A028871 -id:A028871 - cp's OEIS frontend

A130609 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+223)^2 = y^2.

0, 32, 533, 669, 833, 3672, 4460, 5412, 21945, 26537, 32085, 128444, 155208, 187544, 749165, 905157, 1093625, 4366992, 5276180, 6374652, 25453233, 30752369, 37154733, 148352852, 179238480, 216554192, 864664325, 1044678957, 1262170865, 5039633544, 6088835708

Offset: 1

Mohamed Bouhamida, Jun 17 2007

Also values x of Pythagorean triples (x, x+223, y).
Corresponding values y of solutions (x, y) are in A159809.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (227+30*sqrt(2))/223 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (105507+65798*sqrt(2))/223^2 for n mod 3 = 0.

Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159809, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159810 (decimal expansion of (227+30*sqrt(2))/223), A159811 (decimal expansion of (105507+65798*sqrt(2))/223^2).

LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,32,533,669,833,3672,4460}, 70]  (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+446*n+49729), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+446 for n > 6; a(1)=0, a(2)=32, a(3)=533, a(4)=669, a(5)=833, a(6)=3672.
G.f.: x*(32+501*x+136*x^2-28*x^3-167*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 223*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 30 2009

A130610 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+359)^2 = y^2.

Original entry on oeis.org

0, 40, 901, 1077, 1281, 6160, 7180, 8364, 36777, 42721, 49621, 215220, 249864, 290080, 1255261, 1457181, 1691577, 7317064, 8493940, 9860100, 42647841, 49507177, 57469741, 248570700, 288549840, 334959064, 1448777077, 1681792581

Offset: 1

Mohamed Bouhamida, Jun 17 2007

Also values x of Pythagorean triples (x, x+359, y).
Corresponding values y of solutions (x, y) are in A159844.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (363+38*sqrt(2))/359 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (293619+186550*sqrt(2))/359^2 for n mod 3 = 0.

Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159844, A028871, A118337, A130609, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159845 (decimal expansion of (363+38*sqrt(2))/359), A159846 (decimal expansion of (293619+186550*sqrt(2))/359^2).

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+718*n+128881), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+718 for n > 6; a(1)=0, a(2)=40, a(3)=901, a(4)=1077, a(5)=1281, a(6)=6160.
G.f.: x*(40+861*x+176*x^2-36*x^3-287*x^4-36*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 359*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 30 2009

A130645 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+439)^2 = y^2.

Original entry on oeis.org

0, 44, 1121, 1317, 1541, 7644, 8780, 10080, 45621, 52241, 59817, 266960, 305544, 349700, 1557017, 1781901, 2039261, 9076020, 10386740, 11886744, 52899981, 60539417, 69282081, 308324744, 352850640, 403806620, 1797049361, 2056565301, 2353558517, 10473972300

Offset: 1

Mohamed Bouhamida, Jun 20 2007

Also values x of Pythagorean triples (x, x+439, y).
Corresponding values y of solutions (x, y) are in A159890.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (443+42*sqrt(2))/439 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (450483+287918*sqrt(2))/439^2 for n mod 3 = 0.

Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159890, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159891 (decimal expansion of (443+42*sqrt(2))/439), A159892 (decimal expansion of (450483+287918*sqrt(2))/439^2).

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 44, 1121, 1317, 1541, 7644, 8780}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+878*n+192721), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+878 for n > 6; a(1)=0, a(2)=44, a(3)=1121, a(4)=1317, a(5)=1541, a(6)=7644.
G.f.: x*(44+1077*x+196*x^2-40*x^3-359*x^4-40*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 439*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 30 2009

A130646 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.

Original entry on oeis.org

0, 56, 1925, 2181, 2465, 13056, 14540, 16188, 77865, 86513, 96117, 455588, 505992, 561968, 2657117, 2950893, 3277145, 15488568, 17200820, 19102356, 90275745, 100255481, 111338445, 526167356, 584333520, 648929768, 3066729845

Offset: 1

Mohamed Bouhamida, Jun 20 2007

Also values x of Pythagorean triples (x, x+727, y).
Corresponding values y of solutions (x, y) are in A159893.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (731+54*sqrt(2))/727 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1304787+843542*sqrt(2))/727^2 for n mod 3 = 0.

Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159893, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159894 (decimal expansion of (731+54*sqrt(2))/727), A159895 (decimal expansion of (1304787+843542*sqrt(2))/727^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,56,1925,2181,2465,13056,14540},40] (* or *) RecurrenceTable[{a[1]==0,a[2]==56,a[3]==1925,a[4]==2181,a[5] == 2465, a[6] == 13056, a[n] ==6a[n-3]-a[n-6]+1454},a,{n,40}] (* Harvey P. Dale, Jan 16 2013 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+1454*n+528529), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1454 for n > 6; a(1)=0, a(2)=56, a(3)=1925, a(4)=2181, a(5)=2465, a(6)=13056.
G.f.: x*(56+1869*x+256*x^2-52*x^3-623*x^4-52*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 727*A001652(k) for k >= 0.

Edited and one term added by Klaus Brockhaus, Apr 30 2009

A130647 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+839)^2 = y^2.

Original entry on oeis.org

0, 60, 2241, 2517, 2821, 15180, 16780, 18544, 90517, 99841, 110121, 529600, 583944, 643860, 3088761, 3405501, 3754717, 18004644, 19850740, 21886120, 104940781, 115700617, 127563681, 611641720, 674354640, 743497644, 3564911217

Offset: 1

Mohamed Bouhamida, Jun 20 2007

Also values x of Pythagorean triples (x, x+839, y).
Corresponding values y of solutions (x, y) are in A159896.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (843+58*sqrt(2))/839 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1760979+1141390*sqrt(2))/839^2 for n mod 3 = 0.

G. C. Greubel, Table of n, a(n) for n = 1..3895
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159896, A028871, A118337, A130645, A130646, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159897 (decimal expansion of (843+58*sqrt(2))/839), A159898 (decimal expansion of (1760979+1141390*sqrt(2))/839^2).

I:=[0,60,2241,2517,2821,15180,16780]; [n le 7 select I[n] else Self(n-1) +6*Self(n-3) -6*Self(n-4) -Self(n-6) +Self(n=7): n in [1..30]]; // G. C. Greubel, May 17 2018

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,60,2241,2517,2821,15180,16780},30] (* Harvey P. Dale, Jun 19 2014 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+1678*n+703921), print1(n, ",")))}

a(n) = 6*a(n-3) -a(n-6) +1678 for n > 6; a(1)=0, a(2)=60, a(3)=2241, a(4)=2517, a(5)=2821, a(6)=15180.
G.f.: x*(60+2181*x+276*x^2-56*x^3-727*x^4-56*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 839*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 30 2009

A240587 Primes p of the form n^2 + 123456789 where 123456789 is the first zeroless pandigital number.

Original entry on oeis.org

123457189, 123459289, 123465253, 123466789, 123470713, 123481753, 123482389, 123486373, 123489913, 123501733, 123505189, 123510613, 123535189, 123545593, 123564373, 123571033, 123584953, 123587833, 123592213, 123610453, 123631513, 123641689, 123657493

Offset: 1

K. D. Bajpai, Apr 08 2014

			123457189 is a prime and appears in the sequence because 123457189 = 20^2 + 123456789.
123459289 is a prime and appears in the sequence because 123459289 = 50^2 + 123456789.

K. D. Bajpai, Table of n, a(n) for n = 1..1111

Cf. A000040, A002496, A028871, A050289, A056899.

KD := proc() local a; a:=n^2+123456789; if isprime(a) then RETURN (a); fi; end: seq(KD(), n=1..1000);

Select[Table[k^2+123456789,{k,1,3000}],PrimeQ]

A028874 Primes of form k^2 - 3.

Original entry on oeis.org

13, 61, 97, 193, 397, 673, 1021, 1153, 1597, 1933, 2113, 3361, 4093, 4621, 6397, 7393, 7741, 8461, 9601, 12097, 12541, 13921, 15373, 16381, 18493, 19597, 20161, 21313, 26893, 29581, 36097, 37633, 40801, 42433, 43261, 47521, 48397

Offset: 1

Patrick De Geest

nonn

Also primes equal to the product of two consecutive odd numbers (A000466) minus 2. - Giovanni Teofilatto, Feb 11 2010

All terms are of the form 6m + 1. - Zak Seidov, May 01 2014

			61 is prime and equal to 8^2 - 3, so it is in the sequence.
67 is prime but it's 8^2 + 3 = 9^2 - 14, so it is not in the sequence.
9^2 - 3 = 78 but it's composite, so it's not in the sequence either.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
R. J. Mathar, Solutions to the exponential Diophantine 1 + p_1^x + p_2^y + p_3^z = w^2 for distinct primes p_1, p_2, p_3, 2022.
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A002476 (Primes of form 6m + 1), A028871, A028872, A028873.

Primes terms in A082109. Subsequence of A068228. - Klaus Purath, Jan 09 2023

[a: n in [2..300] | IsPrime(a) where a is n^2-3 ]; // Vincenzo Librandi, Nov 08 2014

Select[Range[2, 250]^2 - 3, PrimeQ] (* Harvey P. Dale, Aug 07 2013 *)
Select[Table[n^2 - 3, {n, 2, 300}], PrimeQ] (* Vincenzo Librandi, Nov 08 2014 *)

select(isprime, vector(100,n,n^2-3)) \\ Charles R Greathouse IV, Nov 19 2014

A028872 INTERSECT A000040. - Klaus Purath, Dec 07 2020

a(n) = A028873(n)^2 - 3. - Amiram Eldar, Mar 01 2025

A130608 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+167)^2 = y^2.

Original entry on oeis.org

0, 28, 385, 501, 645, 2668, 3340, 4176, 15957, 19873, 24745, 93408, 116232, 144628, 544825, 677853, 843357, 3175876, 3951220, 4915848, 18510765, 23029801, 28652065, 107889048, 134227920, 166996876, 628823857, 782338053, 973329525, 3665054428, 4559800732

Offset: 1

Mohamed Bouhamida, Jun 17 2007

Also values x of Pythagorean triples (x, x+167, y).
Corresponding values y of solutions (x, y) are in A159777.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (171+26*sqrt(2))/167 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (56211+34510*sqrt(2))/167^2 for n mod 3 = 0.

Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159777, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159778 (decimal expansion of (171+26*sqrt(2))/167), A159779 (decimal expansion of (56211+34510*sqrt(2))/167^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,28,385,501,645,2668,3340},80] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+334*n+27889), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+334 for n > 6; a(1)=0, a(2)=28, a(3)=385, a(4)=501, a(5)=645, a(6)=2668.
G.f.: x*(28+357*x+116*x^2-24*x^3-119*x^4-24*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 167*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 30 2009

A189827 a(n) = d(n-1) + d(n+1), where d(k) is the number of divisors of k.

Original entry on oeis.org

3, 5, 4, 7, 4, 8, 5, 8, 5, 10, 4, 10, 6, 9, 6, 11, 4, 12, 6, 10, 6, 12, 5, 12, 7, 10, 6, 14, 4, 14, 6, 10, 8, 13, 6, 13, 6, 12, 6, 16, 4, 14, 8, 10, 8, 14, 5, 16, 7, 12, 6, 14, 6, 16, 8, 12, 6, 16, 4, 16, 8, 11, 10, 15, 6, 14, 6, 14, 6, 20, 4, 16, 8, 10, 10

Offset: 2

T. D. Noe, Apr 28 2011

nonn

d(n-1) + d(n+1) is a measure of the compositeness of the numbers next to n. Sequence A189825 lists the first occurrence of each number.

It is conjectured that every number greater than 3 occurs an infinite number of times. Note that an infinite number of 4's is equivalent to there being an infinite number of twin primes (A001097). An infinite number of 5's is equivalent to there being an infinite number of primes of the form p^2-2 (A028871) or p^2+2 (A056899) for prime p. An infinite number of 6's is equivalent to there being an infinite number of primes of the form p^3-2 (A066878), p^3+2 (A048636), p*q-2 (A063637), or p*q+2 (A063638), where p and q are distinct primes.

			a(5) = d(4) + d(6) = 3 + 4 = 7.

T. D. Noe, Table of n, a(n) for n = 2..10000

Cf. A175144, A189825.

Table[DivisorSigma[0,n-1] + DivisorSigma[0,n+1], {n, 2, 100}]
First[#]+Last[#]&/@Partition[DivisorSigma[0,Range[80]],3,1] (* Harvey P. Dale, May 27 2013 *)

A234812 Primes p of the form n + 987654321 where 987654321 is the largest zeroless pandigital number.

Original entry on oeis.org

987654323, 987654337, 987654347, 987654359, 987654361, 987654377, 987654379, 987654383, 987654419, 987654439, 987654443, 987654461, 987654463, 987654467, 987654511, 987654539, 987654581, 987654583, 987654601, 987654607, 987654611, 987654673, 987654677, 987654791

Offset: 1

K. D. Bajpai, Apr 19 2014

			987654323 is a prime and appears in the sequence because 987654323 = 2 + 987654321.
987654337 is a prime and appears in the sequence because 987654337 = 16 + 987654321.

K. D. Bajpai, Table of n, a(n) for n = 1..9777

Cf. A000040, A002496, A028871, A050289, A056899, A240587.

KD := proc() local a; a:=n+987654321; if isprime(a) then RETURN (a); fi; end: seq(KD(), n=1..1000);

Select[Table[k + 987654321, {k,1,1000}], PrimeQ]
c=0; a=n+987654321; Do[If[PrimeQ[a], c=c+1; Print[c," ",a]], {n,0,200000}] (* b-file *)

cp's OEIS Frontend

A130609 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+223)^2 = y^2.

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A130610 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+359)^2 = y^2.

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A130645 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+439)^2 = y^2.

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A130646 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.

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A130647 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+839)^2 = y^2.

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A240587 Primes p of the form n^2 + 123456789 where 123456789 is the first zeroless pandigital number.

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A028874 Primes of form k^2 - 3.

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