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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Vincenzo Librandi, Mar 05 2016

Keywords

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)

Examples

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

Links

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.

Programs

  • Magma
    [p: p in PrimesUpTo(50000) | IsSquare(2*p+11)];
  • Mathematica
    Select[Prime[Range[5000]], IntegerQ[Sqrt[2 # + 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

Formula

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

A243888 Primes of the form 2*n^2+26*n+11.

Original entry on oeis.org

71, 107, 191, 239, 347, 1031, 1439, 1667, 1787, 2039, 2447, 2591, 3371, 3539, 5231, 5651, 5867, 6311, 7247, 9311, 9587, 10151, 11027, 11939, 12251, 14207, 14891, 19727, 20939, 21767, 23039, 27539, 30431, 34511, 36107, 39971, 41687, 46439, 47051, 56039, 56711
Offset: 1

Views

Author

Vincenzo Librandi, Jun 16 2014

Keywords

Comments

Subsequence of A068231.
Conjecture: except 107, 2^a(n)-1 is not prime; in other words, these primes are included in A054723.
2*a(n) + 147 is a square. - Vincenzo Librandi, Apr 10 2015

Links

Crossrefs

Cf. A068231.
Cf. Primes of the form 2*n^2+2*(2*k+3)*n+(2*k+1): A176549 (k=0), A221902 (k=1), A154577 (k=2), A154592 (k=3), A154601 (k=4), this sequence (k=5), A243889 (k=6), A217494 (k=7), A243890 (k=8), A221903 (k=9), A217495 (k=10), A217496 (k=11), A217497 (k=12), A217498 (k=13), A243891 (k=14), A243957 (k=15), A217499 (k=16), A217500 (k=17), A217501 (k=18), A217620 (k=19), A243958 (k=20), A217621 (k=21).

Programs

  • Magma
    [a: n in [1..200] | IsPrime(a) where a is 2*n^2+26*n+11];
  • Mathematica
    Select[Table[2 n^2 + 26 n + 11, {n, 800}], PrimeQ]
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