A269789 -id:A269789 - cp's OEIS frontend

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

Vincenzo Librandi, Mar 05 2016

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)

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

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

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.

[p: p in PrimesUpTo(50000) | IsSquare(2*p+11)];

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

lista(nn) = forprime(p=2, nn, if(issquare(2*p+11), print1(p, ", "))); \\ Altug Alkan, Mar 05 2016

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

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

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

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

cp's OEIS Frontend

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

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