally, squarefree composite numbers k of the form 4u+1 for which A336466(k) divides k-1. This follows because on squarefree n, A336466(n) = A053575(n).
No common terms with A016105, because 4xy + 2(x+y) + 1 does not divide 4xy + 3(x+y) + 2 for any distinct x, y >= 0 (where 4x+3 and 4y+3 are the two prime factors of Blum integers).
This can also seen by another way: If this sequence contained any Blum integers, then, because A016105 is a subsequence of A339817, we would have found a composite number n satisfying Lehmer's totient problem y * phi(n) = n-1, for some integer y > 1. But Lehmer proved that such solutions should have at least 7 distinct prime factors, while Blum integers have only two.
Moreover, it seems that none of the terms of A167181 may occur here, and a few of A137409 (i.e., of A125667). See A339875 for those terms.
(End)

			85 = 4*21 + 1 = 5*17, thus phi(85) = 4*16 = 64, the odd part of which is A000265(64) = 1, which certainly divides 85-1, therefore 85 is included as a term.
561 = 4*140 + 1 = 3*11*17, thus phi(561) = 2*10*16 = 320, the odd part of which is A000265(320) = 5, which divides 560, therefore 561 is included.

Antti Karttunen, Table of n, a(n) for n = 1..1316
D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
Wikipedia, Lehmer's totient problem

Cf. A000010, A000265, A016105, A053575, A137409, A167181, A336466.
Subsequence of A005117.
Intersection of A091113 and A339880.
Cf. A339875 (a subsequence).
Cf. also comments in A339817.

odd[n_] := n/2^IntegerExponent[n, 2]; Select[4*Range[45000] + 1, CompositeQ[#] && Divisible[# - 1, odd[EulerPhi[#]]] &] (* Amiram Eldar, Feb 17 2021 *)

A000265(n) = (n>>valuation(n, 2));
isA339870(n) = ((n>1)&&!isprime(n)&&(1==(n%4))&&!((n-1)%A000265(eulerphi(n))));

A093183 Number of consecutive runs of just 1 odd nonprime congruent to 1 mod 4 below 10^n.

Original entry on oeis.org

0, 3, 74, 1114, 13437, 151311, 1642197, 17405273, 181925434, 1883327626, 19364371468, 198115934511, 2019328584101

Offset: 1

Enoch Haga, Mar 30 2004

Split the odd nonprime sequence A014076 into two subsequences A091113 and A091236 with nonprimes labeled 1 mod 4 or 3 mod 4. Add count of nonprimes to sequence if just 1 nonprime congruent to 1 mod 4 occurs before interruption of a nonprime congruent to 3 mod 4.

Otherwise said: count the nonprimes congruent to 1 mod 4 such that the next larger and next smaller odd nonprime is congruent to 3 mod 4. - M. F. Hasler, Sep 30 2018

			a(3) = 74 because 74 single nonprime runs occur below 10^3, each run interrupted by a nonprime congruent to 3 mod 4.
Below 10^2 = 100, there are only a(2) = 3 isolated odd nonprimes congruent to 1 mod 4: 33, 57 and 93. (Credits: _Peter Munn_, SeqFan list.) - _M. F. Hasler_, Sep 30 2018

Cf. A091113, A091236, A093184-A093188, A093397-A093399, A092636.

A014076 := proc(n)
    option remember;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+2 by 2 do
            if not isprime(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
isA091113 := proc(n)
    option remember;
    if modp(n,4) = 1 and not isprime(n) then
        true;
    else
        false;
    end if;
end proc:
isA091236 := proc(n)
    option remember;
    if modp(n,4) = 3 and not isprime(n) then
        true;
    else
        false;
    end if;
end proc:
ct := 0 :
n := 1 :
for i from 2 do
    odnpr := A014076(i) ;
    prev := A014076(i-1) ;
    nxt := A014076(i+1) ;
    if isA091113(odnpr) and isA091236(prev) and isA091236(nxt) then
        ct := ct+1 ;
    end if;
    if odnpr< 10^n and nxt >= 10^n then
        print(n,ct) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Oct 02 2018

A091113 = Select[4 Range[0, 10^5] + 1, ! PrimeQ[#] &];
A091236 = Select[4 Range[0, 10^5] + 3, ! PrimeQ[#] &];
lst = {}; Do[If[Length[s = Select[A091113,Between[{A091236[[i]], A091236[[i + 1]]}]]] == 1, AppendTo[lst, s]], {i, Length[A091236] - 1}]; Table[Count[Flatten[lst], x_ /; x < 10^n], {n, 5}]  (* Robert Price, May 30 2019 *)

a(9)-a(13) from Bert Dobbelaere, Dec 19 2018

A093184 Number of consecutive runs of just 1 odd nonprime congruent to 3 mod 4 below 10^n.

Original entry on oeis.org

0, 4, 76, 1120, 13428, 151342, 1642285, 17405478, 181923798, 1883330510, 19364376037, 198115964781, 2019328569227

Offset: 1

Enoch Haga, Mar 30 2004

			a(3)=76 because 76 single nonprime runs occur below 10^3, each run interrupted by a nonprime congruent to 1 mod 4

Cf. A093183, A093185, A093186, A093187, A093188, A093397, A093398, A093399, A092637.

A091113 = Select[4 Range[0, 10^4] + 1, ! PrimeQ[#] &];
A091236 = Select[4 Range[0, 10^4] + 3, ! PrimeQ[#] &];
lst = {}; Do[If[Length[s = Select[A091236, Between[{A091113[[i]], A091113[[i + 1]]}]]] == 1, AppendTo[lst, s]], {i, Length[A091113] - 1}]; Table[Count[Flatten[lst], x_ /; x < 10^n], {n, 4}]  (* Robert Price, May 30 2019 *)

Generate the odd nonprime sequence with nonprimes labeled 1 mod 4 or 3 mod 4. Add count of nonprimes to sequence if just 1 nonprime congruent to 3 mod 4 occurs before interruption of a nonprime congruent to 1 mod 4.

a(9)-a(13) from Bert Dobbelaere, Dec 19 2018

A093188 Number of consecutive runs of 3 odd nonprimes congruent to 3 mod 4 below 10^n.

Original entry on oeis.org

0, 0, 5, 49, 356, 2678, 21085, 166814, 1345812, 11080939, 92699035, 786630700, 6757485506

Offset: 1

Enoch Haga, Mar 30 2004

			a(3)=5 because 5 nonprime runs of 3 occur below 10^3, each run interrupted by a nonprime congruent to 1 mod 4.

Cf. A093183, A093184, A093185, A093186, A093187, A093397, A093398, A093399, A092643.

A091113 = Select[4 Range[0, 10^4] + 1, ! PrimeQ[#] &];
A091236 = Select[4 Range[0, 10^4] + 3, ! PrimeQ[#] &];
lst = {}; Do[If[Length[s = Select[A091236, Between[{A091113[[i]], A091113[[i + 1]]}]]] == 3, AppendTo[lst, Last[s]]], {i, Length[A091113] - 1}]; Table[Count[lst, x_ /; x < 10^n], {n, 4}]  (* Robert Price, May 31 2019 *)

Generate the odd nonprime sequence with nonprimes labeled 1 mod 4 or 3 mod 4. Add count of nonprimes to sequence if 3 nonprimes congruent to 3 mod 4 occur before interruption of a nonprime congruent to 1 mod 4

a(9)-a(13) from Bert Dobbelaere, Dec 19 2018

A291745 Nonprimes of the form 3*k + 1.

Original entry on oeis.org

1, 4, 10, 16, 22, 25, 28, 34, 40, 46, 49, 52, 55, 58, 64, 70, 76, 82, 85, 88, 91, 94, 100, 106, 112, 115, 118, 121, 124, 130, 133, 136, 142, 145, 148, 154, 160, 166, 169, 172, 175, 178, 184, 187, 190, 196, 202, 205, 208, 214, 217, 220, 226, 232, 235, 238, 244, 247

Offset: 1

Vincenzo Librandi, Aug 31 2017

Subsequence of A018252. A091300 is a subsequence.

A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jul 02 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..1600

Cf. A002476, A016777, A091113, A091300, A291746, A291747, A373977 (characteristic function).

[n: n in [1..400 by 3] | not IsPrime(n)];

DeleteCases[3 Range[0, 300] + 1, _?PrimeQ]

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A014076 Odd nonprimes.

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A016105 Blum integers: numbers of the form p * q where p and q are distinct primes congruent to 3 (mod 4).

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A091236 Nonprimes of form 4k+3.

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A339817 Squarefree numbers k > 1 for which the 2-adic valuation of phi(k) does not exceed the 2-adic valuation of k-1.

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A091300 Nonprimes of the form 6k + 1.

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A339870 Composite numbers k of the form 4u+1 for which the odd part of phi(k) divides k-1.

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A093183 Number of consecutive runs of just 1 odd nonprime congruent to 1 mod 4 below 10^n.