cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

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

A284034 Primes p such that (p^2 - 3)/2 and (p^2 + 1)/2 are twin primes.

Original entry on oeis.org

3, 5, 11, 19, 29, 79, 101, 349, 409, 449, 521, 569, 571, 661, 739, 991, 1091, 1129, 1181, 1459, 1489, 1531, 1901, 2269, 2281, 2341, 2351, 2389, 2549, 2659, 2671, 2719, 2729, 2731, 3109, 4049, 4349, 5279, 5431, 5471, 5531, 5591, 5669, 6329, 6359, 6871, 7559, 7741
Offset: 1

Views

Author

Giuseppe Coppoletta, Mar 19 2017

Keywords

Comments

Primes which correspond to the short leg of an integral right triangle whose hypotenuse is part of a twin prime pair.
Each term p of the sequence must be part of a Pythagorean triple of the form {p, (p^2 - 1)/2, (p^2 + 1)/2} corresponding to {a(n), A284035(n) - 1, A284035(n)}.

Examples

			The prime p = 79 is in the sequence because (p^2-3)/2 = 3119 and (p^2+1)/2 = 3121 are twin primes. Remark that {79, 3120, 3121} is a Pythagorean triple.

Links

Crossrefs

Cf. A284035, A048161, A165635, A284036, A051642.

Programs

  • Mathematica
    Select[Prime@ Range[10^3], Function[p, Times @@ Boole@ Map[PrimeQ[(p^2 + #)/2 ] &, {-3, 1}] == 1]] (* Michael De Vlieger, Mar 20 2017 *)
Select[Prime[Range[1000]],AllTrue[{(#^2-3)/2,(#^2+1)/2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 04 2017 *)
  • PARI
    isok(p) = isprime(p) && isprime((p^2-3)/2) && isprime((p^2+1)/2); \\ Michel Marcus, Mar 31 2017
  • Sage
    [p for p in prime_range(10000) if is_prime((p^2-3)//2) and is_prime((p^2+1)//2)]

A243887 (p^2 - 3)/2 for odd primes p.

Original entry on oeis.org

3, 11, 23, 59, 83, 143, 179, 263, 419, 479, 683, 839, 923, 1103, 1403, 1739, 1859, 2243, 2519, 2663, 3119, 3443, 3959, 4703, 5099, 5303, 5723, 5939, 6383, 8063, 8579, 9383, 9659, 11099, 11399, 12323, 13283, 13943, 14963, 16019, 16379, 18239, 18623, 19403, 19799
Offset: 1

Views

Author

Vincenzo Librandi, Jun 15 2014

Keywords

Links

Crossrefs

Cf. A065091, A165635, A216244.

Programs

  • Magma
    [(p^2-3)/2: p in PrimesInInterval(3, 300)];
  • Mathematica
    Table[(Prime[n]^2 - 3)/2, {n, 2, 100}]

Formula

a(n) = A216244(n) - 1. [Robert Israel, Jun 15 2014]
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