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 1-10 of 12 results. Next

A176549 Primes of the form 2*n^2+6*n+1.

Original entry on oeis.org

37, 109, 541, 757, 1009, 1297, 1621, 2377, 6841, 7561, 8317, 9109, 11701, 12637, 15661, 16741, 19009, 23977, 25309, 28081, 34057, 38917, 40609, 42337, 44101, 47737, 51517, 55441, 57457, 59509, 65881, 70309, 72577, 82009, 84457, 99901
Offset: 1

Views

Author

Vincenzo Librandi, Apr 20 2010

Keywords

Comments

Conjecture: 2^a(n)-1 is not prime; in other words, these primes are included in A054723.
2*a(n) + 7 is a square. - Vincenzo Librandi, Apr 09 2015

Links

Crossrefs

Primes in A059993.
Subsequence of A093838.
Cf. Primes of the form 2*n^2+2*(2*k+3)*n+(2*k+1): this sequence (k=0), A154577 (k=2), A154592 (k=3), A154601 (k=4), A217494 (k=7), A217495 (k=10), A217496 (k=11), A217497 (k=12), A217498 (k=13), A217499 (k=16), A217500 (k=17), A217501 (k=18), A217620 (k=19), A217621 (k=21).

Programs

  • Magma
    [a: n in [0..300] | IsPrime(a) where a is 2*n^2+6*n+1]; // Vincenzo Librandi, Jul 26 2012
  • Mathematica
    Select[Table[2 n^2 + 6 n + 1, {n, 2000}], PrimeQ] (* Vincenzo Librandi, Jul 26 2012 *)

Extensions

Removed an obviously incorrect part of the definition - R. J. Mathar, Apr 21 2010

A217495 Primes of the form 2*n^2 + 46*n + 21.

Original entry on oeis.org

769, 1381, 1741, 2137, 3037, 3541, 4657, 7321, 9697, 22441, 26437, 30757, 35401, 37021, 38677, 47497, 49369, 55201, 61357, 72337, 79357, 81769, 96997, 99661, 105097, 134437, 188869, 207769, 211657, 227569, 256801, 306301, 330241, 469237, 480937, 492781, 510817
Offset: 1

Views

Author

Vincenzo Librandi, Oct 09 2012

Keywords

Comments

Conjecture: 2^a(n)-1 is not prime; in other words, these primes are included in A054723.
2*a(n)+487 is a square. - Vincenzo Librandi, Mar 04 2013

Links

Crossrefs

Cf. Primes of the form 2*n^2+2*(2*k+3)*n+(2*k+1): A176549 (k=0), A154577 (k=2), A154592 (k=3), A154601 (k=4), A217494 (k=7), this sequence (k=10), A217496 (k=11), A217497 (k=12), A217498 (k=13), A217499 (k=16), A217500 (k=17), A217501 (k=18), A217620 (k=19), A217621 (k=21).
Cf. A054723.
Subsequence of A002144.

Programs

  • Magma
    [a: n in [1..500] | IsPrime(a) where a is 2*n^2+46*n+21];
  • Mathematica
    Select[Table[2 n^2 + 46 n + 21, {n, 500}], PrimeQ]

A217500 Primes of the form 2*n^2 + 74*n + 35.

Original entry on oeis.org

191, 863, 1091, 1871, 2963, 3491, 3863, 4451, 9011, 15731, 21191, 21611, 29363, 30851, 35531, 42863, 44651, 45863, 47711, 50231, 52163, 60251, 65963, 68171, 71171, 75011, 100151, 101051, 109331, 112163, 119891, 144611, 147863, 164663, 179951, 204791, 254963
Offset: 1

Views

Author

Vincenzo Librandi, Oct 09 2012

Keywords

Comments

Conjecture: 2^a(n)-1 is not prime; in other words, these primes are included in A054723.
2*a(n) + 1299 is a square. - Vincenzo Librandi, Apr 09 2015

Links

Crossrefs

Cf. Primes of the form 2*n^2+2*(2*k+3)*n+(2*k+1): A176549 (k=0), A154577 (k=2), A154592 (k=3), A154601 (k=4), A217494 (k=7), A217495 (k=10), A217496 (k=11), A217497 (k=12), A217498 (k=13), A217499 (k=16), this sequence (k=17), A217501 (k=18), A217620 (k=19), A217621 (k=21).
Cf. A054723.
Subsequence of A002145.

Programs

  • Magma
    [a: n in [1..600] | IsPrime(a) where a is 2*n^2 + 74*n + 35];
  • Mathematica
    Select[Table[2n^2 + 74n + 35, {n, 600}], PrimeQ]

A217501 Primes of the form 2*n^2 + 78*n + 37.

Original entry on oeis.org

37, 577, 1657, 2089, 2557, 3061, 4177, 4789, 5437, 6121, 6841, 8389, 12889, 17137, 18289, 19477, 21961, 27361, 36541, 38197, 41617, 45181, 47017, 48889, 54721, 56737, 58789, 74161, 78877, 83737, 88741, 91297, 93889, 96517, 99181, 113041, 121789, 124777
Offset: 1

Views

Author

Vincenzo Librandi, Oct 09 2012

Keywords

Comments

Conjecture: 2^a(n)-1 is not prime; in other words, these primes are included in A054723.
2*a(n) + 1447 is a square. - Vincenzo Librandi, Apr 09 2015

Links

Crossrefs

Cf. Primes of the form 2*n^2+2*(2*k+3)*n+(2*k+1): A176549 (k=0), A154577 (k=2), A154592 (k=3), A154601 (k=4), A217494 (k=7), A217495 (k=10), A217496 (k=11), A217497 (k=12), A217498 (k=13), A217499 (k=16), A217500 (k=17), this sequence (k=18), A217620 (k=19), A217621 (k=21).
Cf. A054723.
Subsequence of A002144.

Programs

  • Magma
    [a: n in [0..600] | IsPrime(a) where a is 2*n^2+78*n+37];
  • Mathematica
    Select[Table[2 n^2 + 78 n + 37, {n, 0, 600}], PrimeQ]

A217620 Primes of the form 2*n^2 + 82*n + 39.

Original entry on oeis.org

211, 499, 823, 1579, 2011, 4099, 6043, 6763, 8311, 10903, 11839, 18211, 27283, 28723, 34843, 38119, 41539, 56659, 58711, 76423, 86143, 88663, 93811, 99103, 110119, 121711, 124699, 130783, 149899, 163363, 173839, 181003, 188311, 222979, 227011, 231079, 247711
Offset: 1

Views

Author

Vincenzo Librandi, Oct 09 2012

Keywords

Comments

Conjecture: 2^a(n)-1 is not prime; in other words, these primes are included in A054723.
2*a(n) + 1603 is a square. - Vincenzo Librandi, Apr 09 2015

Links

Crossrefs

Cf. Primes of the form 2*n^2+2*(2*k+3)*n+(2*k+1): A176549 (k=0), A154577 (k=2), A154592 (k=3), A154601 (k=4), A217494 (k=7), A217495 (k=10), A217496 (k=11), A217497 (k=12), A217498 (k=13), A217499 (k=16), A217500 (k=17), A217501 (k=18), this sequence (k=19), A217621 (k=21).
Cf. A054723.
Subsequence of A002145.

Programs

  • Magma
    [a: n in [1..600] | IsPrime(a) where a is 2*n^2+82*n+39];
  • Mathematica
    Select[Table[2 n^2 + 82 n + 39, {n, 600}], PrimeQ]

A217621 Primes of the form 2*n^2 + 90*n + 43.

Original entry on oeis.org

43, 331, 2311, 3931, 7351, 8971, 18043, 19231, 23011, 31543, 33091, 37951, 46771, 50551, 58543, 60631, 81043, 133711, 149731, 173671, 188143, 226843, 251791, 296251, 310291, 319831, 364543, 385351, 395971, 412171, 417643, 439891, 474343, 540871, 625111, 631843
Offset: 1

Views

Author

Vincenzo Librandi, Oct 09 2012

Keywords

Comments

Conjecture: 2^a(n)-1 is not prime; in other words, these primes are included in A054723.
2*a(n) + 1939 is a square. - Vincenzo Librandi, Apr 09 2015

Links

Crossrefs

Cf. Primes of the form 2*n^2+2*(2*k+3)*n+(2*k+1): A176549 (k=0), A154577 (k=2), A154592 (k=3), A154601 (k=4), A217494 (k=7), A217495 (k=10), A217496 (k=11), A217497 (k=12), A217498 (k=13), A217499 (k=16), A217500 (k=17), A217501 (k=18), A217620 (k=19), this sequence (k=21).
Subsequence of A002145.

Programs

  • Magma
    [a: n in [0..700] | IsPrime(a) where a is 2*n^2+90*n+43];
  • Mathematica
    Select[Table[2 n^2 + 90 n + 43, {n, 0, 700}], PrimeQ]

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]

A221902 Primes of the form 2*n^2 + 10*n + 3.

Original entry on oeis.org

31, 103, 211, 751, 1291, 2371, 2803, 3271, 5503, 6151, 8311, 9103, 9931, 17851, 23971, 25303, 32503, 42331, 49603, 51511, 68071, 82003, 94603, 97231, 105331, 119551, 122503, 137803, 157351, 167611, 171103, 174631, 192811, 204151
Offset: 1

Views

Author

Vincenzo Librandi, Jan 31 2013

Keywords

Comments

Conjecture: After the first term, 2^a(n)-1 is not prime; in other words, these primes (except 31) are included in A054723.
2*a(n) + 19 is a square. - Vincenzo Librandi, Apr 10 2015

Links

Crossrefs

Cf. Primes of the form 2*n^2+2*(2*k+3)*n+(2*k+1): A176549 (k=0), A154577 (k=2), A154592 (k=3), A154601 (k=4), A217494 (k=7), A217495 (k=10), A217496 (k=11), A217497 (k=12), A217498 (k=13), A217499 (k=16), A217500 (k=17), A217501 (k=18), A217620 (k=19), A217621 (k=21).
Cf. A054723 (Prime exponents of nonprime Mersenne numbers).

Programs

  • Magma
    [a: n in [1..500] | IsPrime(a) where a is 2*n^2 + 10*n + 3];
  • Mathematica
    Select[Table[2 n^2 + 10 n + 3,{n, 500}],PrimeQ]

A154576 a(n) = 2*n^2 + 14*n + 5.

Original entry on oeis.org

21, 41, 65, 93, 125, 161, 201, 245, 293, 345, 401, 461, 525, 593, 665, 741, 821, 905, 993, 1085, 1181, 1281, 1385, 1493, 1605, 1721, 1841, 1965, 2093, 2225, 2361, 2501, 2645, 2793, 2945, 3101, 3261, 3425, 3593, 3765, 3941, 4121, 4305, 4493, 4685, 4881
Offset: 1

Views

Author

Vincenzo Librandi, Jan 12 2009

Keywords

Comments

Seventh diagonal in A144562.
2*a(n) + 39 is a square.

Links

Crossrefs

Cf. A144562, A154577.

Programs

  • Magma
    I:=[21, 41, 65]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 22 2012
  • Mathematica
    LinearRecurrence[{3, -3, 1}, {21, 41, 65}, 50] (* Vincenzo Librandi, Feb 22 2012 *)
  • PARI
    for(n=1, 40, print1(2*n^2 + 14*n + 5", ")); \\ Vincenzo Librandi, Feb 22 2012

Formula

G.f.: x*(3-x)*(7-5*x)/(1-x)^3. - Bruno Berselli, Dec 07 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 22 2012
Sum_{n>=1} 1/a(n) = 124/1995 + tan(sqrt(39)*Pi/2)*Pi/(2*sqrt(39)). - Amiram Eldar, Feb 25 2023
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.