A154670 -id:A154670 - cp's OEIS frontend

A154671 Averages of twin prime pairs k such that k*3 and k/3 are squares.

12, 108, 192, 432, 1452, 2028, 3468, 4800, 10092, 18252, 106032, 139968, 221952, 284592, 299568, 355008, 549552, 618348, 720300, 786432, 823728, 961068, 995328, 1009200, 1138368, 1190700, 1291008, 1529388, 1537968, 1651692, 1948908

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

nonn

			12*3 = 36 = 6^2, 12/3 = 4 = 2^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A154670.

lst={}; Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n*3)^(1/2); If[Floor[s]==s,AppendTo[lst,n]]],{n,6,10!,6}]; lst...and/or... lst={}; Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n/3)^(1/2); If[Floor[s]==s,AppendTo[lst,n]]],{n,6,10!,6}]; lst
Select[Mean/@Select[Partition[Prime[Range[150000]],2,1],#[[2]]-#[[1]] == 2&],AllTrue[{Sqrt[#/3],Sqrt[3#]},IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 06 2015 *)

for(i=1,999, isprime(12*i^2+1) && isprime(12*i^2-1) && print1(12*i^2",")) \\ M. F. Hasler, Jan 15 2009

a(n) = 12*A154331(n)^2. - M. F. Hasler, Jan 15 2009

A154672 Numbers n = 5k^2 such that n - 1 and n + 1 are (twin) primes (thus k=6m).

Original entry on oeis.org

180, 1620, 8820, 35280, 87120, 151380, 302580, 380880, 691920, 737280, 808020, 1393920, 5020020, 5767380, 7712820, 9604980, 10281780, 11160180, 12450420, 12736080, 14723280, 15138000, 17186580, 17860500, 18663120, 18779220, 19129680, 21300480

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

nonn

Original definition: Averages of twin prime pairs n such that n*5 and n/5 are squares.

Obviously, n*5 is a square iff n/5 is a square, say k^2. But n=5k^2 can't be the average of a twin prime pair unless it's a multiple of 6, thus k=6m and n=5*36*m^2. - M. F. Hasler, Apr 11 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000290, A014574, A154670, A154671, A154673, A154674, A154675, A154676.

lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n*5)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,10!,6}];lst (*...and/or...*) lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n/5)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,10!,6}];lst

forstep(k=0,1e4,6, isprime(k^2*5+1) & isprime(k^2*5-1) & print1(k^2*5,",")) \\ M. F. Hasler, Apr 11 2009

A154672 = 5*A000290 intersect A014574 = 180*A000290 intersect A014574. - M. F. Hasler, Apr 11 2009

Edited and extended by M. F. Hasler, Apr 11 2009

A154673 Averages of twin prime pairs k such that k*7 and k/7 are squares.

Original entry on oeis.org

2268, 6300, 30492, 49392, 81648, 100800, 170352, 183708, 197568, 211932, 242172, 363888, 444528, 681408, 847728, 1032192, 1342908, 1694448, 1820700, 2041200, 3049200

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Harvey P. Dale)

Cf. A154670, A154671, A154672.

lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n*7)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,10!,6}];lst...and/or... lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n/7)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,10!,6}];lst
Select[Mean/@Select[Partition[Prime[Range[250000]],2,1],#[[2]]-#[[1]] == 2&],AllTrue[{Sqrt[7#],Sqrt[#/7]},IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 22 2018 *)
Select[7*Range[10^3]^2, And @@ PrimeQ[# + {-1, 1}] &] (* Amiram Eldar, Dec 25 2019 *)

A154676 Numbers n = 103k^2 such that (n-1,n+1) is a twin prime pair (thus k = 6m).

Original entry on oeis.org

2317500, 12047292, 26163648, 43250112, 47347452, 61704828, 168228252, 333720000, 351755712, 426127068, 513127872, 840143808, 979638768, 998790588, 1089276912, 1330434108, 1357220700, 1388809152, 1694467008, 1927570428, 1986835392, 2035992348, 2136108348, 2858437872, 3070594800, 3241626300, 3903322608

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 14 2009

Original definition: Averages of twin prime pairs n such that n*103 and n/103 are squares.
All terms are of the form 3708*k^2. - Zak Seidov, Jan 15 2009
Obviously n*103 is a square iff n/103 is a square, say k^2. But n=103k^2 can't be the average of a twin prime pair unless it's a multiple of 6, thus k=6m and n=103*36*m^2. - M. F. Hasler, Apr 11 2009

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A154670, A154671, A154672, A154673, A154674, A154675.

select(t -> isprime(t+1) and isprime(t-1), [seq(3708*i^2, i=1..2000)]); # Robert Israel, Mar 13 2019

lst={}; Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n*103)^(1/2); If[Floor[s]==s,AppendTo[lst,n]]],{n,9!,2*11!,6}]; lst (*...and/or...*) lst={}; Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n/103)^(1/2); If[Floor[s]==s,AppendTo[lst,n]]],{n,9!,2*11!,6}]; lst
Select[3708*Range[1200]^2,AllTrue[#+{1,-1},PrimeQ]&] (* Harvey P. Dale, May 15 2025 *)

forstep(k=0,1e4,6, isprime(k^2*103+1) & isprime(k^2*103-1) & print1(k^2*103,",")) \\ M. F. Hasler, Apr 11 2009

Edited and extended by M. F. Hasler, Apr 11 2009

A037072 Squares which are the sum of twin prime pairs.

Original entry on oeis.org

36, 144, 1764, 2304, 5184, 7056, 8100, 30276, 41616, 69696, 93636, 138384, 166464, 207936, 224676, 298116, 352836, 360000, 412164, 562500, 725904, 777924, 876096, 944784, 956484, 1077444, 1299600, 1468944, 1617984, 1920996, 2160900, 2286144, 2304324, 2509056

Offset: 1

G. L. Honaker, Jr.

nonn

There are exactly 5^2 squares less than or equal to 1000^2 which are the sum of twin prime pairs.

			36 (square) = 6^2 = 17 + 19 (twin prime pair).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000290, A001359, A006512, A152786, A154670.

[k^2:k in [2..1700 by 2]| IsPrime(k^2 div 2 -1) and IsPrime(k^2 div 2 +1)]; // Marius A. Burtea, Jan 01 2020

lst={};Do[p=n^2;If[PrimeQ[p/2-1]&&PrimeQ[p/2+1], AppendTo[lst, p]], {n, 0, 7!, 2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 25 2008 *)

a(n) = 2 * A154670(n) = A152786(n)^2. - Amiram Eldar, Jan 01 2020

More terms from Amiram Eldar, Jan 01 2020

A154674 Averages of twin prime pairs k such that k*11 and k/11 are squares.

Original entry on oeis.org

990000, 1940400, 2227500, 6187500, 35640000, 54212400, 57182400, 74933100, 159677100, 164745900, 167310000, 194040000, 289485900, 380318400, 424205100, 496742400, 664101900, 684773100, 902919600, 976100400, 982327500, 1007433900

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A154670, A154671, A154672, A154673, A154774.

lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n*11)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,11!,6}];lst...and/or... lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n/11)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,11!,6}];lst
Select[11*Range[10^4]^2, And @@ PrimeQ[# + {-1, 1}] &] (* Amiram Eldar, Dec 25 2019 *)

for(i=1,999, isprime(9900*i^2+1) && isprime(9900*i^2-1) && print1(9900*i^2",")) \\ M. F. Hasler, Jan 15 2009

a(n) = 9900*A154774(n)^2. - M. F. Hasler, Jan 15 2009

More terms from M. F. Hasler, Jan 15 2009

A154675 Averages of twin prime pairs k such that k*13 and k/13 are squares.

Original entry on oeis.org

1872, 7488, 11700, 825552, 990288, 1123668, 1629108, 3146832, 3302208, 4680000, 6627348, 7667712, 8783892, 15502032, 16017300, 16365492, 17252352, 25407252, 32617728, 42401268, 42966612, 54100800, 66163968, 71182800, 73019700

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

nonn

Terms of this sequence must be of the form 13(6m)^2, the values for m are listed in A154775. - M. F. Hasler, Jan 15 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A154670, A154671, A154672, A154673, A154674, A154775.

lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n*13)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,11!,6}];lst
(* ... and/or ... *)
lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n/13)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,11!,6}];lst
Select[13*Range[10^3]^2, And @@ PrimeQ[# + {-1, 1}] &] (* Amiram Eldar, Dec 25 2019 *)

for(i=1,499, isprime(468*i^2+1) && isprime(468*i^2-1) && print1(468*i^2",")) \\ M. F. Hasler, Jan 15 2009

a(n) = 468 A154775(n)^2 - M. F. Hasler, Jan 15 2009

More terms from M. F. Hasler, Jan 15 2009

A173165 Numbers k such that 2*k^2 -+ 1 is a twin prime pair.

Original entry on oeis.org

3, 6, 21, 24, 36, 42, 45, 87, 102, 132, 153, 186, 204, 228, 237, 273, 297, 300, 321, 375, 426, 441, 468, 486, 489, 519, 570, 606, 636, 693, 735, 756, 759, 792, 885, 918, 951, 990, 993, 1065, 1098, 1128, 1131, 1158, 1191, 1326, 1344, 1359, 1386, 1407, 1443

Offset: 1

Juri-Stepan Gerasimov, Feb 11 2010

nonn

A001105(k+1) -+ 1 is a twin prime pair.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001105, A014574, A037073, A154670.

Select[Range[1500], And @@ PrimeQ[2*#^2 + {-1, 1}] &] (* Amiram Eldar, Dec 20 2019 *)

a(n) = 3*A037073(n) = A154670(n)/6.

Entries checked by R. J. Mathar, Mar 09 2010

A256917 Primes which are not the sums of two consecutive nonsquares.

Original entry on oeis.org

2, 3, 7, 17, 19, 31, 71, 73, 97, 127, 163, 199, 241, 337, 449, 577, 647, 881, 883, 967, 1151, 1153, 1249, 1459, 1567, 1801, 2179, 2311, 2591, 2593, 2887, 3041, 3361, 3527, 3529, 3697, 4049, 4051, 4231, 4801, 4999, 5407, 6271, 6961, 7687, 7937, 8191, 8713, 9521, 10369, 10657

Offset: 1

Juri-Stepan Gerasimov, Apr 23 2015

The union of 2 and A066436 and A090698.

The sums of two consecutive nonsquares are 5, 8, 11, 13, 15, 18, 21, 23, 25, 27, 29, 32, 35, 37, ...

			2, 3, 7 are in this sequence because first three sums of two consecutive nonsquares are 5, 8, 11 and 2, 3, 7 are primes.

Colin Barker and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 800 terms from Barker)

Cf. A000037, A066436, A090698, A154670.

Union[{2},Select[Table[2n^2-1,{n,0,1000}],PrimeQ],Select[Table[2n^2+1,{n,0,1000}],PrimeQ]] (* Ivan N. Ianakiev, Apr 24 2015 *)
Module[{nn=11000,ns},ns=Total/@Partition[Select[Range[nn],!IntegerQ[Sqrt[#]]&],2,1]; Complement[ Prime[Range[PrimePi[Last[ns]]]],ns]] (* Harvey P. Dale, Mar 06 2024 *)

a256917(maxp) = {
  ps=[2];
  k=1; while((t=2*k^2-1)<=maxp, k++; if(isprime(t), ps=setunion(ps, [t])));
  k=1; while((t=2*k^2+1)<=maxp, k++; if(isprime(t), ps=setunion(ps, [t])));
  ps
}
a256917(11000) \\ Colin Barker, Apr 23 2015

list(lim)=my(v=List([2]),t); for(k=2,sqrtint((lim+1)\2), if(isprime(t=2*k^2-1), listput(v,t))); for(k=1,sqrtint((lim-1)\2), if(isprime(t=2*k^2+1), listput(v,t))); Set(v) \\ Charles R Greathouse IV, Apr 23 2015

cp's OEIS Frontend

A154671 Averages of twin prime pairs k such that k*3 and k/3 are squares.

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A154672 Numbers n = 5*k^2 such that n - 1 and n + 1 are (twin) primes (thus k=6*m).

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A154673 Averages of twin prime pairs k such that k*7 and k/7 are squares.

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Mathematica

A154676 Numbers n = 103*k^2 such that (n-1,n+1) is a twin prime pair (thus k = 6*m).

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Maple

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A037072 Squares which are the sum of twin prime pairs.

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Magma

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A154674 Averages of twin prime pairs k such that k*11 and k/11 are squares.

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Mathematica

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A154675 Averages of twin prime pairs k such that k*13 and k/13 are squares.

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Mathematica

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Extensions

A154672 Numbers n = 5k^2 such that n - 1 and n + 1 are (twin) primes (thus k=6m).

A154676 Numbers n = 103k^2 such that (n-1,n+1) is a twin prime pair (thus k = 6m).