A339698
Primes of the form p^2 - p*q + q^2, where p and q are consecutive primes.
Original entry on oeis.org
7, 19, 3163, 23743, 28927, 70783, 141403, 198943, 223837, 265333, 283267, 329503, 1136383, 1223263, 1254427, 1488427, 2238043, 2421163, 3625243, 3904603, 4709143, 4884127, 5216683, 5784133, 7376683, 8065627, 8797183, 10660333, 11242717, 12348223, 16613803, 18594019, 19202167, 19999027
Offset: 1
a(2) = 19 = 3^2 - 3*5 + 5^2 is the only prime obtained with a pair of twin primes: (3, 5). - _Bernard Schott_, Dec 23 2020
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Select[Map[#1^2 - #1 #2 + #2^2 & @@ # &, Partition[Prime@ Range[610], 2, 1]], PrimeQ] (* Michael De Vlieger, Dec 13 2020 *)
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forprime(p=1, 1e4, my(q=nextprime(p+1), x=p^2-p*q+q^2); if(ispseudoprime(x), print1(x, ", "))) \\ Felix Fröhlich, Dec 14 2020
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first(n) = { my(q = 2, p, res = vector(n), t = 0); forprime(p = 3, oo, c = p^2 - p*q + q^2; if(isprime(c), t++; res[t] = c; if(t >= n, return(res) ) ); q = p; ) } \\ David A. Corneth, Dec 19 2020
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
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.
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Select[Table[(Prime[n] + Prime[n + 1])^3 + 3, {n, 500}], PrimeQ[#] &]
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s=[]; for(k=1, 500, t=(prime(k) + prime(k+1))^3 + 3; if(isprime(t), s=concat(s, t))); s
A243904
Semiprimes of the form p^2 + pq + q^2, where p, q are consecutive primes.
Original entry on oeis.org
49, 247, 679, 973, 2701, 5293, 7509, 10801, 12297, 15553, 17337, 25963, 29407, 33079, 34993, 36967, 43249, 53877, 67501, 71157, 76809, 97201, 117613, 155953, 181573, 225237, 270049, 292033, 297679, 314977, 350917, 380217, 477607, 492091, 514213, 632047, 648679
Offset: 1
247 is in the sequence because 7^2 + 7*11 + 11^2 = 247 = 13*19, which is semiprime.
679 is in the sequence because 13^2 + 13*17 + 17^2 = 679 = 7*97, which is semiprime.
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with(numtheory): A243904:= proc() local k, p, q; p:=ithprime(n); q:=ithprime(n+1); k:=p^2 + p*q + q^2; if bigomega(k)=2 then RETURN (k); fi; end: seq(A243904 (), n=1..200);
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Select[Table[Prime[n]^2 + Prime[n] Prime[n + 1] + Prime[n + 1]^2, {n, 100}], PrimeOmega[#] == 2 &]
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issemi(n)=bigomega(n)==2
list(lim)=my(v=List(),p=3,t); forprime(q=5,, t=p^2+p*q+q^2; if(t>lim, break); if(issemi(t), listput(v,t)); p=q); Vec(v) \\ Charles R Greathouse IV, Jul 05 2017
A252231
Primes of the form (p+q)^2 + pq, where p and q are consecutive primes.
Original entry on oeis.org
31, 79, 179, 401, 719, 1619, 3371, 8819, 12491, 15671, 23801, 25919, 28871, 32801, 95219, 118571, 154871, 161999, 190121, 266801, 322571, 364499, 375371, 449951, 524831, 725801, 772229, 796001, 820109, 994571, 1026029, 1053401, 1081121, 1225109, 1326089, 1415039
Offset: 1
79 is in the sequence because (3+5)^2 + 3*5 = 79, which is prime.
401 is in the sequence because (7+11)^2 + 7*11 = 401, which is prime.
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count:= 0:
p:= 2:
while count < 100 do
q:= nextprime(p);
x:= (p+q)^2+p*q;
if isprime(x) then
count:= count+1;
a[count]:= x;
fi;
p:= q;
od:
seq(a[i],i=1..count); # Robert Israel, Dec 16 2014
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Select[Table[(Prime[n] + Prime[n+1])^2 + Prime[n]Prime[n+1], {n,100}], PrimeQ[#] &]
Select[Total[#]^2+Times@@#&/@Partition[Prime[Range[100]],2,1],PrimeQ] (* Harvey P. Dale, Sep 06 2020 *)
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s=[]; for(k=1, 100, p=prime(k); q=prime(k+1); t=(p+q)^2 + p*q; if(isprime(t), s=concat(s, t))); s
A270249
Greater of a pair of twin primes (r,s=r+2) where s is of the form p^2 + pq + q^2 and p and q are also twin primes.
Original entry on oeis.org
109, 433, 2056753, 3121201, 3577393, 26462701, 37340353, 43823053, 128786113, 202705201, 304093873, 888345793, 1005988033, 1399680001, 1537437133, 2282300173, 2310187501, 2444964913, 2929312513, 3564542701, 5831255233, 7950571201, 8512439473, 9346947373, 9648752833, 12627464653, 15624660673
Offset: 1
109 is a term because 109 and 107 are twin primes and 109 = 5^2 + 5*7 + 7^2, 5 and 7 are also twin primes.
433 is a term because 433 and 431 are twin primes and 433 = 11^2 + 11*13 + 13^2, 11 and 13 are also twin primes.
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t(n, p=3) = {while( p+2 < (p=nextprime( p+1 )) || n-->0, ); p-2}
for(n=1, 1e3, if(ispseudoprime(P=(3*t(n)^2 + 6*t(n) + 4)) && ispseudoprime(P-2), print1(P, ", ")));
-
from itertools import islice
from sympy import isprime, nextprime
def A270249_gen(): # generator of terms
p, q = 2, 3
while True:
if q-p == 2 and isprime(s:=3*p*q+4) and isprime(s-2):
yield s
p, q = q, nextprime(q)
A270249_list = list(islice(A270249_gen(),20)) # Chai Wah Wu, Feb 27 2023
Original entry on oeis.org
109, 433, 172801, 238573, 363313, 640333, 1145773, 1968301, 2056753, 3121201, 3577393, 6588973, 11197873, 13079233, 13381633, 15431473, 21676033, 26462701, 34476301, 37340353, 43823053, 48481201, 54749953, 56454733, 90816013, 96038893, 102667501, 128786113
Offset: 1
172801 is a term because 172801 = (241^3 - 239^3)/2, and 172801, 239 and 241 are all primes.
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from itertools import islice
from sympy import isprime, nextprime
def A360490_gen(): # generator of terms
p, q = 3**3, 5
while True:
if isprime(k:=(m:=q**3)-p>>1):
yield k
p, q = m, nextprime(q)
A360490_list = list(islice(A360490_gen(),20)) # Chai Wah Wu, Feb 27 2023
Showing 1-6 of 6 results.
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