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.

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A155007 Primes p such that (p-3)*(p+3)-+3*p are primes.

Original entry on oeis.org

7, 17, 37, 113, 157, 227, 283, 293, 313, 347, 443, 587, 787, 883, 1063, 1097, 1237, 1303, 1327, 1427, 1567, 1723, 1933, 1973, 2087, 2347, 2467, 2687, 2777, 3457, 3593, 4447, 4703, 4793, 4967, 5737, 5827, 6317, 6607, 6793, 6857, 8297, 8563, 8803, 9433
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

Keywords

Comments

4*10-3*7=19, 4*10+3*7=61, ...

Crossrefs

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

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[(p-3)*(p+3)-3*p]&&PrimeQ[(p-3)*(p+3)+3*p],AppendTo[lst,p]],{n,7!}];lst

A155008 Primes p such that (p-a)*(p+a)-+a*p 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

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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)-+a*p 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

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

A275068 Squarefree numbers in A022344.

Original entry on oeis.org

1, 5, 11, 19, 29, 31, 41, 55, 59, 61, 71, 79, 89, 95, 101, 109, 131, 139, 145, 149, 151, 155, 179, 181, 191, 199, 205, 209, 211, 229, 239, 241, 251, 269, 271, 281, 295, 305, 311, 319, 331, 341, 349, 355, 359, 379, 389, 395, 401, 409, 419, 421, 431, 439, 445
Offset: 0

Views

Author

Clark Kimberling, Jul 15 2016

Keywords

Comments

The final digit of every number is 1, 5, or 9. As a set, A022344 consists of the numbers m*F^2, where m is in (1,5,11,19,...) and F is a Fibonacci number.
The restriction here to squarefree numbers excludes any of Wechsler's J determinants that derive from rows of the Wythoff array where all terms share a common factor, but there are also nonsquarefree numbers that are determinants of other rows: for example, 121 is the J determinant of row 45 (..., 3, 13, 16, 29, 45, 74, 119, 193, ...). Compare with A089270, which includes 121 and other such numbers. - Peter Munn, Aug 20 2025

Examples

			A022344 = (1,5,4,9,16,11,19,11,20,31,19,31,45,29,... ), and deletion of 4,9,16,20, ... leaves (1,5,11,19,29,31,...).

Links

Crossrefs

Cf. A000045, A022344, A035513, A089270.

Programs

  • Mathematica
    g = GoldenRatio; a[n_] := Floor[(n + 1)*g]^2 - n*Floor[(n + 1)*g] - n^2;
u = Table[a[n], {n, 0, 200}]  (* A022344 *)
Union[Select[u, SquareFreeQ[#] &]]  (* A275068 *)

A336403 Multiplicative closure of A045468: numbers which are the product of zero or more primes which are 1 or 4 mod 5.

Original entry on oeis.org

1, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 121, 131, 139, 149, 151, 179, 181, 191, 199, 209, 211, 229, 239, 241, 251, 269, 271, 281, 311, 319, 331, 341, 349, 359, 361, 379, 389, 401, 409, 419, 421, 431, 439, 449, 451, 461, 479, 491, 499, 509
Offset: 1

Views

Author

David Friend, Jul 20 2020

Keywords

Comments

The subsequence of A089270 which excludes terms divisible by 5.

Links

Crossrefs

Cf. A031363 and its subset A089270 and its subset A038872 and its subset A045468.

Formula

Numbers of the form r^2 + 3*r*s + s^2 not divisible by 5, where r and s are relatively prime and r > s >= 0.

Extensions

New name and general cleanup by Charles R Greathouse IV, Sep 09 2022

A087781 Number of non-congruent solutions to x^2 - x - 1 == 0 mod n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0
Offset: 1

Views

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Keywords

Comments

Sequence A089270 gives the positions of the nonzero terms. The term a(n) gives the number of primitive solutions (x,y) of the equation x^2 + xy - y^2 = n.

Links

Crossrefs

Cf. A000086, A034444.

Extensions

More terms from David Wasserman, Jun 17 2005

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

A356716 a(n) is the integer w such that (c(n)^2, -d(n)^2, -w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 11^3, where c(n) = F(n+2) + (-1)^n * F(n-3), d(n) = F(n+1) + (-1)^n * F(n-4) and F(n) is the n-th Fibonacci number (A000045).

Original entry on oeis.org

5, 19, 31, 101, 179, 655, 1189, 4451, 8111, 30469, 55555, 208799, 380741, 1431091, 2609599, 9808805, 17886419, 67230511, 122595301, 460804739, 840280655, 3158402629, 5759369251, 21648013631, 39475304069, 148377692755, 270567759199, 1016995835621, 1854499010291
Offset: 1

Views

Author

XU Pingya, Aug 24 2022

Keywords

Comments

Conjecture:
(i) For all k > 2, 2*x^3 + 2*y^3 + z^3 = A089270(k)^3 have primitive solutions form (c(n)^2, -d(n)^2, -w(n)) with d(n) = 3*d(n-2) - d(n-4), c(n) = d(n+2) - d(n) and w(n) = 8*w(n-2) - 8*w(n-4) + w(n-6).
(ii) This sequence is a subsequence of A089270.
From XU Pingya, Jun 07 2024: (Start)
Several positive examples of conjecture:
When A089270(4,5,6,7) = {19,29,31,41}, d(n) can be taken as:
(1/2) * (F(n+3) + (-1)^n * F(n-6));
((1-(-1)^n)/2) * (F(n+3) + F(n-4)) + ((1+(-1)^n)/2) * (F(n+3) - F(n-4));
((1-(-1)^n)/2) * (2*F(n-1) + 3*F(n-3)) + ((1+(-1)^n)/2) * (3*F(n-2) + 2*F(n-4));
and
((1-(-1)^n)/2) * (2*F(n+1) + F(n-5)) + ((1+(-1)^n)/2) * (F(n+2) + 2*F(n-4)).
When A089270(17) = 121, d(n) can be taken as d(1,2,3,4) = {-3,0,7,11}. (End)
From XU Pingya, Jul 17 2024: (Start)
Furthermore, we observe that if (x, y) (y < x/2) is the solution of the Diophantine equation x^2 + x * y - y^2 = A089270(k). Let
d(2*n-1) = x * F(2*n-2) - y * F(2*n-3), c(2*n-1) = d(2*n+1) - d(2*n-1);
d(2*n) = x * F(2*n-2) + y * F(2*n-1), c(2*n) = d(2*n+2) - d(2*n).
Then such c(n) and d(n) satisfy the conjecture. (End)

Examples

			For n=3, 2 * ((F(5) - F(0))^2)^3 + 2 * (-(F(4) - F(-1))^2)^3 + (-31)^3 = 2 * 25^3 - 2 * 4^3 - 31^3 = 1331, a(3) = 31.

Links

Crossrefs

Cf. A000045, A081016, A089270, A206351, A228208, A237132.
Cf. also A337928, A354336, A356717.

Programs

  • Mathematica
    Table[(-1331+2*((Fibonacci[n+2]+(-1)^n*Fibonacci[n-3]))^6-2*(Fibonacci[n+1]+(-1)^n*Fibonacci[n-4])^6)^(1/3), {n,28}]

Formula

a(n) = (-1331 + 2 * A237132(n)^6 - 2 * A228208(n-1)^6)^(1/3).
a(n) = ((1-(-1)^n)/2) * (-1 + 6 * Sum_{k=0..n-1} Fibonacci(4*k-1) + 14 * Sum_{k=0..n-2} Fibonacci(4*k+1)) + ((1+(-1)^n)/2) * (-1 + 6 * Sum_{k=0..n-1} Fibonacci(4*k-1) + 14 * Sum_{k=0..n-1} Fibonacci(4*k+1)).
a(n) = ((1-(-1)^n)/2) * (-1 + 6*A206351(n) + 14*A081016(n-2)) + ((1+(-1)^n)/2) * (-1 + 6*A206351(n) + 14*A081016(n-1)).
From Stefano Spezia, Aug 25 2022: (Start)
G.f.: x*(5 + 14*x - 23*x^2 - 28*x^3 - x^4)/((1 - x)*(1 - 3*x + x^2)*(1 + 3*x + x^2)).
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)
From XU Pingya, Jul 17 2024: (Start)
a(2*n-1) = (F(2*n) + F(2*n-2) + F(2*n-5))^2 + (F(2*n) + F(2*n-2) + F(2*n-5)) * (F(2*n-2) + F(2*n-4) + F(2*n-7)) - (F(2*n-2) + F(2*n-4) + F(2*n-7))^2;
a(2*n) = (F(2*n+2) + F(2*n-3))^2 + (F(2*n+2) + F(2*n-3)) * (F(2*n) + F(2*n-5)) - (F(2*n) + F(2*n-5))^2. (End)

A356717 a(n) is the integer w such that (c(n)^2, -d(n)^2, w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 11^3, where c(n) = F(n+2) + (-1)^n * F(n-3), d(n) = F(n+3) + (-1)^n * F(n-2) and F(n) is the n-th Fibonacci number (A000045).

Original entry on oeis.org

1, 29, 59, 241, 445, 1691, 3089, 11629, 21211, 79745, 145421, 546619, 996769, 3746621, 6831995, 25679761, 46827229, 176011739, 320958641, 1206402445, 2199883291, 8268805409, 15078224429, 56675235451, 103347687745, 388457842781, 708355589819, 2662529664049
Offset: 1

Views

Author

XU Pingya, Aug 24 2022

Keywords

Examples

			For n=3, 2 * ((F(5) - F(0))^2)^3 + 2 * (-(F(6) - F(1))^2)^3 + 59^3 = 2 * 25^3 - 2 * 49^3 + 59^3 = 1331, a(3) = 59.

Links

Crossrefs

Cf. A000045, A081016, A081018, A089270, A228208, A237132.
Cf. also A337929, A354337, A356716.

Programs

  • Mathematica
    Table[(1331-2*((Fibonacci[n+2]+(-1)^n*Fibonacci[n-3]))^6+2*(Fibonacci[n+3]+(-1)^n*Fibonacci[n-2])^6)^(1/3), {n,28}]

Formula

a(n) = (1331 - 2 * A237132(n)^6 + 2 * A228208(n+1)^6)^(1/3).
a(n) = ((1-(-1)^n)/2) * (-5 + 14 * Sum_{k=1..n-1} Fibonacci(4*k-1) + 6 * Sum_{k=0..n-1} Fibonacci(4*k+1)) + ((1+(-1)^n)/2) * (-5 + 14 * Sum_{k=1..n} Fibonacci(4*k-1) + 6 * Sum_{k=0..n-1} Fibonacci(4*k+1)).
a(n) = ((1-(-1)^n)/2) * (-5 + 14 * A081018(n-1) + 6 * A081016(n-1)) + ((1+(-1)^n)/2) * (-5 + 14 * A081018(n) + 6 * A081016(n-1)).
From Stefano Spezia, Aug 25 2022: (Start)
G.f.: x*(1 + 28*x + 23*x^2 - 14*x^3 - 5*x^4)/((1 - x)*(1 - 3*x + x^2)*(1 + 3*x + x^2)).
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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 *)
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