A269784 Primes p such that 2*p + 11 is a square.

7, 19, 79, 107, 139, 307, 359, 607, 919, 1399, 1619, 1979, 2239, 2659, 3607, 3779, 4507, 5507, 6379, 6607, 7559, 8059, 8839, 10799, 11699, 12007, 15307, 17107, 20599, 21419, 22679, 23539, 24859, 25307, 25759, 32507, 35107, 40039, 41179, 46507, 47119

Offset

1,1

Comments

Primes of the form 2*n^2 + 10*n + 7.

From Connor Murray, Mar 28 2022: (Start)

Terms appear to all be the difference of a product of consecutive sums and a sum of consecutive products:

(((1+2)*(3+4))-((1*2)+(3*4))) = (21-14) = 7

(((2+3)*(4+5))-((2*3)+(4*5))) = (45-26) = 19

(((5+6)*(7+8))-((5*6)+(7*8))) = (165-86) = 79

(((6+7)*(8+9))-((6*7)+(8*9))) = (221-114) = 107

(((7+8)*(9+10))-((7*8)+(9*10))) = (285-146) = 139

(((11+12)*(13+14))-((11*12)+(13*14))) = (621-314) = 307

(((12+13)*(14+15))-((12*13)+(14*15))) = (725-366) = 359

(((16+17)*(18+19))-((16*17)+(18*19))) = (1221-614) = 607

(((20+21)*(22+23))-((20*21)+(22*23))) = (1845-926) = 919

(((25+26)*(27+28))-((25*26)+(27*28))) = (2805-1406) = 1399

(((27+28)*(29+30))-((27*28)+(29*30))) = (3245-1626) = 1619

(((30+31)*(32+33))-((30*31)+(32*33))) = (3965-1986) = 1979

(((32+33)*(34+35))-((32*33)+(34*35))) = (4485-2246) = 2239

(((35+36)*(37+38))-((35*36)+(37*38))) = (5325-2666) = 2659

(((41+42)*(43+44))-((41*42)+(43*44))) = (7221-3614) = 3607

(((42+43)*(44+45))-((42*43)+(44*45))) = (7565-3786) = 3779

(((46+47)*(48+49))-((46*47)+(48*49))) = (9021-4514) = 4507

(((51+52)*(53+54))-((51*52)+(53*54))) = (11021-5514) = 5507

(((55+56)*(57+58))-((55*56)+(57*58))) = (12765-6386) = 6379

(((56+57)*(58+59))-((56*57)+(58*59))) = (13221-6614) = 6607

(((60+61)*(62+63))-((60*61)+(62*63))) = (15125-7566) = 7559

(((62+63)*(64+65))-((62*63)+(64*65))) = (16125-8066) = 8059

(((65+66)*(67+68))-((65*66)+(67*68))) = (17685-8846) = 8839

(((72+73)*(74+75))-((72*73)+(74*75))) = (21605-10806) = 10799

(((75+76)*(77+78))-((75*76)+(77*78))) = (23405-11706) = 11699

(((76+77)*(78+79))-((76*77)+(78*79))) = (24021-12014) = 12007

(((86+87)*(88+89))-((86*87)+(88*89))) = (30621-15314) = 15307

(((91+92)*(93+94))-((91*92)+(93*94))) = (34221-17114) = 17107

(((100+101)*(102+103))-((100*101)+(102*103))) = (41205-20606) = 20599

(((102+103)*(104+105))-((102*103)+(104*105))) = (42845-21426) = 21419

(((105+106)*(107+108))-((105*106)+(107*108))) = (45365-22686) = 22679

(((107+108)*(109+110))-((107*108)+(109*110))) = (47085-23546) = 23539

(((110+111)*(112+113))-((110*111)+(112*113))) = (49725-24866) = 24859

(((111+112)*(113+114))-((111*112)+(113*114))) = (50621-25314) = 25307

(((112+113)*(114+115))-((112*113)+(114*115))) = (51525-25766) = 25759

(((126+127)*(128+129))-((126*127)+(128*129))) = (65021-32514) = 32507

(((131+132)*(133+134))-((131*132)+(133*134))) = (70221-35114) = 35107

(((140+141)*(142+143))-((140*141)+(142*143))) = (80085-40046) = 40039

(((142+143)*(144+145))-((142*143)+(144*145))) = (82365-41186) = 41179

(((151+152)*(153+154))-((151*152)+(153*154))) = (93021-46514) = 46507

(((152+153)*(154+155))-((152*153)+(154*155))) = (94245-47126) = 47119 (End)

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Formula

a(n) >> n^2 log n. - Charles R Greathouse IV, Aug 23 2022

Example

a(1) = 7 because 2*7+11 = 25.
a(2) = 19 because 2*19+11 = 49.

Mathematica

Select[Prime[Range[5000]], IntegerQ[Sqrt[2 # + 11]] &]

Prog

(Magma) [p: p in PrimesUpTo(50000) | IsSquare(2*p+11)];
(PARI) lista(nn) = forprime(p=2, nn, if(issquare(2*p+11), print1(p, ", "))); \\ Altug Alkan, Mar 05 2016
(PARI) list(lim)=my(v=List(), p); forstep(n=5, sqrtint(lim\1*2+11), 2, if(isprime(p=(n^2-11)/2), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Mar 28 2022
(Python)
from sympy import isprime
A269784_list, j = [], -5
for i in range(10**5):
    A269784_list.extend([j] if isprime(j) else [])
    j += 4*(i+1) # Chai Wah Wu, Mar 09 2016
(Python)
from gmpy2 import is_prime, is_square
for p in range(3, 10**6, 2):
    if(not is_square(2*p+11)):continue
    elif(is_prime(p)):print(p)
# Soumil Mandal, Apr 07 2016

Crossrefs

Cf. primes p such that 2*p + k is a square: A165635 (k=3), A176549 (k=7), A201713 (k=10), this sequence (k=11), A201714 (k=14), A176470 (k=15), A155702 (k=18), A221902 (k=19) A269785 (k=23), A269786 (k=31), A176557 (k=35), A154577 (k=39), A269787 (k=43), A269788 (k=47), A269789 (k=59), A154592 (k=67), A269790 (k=79), A155770 (k=83), A154601 (k=103).

Subsequence of A002145.

Sequence in context: A144723 A062551 A155390 * A237436 A300556 A267378

Adjacent sequences: A269781 A269782 A269783 * A269785 A269786 A269787

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 05 2016

Status

approved