A007645 -id:A007645 - cp's OEIS frontend

A182312 Primes of the form a^2 + b^2 such that both a^2 + b^2 - ab and a^2 + b^2 + ab are prime.

5, 13, 37, 109, 193, 421, 457, 541, 613, 709, 757, 1033, 1117, 1201, 1549, 1597, 1621, 1789, 2137, 2293, 2377, 2437, 2797, 3061, 3109, 3313, 3361, 3469, 4153, 4621, 4657, 4729, 5077, 5233, 5569, 5653, 6421, 6469, 6637, 6997, 7417, 7561, 7681, 7753, 8101, 8689

Offset: 1

Thomas Ordowski, Apr 24 2012

nonn

			The prime 13 = 2^2 + 3^2 is a term, since 13 - 2*3 = 7 is prime and 13 + 2*3 = 19 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A002313.

Cf. A007645.

prsQ[{a_,b_}]:=Module[{c=a^2+b^2,d=a*b},And@@PrimeQ[c+{0,d,-d}]]; Sort[#[[1]]^2+#[[2]]^2&/@Select[Subsets[Range[100],{2}],prsQ]] (* Harvey P. Dale, Apr 27 2014 *)

list(lim)=my(v=List(), t); for(a=1, sqrt(lim), forstep(b=1+a%2, min(a, sqrt(lim-a^2)), 2, if(isprime(t=a^2+b^2) && isprime(t-a*b) && isprime(t+a*b), listput(v, t)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Apr 25 2012

a(n) == 1 (mod 4). - Thomas Ordowski, Mar 13 2018

a(6)-a(46) from Charles R Greathouse IV, Apr 25 2012

A227680 Numbers whose sum of semiprime divisors is a prime number.

Original entry on oeis.org

30, 36, 42, 66, 70, 72, 78, 105, 108, 114, 130, 144, 154, 165, 174, 182, 196, 210, 216, 222, 231, 238, 246, 255, 273, 282, 285, 286, 288, 310, 318, 324, 345, 357, 366, 370, 385, 392, 399, 418, 430, 432, 434, 441, 442, 455, 462, 465, 474, 483, 494, 498, 518

Offset: 1

Michel Lagneau, Jul 19 2013

nonn

There exists a subsequence of infinite squares {36, 144, 196, 324, 441, 576, 676, 784, 1089, 1225, 1296, 1764,...} because the numbers of the form n = (p*q)^2 with p and q primes are in the sequence if p^2 + p*q + q^2 is prime (subsequence of A007645), and the numbers p^2, p*q and q^2 are the three possible semiprime divisors of n. This numbers of the sequence are 6^2, 14^2, 21^2, 26^2, 33^2, 35^2, 51^2, 69^2,...
The numbers of the form n = (p^a*q^v)^2 are also in the sequence => the sequence is infinite.
There exists a subsequence of numbers having three distinct prime divisors p, q and r such that  p*q+q*r+r*p is prime (see A087054). This numbers are 30, 42, 66, 70, 78, 105, 114, ...

			30 is in the sequence because the semiprime divisors of 30 are 2*3, 2*5 and 3*5 and the sum 6+10+15 = 31 is a prime number.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007645 (primes of the form x^2 + xy + y^2).

Cf. A087054 (primes of the form p*q + q*r + r*p where p, q and r are distinct prime numbers).

with(numtheory):for n from 2 to 600 do:x:=divisors(n):n1:=nops(x): y:=factorset(n):n2:=nops(y):s1:=0:s2:=0:for i from 1 to n1 do: if bigomega(x[i])=2 then s1:=s1+x[i]:else fi:od: s2:=sum('y[i]', 'i'=1..n2):if type(s1,prime)=true then printf(`%d, `,n):else fi:od:

semipSigma[n_] := DivisorSum[n, # &, PrimeOmega[#] == 2 &]; Select[Range[500], PrimeQ @ semipSigma[#] &] (* Amiram Eldar, May 10 2020 *)

A230834 Loeschian semiprimes: semiprimes of the form x^2 + x*y + y^2.

Original entry on oeis.org

4, 9, 21, 25, 39, 49, 57, 91, 93, 111, 121, 129, 133, 169, 183, 201, 217, 219, 237, 247, 259, 289, 291, 301, 309, 327, 361, 381, 403, 417, 427, 453, 469, 471, 481, 489, 511, 529, 543, 553, 559, 579, 589, 597, 633, 669, 679, 687, 703, 721, 723, 763, 793, 813, 817, 831, 841, 849, 871

Offset: 1

Irina Gerasimova, Oct 31 2013

nonn

Intersection of A001358 and A003136.

Loeschian primes equals generalized cuban primes (A007645).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

list(lim)=my(v=List()); forprime(p=2,lim\2, forprime(q=2, min(lim\p,p), if((p%3<2 && q%3<2) || p==q, listput(v,p*q)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Oct 31 2013

A252017 Primes of the form (p + q)^3 + 3, where p and q are consecutive primes.

Original entry on oeis.org

140611, 1000003, 68921003, 81746507, 105154051, 360944131, 709732291, 1643032003, 8072216219, 8390176771, 10021812419, 10823192131, 11239424003, 14526784003, 15363967259, 17014253251, 23689358851, 24693014531, 26784575491, 27270901003, 27928443307, 36594368003

Offset: 1

K. D. Bajpai, Dec 13 2014

nonn

			140611 is in the sequence because (23 + 29)^3 + 3 = 140611 which is prime: 23 and 29 are consecutive primes.
81746507 is in the sequence because (211 + 223)^3 + 3 = 81746507 which is prime: 211 and 223 are consecutive primes.

K. D. Bajpai, Table of n, a(n) for n = 1..12289

Cf. A000040, A007645, A003136, A243761.

Select[Table[(Prime[n] + Prime[n + 1])^3 + 3, {n, 500}], PrimeQ[#] &]

s=[]; for(k=1, 500, t=(prime(k) + prime(k+1))^3 + 3; if(isprime(t), s=concat(s, t))); s

A264732 Löschian numbers (A003136) which are the sum of 2 nonzero squares.

Original entry on oeis.org

13, 25, 37, 52, 61, 73, 97, 100, 109, 117, 148, 157, 169, 181, 193, 208, 225, 229, 241, 244, 277, 289, 292, 313, 325, 333, 337, 349, 373, 388, 397, 400, 409, 421, 433, 436, 457, 468, 481, 541, 549, 577, 592, 601, 613, 625, 628, 637, 657, 661, 673, 676, 709, 724, 733

Offset: 1

Altug Alkan, Nov 22 2015

nonn

n is in the sequence iff 4*n is.
If a(n) is a prime number, a(n) mod 12 = 1.
Prime terms of sequence are listed in A068228 that lists generalized cuban primes (A007645) which are the sum of 2 nonzero squares.
Also positive numbers of the form x^2 - 3*y^2 (A084916) that are the sum of 2 nonzero squares. - Frank M Jackson, Oct 13 2019

			a(1) = 13 because 13 = 3^2 + 3*1 + 1^2 = 3^2 + 2^2.
a(2) = 25 because 25 = 5^2 + 5*0 + 0^2 = 4^2 + 3^2.
a(3) = 37 because 37 = 4^2 + 4*3 + 3^2 = 6^2 + 1^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000404, A003136, A068228, A084916.

Select[Range@750, Length[PowersRepresentations[#, 2, 2] /. {0, }->Nothing]>0 && Reduce[#==x^2+x*y+y^2, {x, y}, Integers]=!=False &] (* _Frank M Jackson, Oct 13 2019 *)

isok(n) = { for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
is(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
for(n=1, 1e3, if( is(n) && isok(n), print1(n, ", ")))

A108768 Primes that appear in the sequence p:=x^2+x+1, sieved with a quadratic sieve construction.

Original entry on oeis.org

3, 7, 13, 7, 31, 43, 19, 73, 13, 37, 19, 157, 61, 211, 241, 307, 127, 421, 463, 79, 601, 31, 37, 757, 271, 67, 331, 151, 1123, 397, 97, 43, 67, 1483, 223, 547, 1723, 139, 631, 283, 109, 103, 61, 181, 2551, 379, 919, 409, 2971, 79, 103, 3307, 163, 3541, 523, 97, 3907

Offset: 1

Bernhard Helmes (pi(AT)devalco.de), Jun 24 2005

This sequence appears in a website available on web.archive (see Quadratic Sieve Construction link). There is a single appearance of the first term 3, while all other primes appear twice. See A256148 for a version of the sequence consistent with the current version of the website where each prime appears only once. - Ray Chandler, Jul 05 2015

Bernhard Helmes, Prime sieving on the polynomial f(n)=n^2+n+1.
Bernhard Helmes, Quadratic Sieve Construction (web.archive).

Cf. A002061, A002383, A007645, A162471, A256148.

// from Quadratic Sieve Construction link.
liste_max:=10000;
for x from 1 to liste_max do
    liste_x[x]:=x^2+x+1;
    liste_prim[x]:=1;
end_for;
x:=1;
while (x1) then
     print ("Prim ", p, "x = ", x, isprime (p)) ;
     // Aussiebung
     while (stelleRay Chandler, Jul 05 2015

A112770 Products of pairs of terms from A003627.

Original entry on oeis.org

4, 10, 22, 25, 34, 46, 55, 58, 82, 85, 94, 106, 115, 118, 121, 142, 145, 166, 178, 187, 202, 205, 214, 226, 235, 253, 262, 265, 274, 289, 295, 298, 319, 334, 346, 355, 358, 382, 391, 394, 415, 445, 451, 454, 466, 478, 493, 502, 505, 514, 517, 526, 529, 535

Offset: 1

Jonathan Vos Post and Ray Chandler, Oct 15 2005

It would be incorrect to call these Eisenstein semiprimes. For the Eisenstein primes see A055664. - N. J. A. Sloane, Feb 06 2008. For

Conway, J. H. and Guy, R. K., The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
Wagon, S. "Eisenstein Primes." Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.

Eric Weisstein's World of Mathematics, Eisenstein Integer.
Eric Weisstein's World of Mathematics, Eisenstein Prime.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A003627.

{a(n)} = {p*q: p and q both elements of A007645} = {p*q: p and q both of form 3*m^2 * n^2 for integers m, n}.

Definition corrected by N. J. A. Sloane, Feb 06 2008

A120124 Smallest prime p such that p*10^n + 1 is a prime.

Original entry on oeis.org

3, 7, 3, 7, 7, 61, 3, 7, 7, 3, 19, 37, 109, 79, 97, 13, 37, 19, 73, 103, 97, 283, 157, 61, 19, 61, 1213, 3, 163, 691, 367, 163, 181, 157, 241, 3, 103, 733, 151, 283, 337, 193, 211, 163, 7, 73, 307, 61, 223, 1549, 31, 127, 13, 547, 103, 151, 193, 811, 337, 19, 1021, 151

Offset: 1

Alexander Adamchuk, Aug 15 2006

nonn

All terms belong to A007645. All terms also belong to A055664. Also many terms including the first 14 smallest primes from 3 to 139 {3,7,13,19,31,37,43,61,73,79,97,103,127,139} belong tpA023203. The smallest term that differs from A023203 is 151.

			a(1) = 3 because 31 = 3*10 + 1 is the smallest prime of form p*10 + 1, where p is a prime.
a(2) = 7 because 701 = 7*100 + 1 is the smallest prime of form p*100 + 1.

Robert Israel, Table of n, a(n) for n = 1..1000 (first 300 terms from _Vincenzo Librandi_)

Cf. A121172, A030430, A062800, A007645, A055664, A023203.

Primes:= select(isprime,[$1..10^5]):
for n from 1 to 1000 do
   for p in Primes do
      if isprime(p*10^n+1) then
        A[n]:= p
      fi
    od
od:
seq(A[n],n=1..1000); # Robert Israel, May 29 2014

prs=Prime[Range[2000]];Table[i=1;While[!PrimeQ[First[Take[prs,{i}]] 10^n+1],i++];Prime[i],{n,200}] (* Harvey P. Dale, May 15 2011 *)

A139493 Primes of the form x^2 + 9x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

11, 23, 37, 53, 67, 71, 113, 137, 163, 179, 191, 317, 331, 379, 389, 401, 421, 443, 449, 463, 487, 499, 599, 617, 631, 641, 653, 683, 709, 751, 757, 823, 863, 883, 907, 911, 947, 977, 991, 1061, 1087, 1093, 1103, 1171, 1213, 1303, 1367, 1373, 1409, 1423

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

This is a member of the family of sequences of primes of the forms x^2 + kxy + y^2.

See for k=1 A007645 = x^2+3y^2, k=2 squares no primes, k=3 A038872, k=4 A068228 = x^2+9y^2, k=5 A139492, k=6 A007519 = x^2+8y^2, k=7 A033212 = x^2+15y^2, k=8 A107152 = x^2+45y^2, k=9 A139493, k=10 A107008 = x^2+24y^2, k=11 A139494, k=12 A139495, k=13 A139496, k=14* = 10 A107008 = x^2+24y^2, k=15 A139497, k=16 A033215 = x^2+21y^2, k=17 A139498, k=18 A107145 = x^2+40y^2, k=19 A139499, k=20 A139500, k=21 A139501, k=22 A139502, k=23 A139503, k=24 A139504, k=25 A139505, k=26,A139506, k=27 A139507, k=28 A139508, k=29 A139509, k=30 A139510, k=31 A139511, k=32 A139512

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 9; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

A139495 Primes of the form x^2 + 12x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

29, 109, 149, 281, 389, 401, 421, 449, 541, 569, 641, 701, 709, 809, 821, 1009, 1061, 1129, 1201, 1229, 1289, 1381, 1409, 1429, 1481, 1549, 1621, 1709, 1789, 1801, 1901, 2069, 2081, 2129, 2221, 2269, 2381, 2389, 2521, 2549, 2689, 2741, 2801, 2909, 2969

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 12; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)
With[{nn=50},Take[Union[Select[#[[1]]^2+12#[[1]]#[[2]]+#[[2]]^2&/@ Tuples[ Range[ nn],2],PrimeQ]],nn]] (* Harvey P. Dale, Dec 18 2015 *)

cp's OEIS Frontend

A182312 Primes of the form a^2 + b^2 such that both a^2 + b^2 - a*b and a^2 + b^2 + a*b are prime.

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A227680 Numbers whose sum of semiprime divisors is a prime number.

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A230834 Loeschian semiprimes: semiprimes of the form x^2 + x*y + y^2.

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PARI

A252017 Primes of the form (p + q)^3 + 3, where p and q are consecutive primes.

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Mathematica

PARI

A264732 Löschian numbers (A003136) which are the sum of 2 nonzero squares.

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Mathematica

PARI

A108768 Primes that appear in the sequence p:=x^2+x+1, sieved with a quadratic sieve construction.

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MuPAD

A112770 Products of pairs of terms from A003627.

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Extensions

A120124 Smallest prime p such that p*10^n + 1 is a prime.

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A182312 Primes of the form a^2 + b^2 such that both a^2 + b^2 - ab and a^2 + b^2 + ab are prime.