A141158 -id:A141158 - cp's OEIS frontend

A141772 Primes of the form 3x^2 + 5xy - 5y^2 (as well as of the form 7x^2 + 13xy + 3y^2).

3, 5, 7, 17, 23, 37, 73, 97, 107, 113, 163, 167, 173, 193, 197, 227, 233, 277, 283, 313, 317, 337, 347, 367, 397, 487, 503, 547, 607, 617, 643, 653, 673, 677, 683, 743, 787, 823, 827, 853, 857, 877, 887, 907, 947, 983, 997, 1013, 1093, 1117, 1153, 1163, 1187

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 04 2008

nonn

Discriminant = 85. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.

			a(1) = 3 because we can write 3 = 3*1^2 + 5*1*0 - 5*0^2 (or 3 = 7*0^2 + 13*0*1 + 3*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A141773 (d=85). See also A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141158 (d=20). A141159, A141160 (d=21). A141170, A141171 (d=24). A141172, A141173 (d=28). A141174, A141175 (d=32). A141176, A141177 (d=33). A141178 (d=37). A141179, A141180 (d=40). A141181 (d=41). A141182, A141183 (d=44). A033212, A141785 (d=45). A068228, A141187 (d=48). A141188 (d=52). A141189 (d=53). A141190, A141191 (d=56). A141192, A141193 (d=57). A107152, A141302, A141303, A141304 (d=60). A141215 (d=61). A141111, A141112 (d=65). A141750 (d=73). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).

More terms from Colin Barker, Apr 04 2015

Typo in crossrefs fixed by Colin Barker, Apr 05 2015

A155008 Primes p such that (p-a)(p+a)-+ap are primes,a=4.

Original entry on oeis.org

3, 5, 7, 11, 19, 29, 31, 59, 101, 139, 239, 271, 829, 1031, 1201, 1439, 1511, 1531, 2251, 2609, 3929, 4349, 4969, 5449, 5639, 5711, 5801, 5881, 5981, 6521, 6569, 6701, 6949, 6959, 8221, 8831, 9001, 9181, 9209, 9419, 9511, 9929, 10139, 10711, 11839, 11981

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

3*11-28=5, 3*11+28=61, ...

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006, A155007

lst={};Do[p=Prime[n];If[PrimeQ[(p-4)*(p+4)-4*p]&&PrimeQ[(p-4)*(p+4)+4*p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]],AllTrue[(#-4)(#+4)+{4#,-4#},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 27 2020 *)

A155009 Primes p such that (p-a)(p+a)-+ap are primes,a=5.

Original entry on oeis.org

2, 7, 11, 17, 19, 23, 41, 43, 61, 67, 107, 109, 131, 137, 179, 197, 263, 269, 331, 353, 397, 641, 677, 743, 859, 941, 1163, 1171, 1213, 1303, 1319, 1433, 1447, 1453, 1543, 1601, 1783, 2221, 2351, 2371, 2417, 2503, 2657, 2689, 2791, 2797, 2909, 3037, 3301

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

1*12-35=-23, 1*12+35=47; 6*16-55=96-55=41, 6*16-55=96+55=151, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006, A155007, A155008

lst={};Do[p=Prime[n];If[PrimeQ[(p-5)*(p+5)-5*p]&&PrimeQ[(p-5)*(p+5)+5*p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[500]],AllTrue[(#-5)(#+5)+{5#,-5#},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 01 2016 *)

A141778 Primes of the form 4x^2 + 3x*y - 5y^2 (as well as of the form 8x^2 + 11xy + y^2).

Original entry on oeis.org

2, 5, 11, 17, 47, 53, 67, 71, 73, 79, 89, 97, 107, 109, 131, 139, 157, 167, 173, 179, 199, 223, 227, 233, 251, 257, 263, 269, 271, 277, 283, 307, 311, 317, 331, 347, 367, 373, 401, 409, 443, 449, 461, 463, 467, 479, 487, 509, 523, 587, 601, 607, 613, 619, 631

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 04 2008

nonn

Discriminant = 89. Class = 1. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.

A subsequence of (and may possibly coincide with) A038977. - R. J. Mathar, Jul 22 2008

			a(1) = 2 because we can write 2 = 4*1^2 + 3*1*1 - 5*1^2.

Z. I. Borevich and I. R. Shafarevich, Number Theory.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

See also A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141158 (d=20). A141159, A141160 (d=21). A141170, A141171 (d=24). A141172, A141173 (d=28). A141174, A141175 (d=32). A141176, A141177 (d=33). A141178 (d=37). A141179, A141180 (d=40). A141181 (d=41). A141182, A141183 (d=44). A033212, A141785 (d=45). A068228, A141187 (d=48). A141188 (d=52). A141189 (d=53). A141190, A141191 (d=56). A141192, A141193 (d=57). A107152, A141302, A141303, A141304 (d=60). A141215 (d=61). A141111, A141112 (d=65). A141750 (d=73). A141772, A141773 (d=85). A141776, A141777 (d=88). A141778 (d=89). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).

Typo in crossrefs fixed by Colin Barker, Apr 05 2015

A154942 Primes p such that (p-1)p(p+1)-p-2 and (p-1)p(p+1)+p+2 are primes.

Original entry on oeis.org

3, 5, 29, 71, 113, 263, 1103, 2339, 3203, 3413, 3593, 3659, 3719, 4421, 5939, 6269, 7841, 9011, 9029, 13121, 13841, 14423, 15671, 17033, 19073, 22079, 22811, 26783, 27851, 28949, 29303, 30839, 31973, 32063, 32141, 34301, 38543, 38873, 39119

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

nonn

2*3*4=24-3-2=19, 2*3*4=24+3+2=29, ...

Cf. A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939

lst={};Do[p=Prime[n];If[PrimeQ[(p-1)*p*(p+1)-p-2]&&PrimeQ[(p-1)*p*(p+1)+p+2],AppendTo[lst,p]],{n,8!}];lst
prQ[n_]:=Module[{x=n^3-n,y=n+2},And@@PrimeQ[{x+y,x-y}]]; Select[Prime[ Range[4200]],prQ] (* Harvey P. Dale, Jun 21 2012 *)

A266297 Numbers whose last digit is a square.

Original entry on oeis.org

0, 1, 4, 9, 10, 11, 14, 19, 20, 21, 24, 29, 30, 31, 34, 39, 40, 41, 44, 49, 50, 51, 54, 59, 60, 61, 64, 69, 70, 71, 74, 79, 80, 81, 84, 89, 90, 91, 94, 99, 100, 101, 104, 109, 110, 111, 114, 119, 120, 121, 124, 129, 130, 131, 134, 139, 140, 141, 144, 149

Offset: 1

Wesley Ivan Hurt, Dec 26 2015

Numbers ending in 0, 1, 4 and 9.
Union of A008592, A017281, A017317 and A017377. - Hurt
None of these numbers are prime in Z[phi] (where phi = 1/2 + sqrt(5)/2 is the golden ratio), since the numbers in this sequence that are prime in Z can be expressed in the form (a - b sqrt(5))(a + b sqrt(5)). - Alonso del Arte, Dec 30 2015
Union of A197652 and A016897. - Wesley Ivan Hurt, Dec 31 2015
Union of A146763 and A090771. - Wesley Ivan Hurt, Jan 01 2016

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A008592, A016897, A017281, A017317, A017377, A090771, A091084, A141158, A146763, A197652.

[(10*n-11+(-1)^n+(4+2*(-1)^n)*(-1)^((2*n-1+(-1)^n) div 4))/4: n in [1..60]]; // Vincenzo Librandi, Dec 27 2015

A266297:=n->(10*n-11+(-1)^n+(4+2*(-1)^n)*(-1)^((2*n-1+(-1)^n)/4))/4: seq(A266297(n), n=1..100);

Table[(10 n - 11 + (-1)^n + (4 + 2 (-1)^n)*(-1)^((2 n - 1 + (-1)^n)/4))/4, {n, 50}] (* G. C. Greubel, Dec 27 2015 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 4, 9, 10}, 60] (* Vincenzo Librandi, Dec 27 2015 *)
CoefficientList[Series[x*(1 + 3*x + 5*x^2 + x^3)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Dec 30 2015 *)
Flatten[Table[10n + {0, 1, 4, 9}, {n, 0, 19}]] (* Alonso del Arte, Dec 30 2015 *)
Select[Range[0,150],MemberQ[{0,1,4,9},Mod[#,10]]&] (* Harvey P. Dale, Jul 30 2019 *)

is(n) = issquare(n%10); \\ Altug Alkan, Dec 29 2015

G.f.: x^2*(1 + 3*x + 5*x^2 + x^3)/((x - 1)^2*(1 + x + x^2 + x^3)).
a(n) = a(n - 1) + (n - 4) - a(n - 5) for n > 5.
a(n) = (10n - 11 + (-1)^n + (4 + 2(-1)^n) * (-1)^((2n - 1 + (-1)^n)/4))/4.
a(n+1) - a(n) = A091084(n+1) for n>0.
Sum_{n>=2} (-1)^n/a(n) = (14*sqrt(5)*arccoth(sqrt(5)) - 2*Pi*sqrt(1-2/sqrt(5)) + 16*log(2) + 5*log(5))/40. - Amiram Eldar, Jul 30 2024

A154941 Sophie Germain primes in A154939.

Original entry on oeis.org

3, 5, 11, 131, 419, 1409, 2069, 3449, 3761, 3911, 6899, 7079, 7151, 9539, 9791, 10529, 10691, 11321, 11831, 14741, 15269, 17291, 22079, 27281, 27809, 30449, 34439, 45131, 48479, 52289, 54251, 64439, 70901, 75389, 78839, 85691, 101411, 102911

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

nonn

2*3+1=7, 5*2+1=11, ...

Cf. A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939

lst={};Do[p=Prime[n];If[PrimeQ[(p-1)*(p+1)-p]&&PrimeQ[(p-1)*(p+1)+p],If[PrimeQ[p*2+1],AppendTo[lst,p]]],{n,8!}];lst
Select[Prime[Range[10000]],AllTrue[{2#+1,(#-1)(#+1)+#,(#-1)(#+1)-#},PrimeQ]&] (* Harvey P. Dale, Sep 21 2023 *)

A154944 Primes p such that (p-1)p(p+1)-p+2 and (p-1)p(p+1)+p-2 are primes.

Original entry on oeis.org

19, 37, 67, 151, 367, 859, 1471, 2791, 2971, 3061, 4357, 4447, 4507, 6367, 7159, 7237, 7591, 8311, 8647, 11617, 12211, 12601, 13249, 14947, 15271, 15661, 16699, 18097, 19777, 20149, 20347, 20947, 21019, 22741, 23311, 23857, 24019, 25867, 26701

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

nonn

Cf. A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A154942

lst={};Do[p=Prime[n];If[PrimeQ[(p-1)*p*(p+1)-p+2]&&PrimeQ[(p-1)*p*(p+1)+p-2],AppendTo[lst,p]],{n,8!}];lst

A155010 Primes p such that (p-a)(p+a)-+ap and (p-b)(p+b)-+bp are primes, a=2,b=3.

Original entry on oeis.org

7, 37, 587, 28703, 35677, 36857, 99367, 326707, 361687, 578167, 613573, 619007, 656407, 688783, 702203, 713467, 874823, 922027, 940573, 1045763, 1057907, 1244687, 1371157, 1419697, 1555187, 1665767, 1687187, 1687327, 1799453

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006, A155007, A155008, A155009

lst={};Do[p=Prime[n];If[PrimeQ[(p-2)*(p+2)-2*p]&&PrimeQ[(p-2)*(p+2)+2*p]&&PrimeQ[(p-3)*(p+3)-3*p]&&PrimeQ[(p-3)*(p+3)+3*p],AppendTo[lst,p]],{n,9!}];lst
Select[Prime[Range[200000]],AllTrue[Flatten[{(#-2)(#+2)+{2#,-2#},(#-3)(#+3)+ {3#,-3#}}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 26 2015 *)

cp's OEIS Frontend

A141772 Primes of the form 3*x^2 + 5*x*y - 5*y^2 (as well as of the form 7*x^2 + 13*x*y + 3*y^2).

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A155008 Primes p such that (p-a)*(p+a)-+a*p are primes,a=4.

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A155009 Primes p such that (p-a)*(p+a)-+a*p are primes,a=5.

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A141778 Primes of the form 4*x^2 + 3*x*y - 5*y^2 (as well as of the form 8*x^2 + 11*x*y + y^2).

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A154942 Primes p such that (p-1)*p*(p+1)-p-2 and (p-1)*p*(p+1)+p+2 are primes.

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A266297 Numbers whose last digit is a square.

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Magma

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A154941 Sophie Germain primes in A154939.

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Mathematica

A154944 Primes p such that (p-1)*p*(p+1)-p+2 and (p-1)*p*(p+1)+p-2 are primes.

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Mathematica

A155010 Primes p such that (p-a)*(p+a)-+a*p and (p-b)*(p+b)-+b*p are primes, a=2,b=3.

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A141772 Primes of the form 3x^2 + 5xy - 5y^2 (as well as of the form 7x^2 + 13xy + 3y^2).

A155008 Primes p such that (p-a)(p+a)-+ap are primes,a=4.

A155009 Primes p such that (p-a)(p+a)-+ap are primes,a=5.

A141778 Primes of the form 4x^2 + 3x*y - 5y^2 (as well as of the form 8x^2 + 11xy + y^2).

A154942 Primes p such that (p-1)p(p+1)-p-2 and (p-1)p(p+1)+p+2 are primes.

A154944 Primes p such that (p-1)p(p+1)-p+2 and (p-1)p(p+1)+p-2 are primes.

A155010 Primes p such that (p-a)(p+a)-+ap and (p-b)(p+b)-+bp are primes, a=2,b=3.