A037073 -id:A037073 - cp's OEIS frontend

A173167 Values n such that n and n+1 are both in A037073.

1, 7, 14, 99, 162, 252, 330, 376, 750, 966, 1022, 1079, 1637, 2156, 2288, 2352, 3389, 4530, 4633, 4975, 5514, 6125, 6126, 6622, 6938, 7215, 7489, 8182, 8252, 8293, 10528, 10837, 11332

Offset: 1

Juri-Stepan Gerasimov, Feb 11 2010

nonn

			a(1)=1 because 18*(A037073(1)+1/2-1/2)^2-1=17=prime, 18*(A037073(1)+1/2-1/2)^2+1=19=prime, 18*(A037073(1)+1/2+1/2)^2-1=71=prime and 18*(A037073(1)+1/1+1/2)^2+1=73=prime.

Cf. A037073.

Definition corrected and sequence extended beyond a(8) by R. J. Mathar, Mar 09 2010

A154670 Averages of twin prime pairs k such that k*2 and k/2 are squares.

Original entry on oeis.org

18, 72, 882, 1152, 2592, 3528, 4050, 15138, 20808, 34848, 46818, 69192, 83232, 103968, 112338, 149058, 176418, 180000, 206082, 281250, 362952, 388962, 438048, 472392, 478242, 538722, 649800, 734472, 808992, 960498, 1080450, 1143072

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

nonn

			18/2 = 9 = 3^2, 18*2 = 36 = 6^2.

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

Cf. A037073.

lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=Sqrt[n*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=Sqrt[n/2];If[Floor[s]==s,AppendTo[lst,n]]],{n,6,10!,6}];lst

for(i=1,999, isprime(18*i^2+1) && isprime(18*i^2-1) && print1(18*i^2",")) \\ M. F. Hasler, Jan 15 2009

a(n) = 18*A037073(n)^2. - M. F. Hasler, Jan 15 2009

A152786 Integers k such that (k^2)/2 is the arithmetic mean of a pair of twin primes.

Original entry on oeis.org

6, 12, 42, 48, 72, 84, 90, 174, 204, 264, 306, 372, 408, 456, 474, 546, 594, 600, 642, 750, 852, 882, 936, 972, 978, 1038, 1140, 1212, 1272, 1386, 1470, 1512, 1518, 1584, 1770, 1836, 1902, 1980, 1986, 2130, 2196, 2256, 2262, 2316, 2382, 2652, 2688, 2718

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 12 2008

nonn

Square roots of A054735 where these are integer.

			6 is a term since (6^2)/2 = 18 = mean(17, 19).
12 is a term since (12^2)/2 = 72 = mean(71,73).
42 is a term since (42^2)/2 = 882 = mean(881,883).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..4288 from Zak Seidov)
Zak Seidov, A152786 = 6*A037073: near-duplicates?, seqfan list, Aug 20 2010.

Cf. A014574, A037073, A054735, A152788 (cubic version).

Subsequence of A074924. - Zak Seidov, Feb 01 2013

[k:k in [2..2800 by 2]| IsPrime(k*k div 2 -1) and IsPrime(k*k div 2 +1)]; // Marius A. Burtea, Jan 01 2020

isa := n -> isprime(n) and isprime(n+2) and issqr(2*n+2):
select(isa, [$4..1000000]): map(n -> sqrt(2*n+2), %); # Peter Luschny, Jan 05 2020

lst={};Do[p1=Prime[n];p2=Prime[n+1];If[p2-p1==2,e=(2*(p1+1))^(1/2);i=Floor[e]; If[e==i,AppendTo[lst,i]]],{n,3*9!}];lst
(* Second program: *)
Select[Map[Sqrt[2 #] &, Mean /@ Select[Partition[Prime@ Range[10^6], 2, 1], Subtract @@ # == -2 &]], IntegerQ] (* Michael De Vlieger, Feb 18 2018 *)

forstep(n=6,1e3,6,if(isprime(n^2/2-1)&&isprime(n^2/2+1),print1(n", "))) \\ Charles R Greathouse IV, Feb 01 2013

{n: n^2 = A054735(i), any i}. - R. J. Mathar, Dec 12 2008

a(n) = 6*A037073(n). [Zak Seidov, seqfan list, Aug 20 2010] [From R. J. Mathar, Sep 07 2010]

Edited by R. J. Mathar, Dec 12 2008

A154331 Numbers m such that 12 m^2 is the average of a twin prime pair.

Original entry on oeis.org

1, 3, 4, 6, 11, 13, 17, 20, 29, 39, 94, 108, 136, 154, 158, 172, 214, 227, 245, 256, 262, 283, 288, 290, 308, 315, 328, 357, 358, 371, 403, 413, 414, 420, 475, 491, 510, 522, 536, 543, 546, 556, 559, 561, 581, 585, 622, 630, 633, 647, 666, 669, 676, 699, 735

Offset: 1

M. F. Hasler, Jan 15 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A037073, A154671.

[m:m in [1..740]| IsPrime(12*m^2-1) and IsPrime(12*m^2+1)]; // Marius A. Burtea, Jan 23 2020

Select[Range[800],AllTrue[12#^2+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 27 2014 *)

for(i=1,999, isprime(12*i^2+1) & isprime(12*i^2-1) & print1(i","))

A154772 Numbers m such that 180 m^2 is the average of a twin prime pair.

Original entry on oeis.org

1, 3, 7, 14, 22, 29, 41, 46, 62, 64, 67, 88, 167, 179, 207, 231, 239, 249, 263, 266, 286, 290, 309, 315, 322, 323, 326, 344, 350, 353, 354, 372, 392, 421, 444, 454, 458, 496, 505, 553, 560, 561, 571, 585, 613, 636, 647, 661, 669, 682, 745, 788, 790, 791, 815

Offset: 1

M. F. Hasler, Jan 15 2009

nonn

Inspired by Z. Seidov's post to the SeqFan list, cf. link. This yields A154672 as 180 a(n)^2. Indeed, if N is such that N/5 is a square, then M=5m^2 and this can't by the average of a twin prime pair unless m=6a.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Zak Seidov, "A154676", Jan 15 2009

Cf. A014574, A037073, A154331, A154672.

Select[Range[10^3], And @@ PrimeQ[180#^2 + {-1, 1}] &] (* Amiram Eldar, Dec 25 2019 *)

for(i=1,999, isprime(180*i^2+1) & isprime(180*i^2-1) & print1(i","))

a(n) = sqrt(A154672(n)/180)

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

A226539 Numbers which are the sum of two squared primes in exactly two ways (ignoring order).

Original entry on oeis.org

338, 410, 578, 650, 890, 1010, 1130, 1490, 1730, 1802, 1898, 1970, 2330, 2378, 2738, 3050, 3170, 3530, 3650, 3842, 3890, 4010, 4658, 4850, 5018, 5090, 5162, 5402, 5450, 5570, 5618, 5690, 5858, 6170, 6410, 6530, 6698, 7010, 7178, 7202, 7250, 7850, 7970, 8090

Offset: 1

Henk Koppelaar, Jun 10 2013

nonn

			338 = 7^2 + 17^2 = 13^2 + 13^2;
410 = 7^2 + 19^2 = 11^2 + 17^2.

Stan Wagon, Mathematica in Action, Springer, 2000 (2nd ed.), Ch. 17.5, pp. 375-378.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A054735 (restricted to twin primes), A037073, A069496.
Cf. A045636 (sum of two squared primes: a superset).
Cf. A214511 (least number having n representations).
Cf. A226562 (restricted to sums decomposed in exactly three ways).

Prime2PairsSum := p -> select(x ->`if`(andmap(isprime, x),true,false), numtheory:-sum2sqr(p)):
for n from 2 to 10^6 do
  if nops(Prime2PairsSum(n)) = 2 then print(n, Prime2PairsSum(n)) fi;
od;

Select[Range@10000, Length[Select[ PowersRepresentations[#, 2, 2], And @@ PrimeQ[#] &]] == 2 &] (* Giovanni Resta, Jun 11 2013 *)

select( is_A226539(n)={#[0|t<-sum2sqr(n),isprime(t[1])&&isprime(t[2])]==2}, [1..10^4]) \\ For more efficiency, apply selection to A045636. See A133388 for sum2sqr(). - M. F. Hasler, Dec 12 2019

a(25)-a(44) from Giovanni Resta, Jun 11 2013

A226562 Numbers which are the sum of two squared primes in exactly three ways (ignoring order).

Original entry on oeis.org

2210, 3770, 5330, 6290, 12818, 16490, 18122, 19370, 24050, 24650, 26690, 32810, 33410, 34970, 36530, 39650, 39770, 44642, 45050, 45890, 49010, 50690, 51578, 57770, 59450, 61610, 63050, 66170, 67490, 72410, 73610, 74210, 80330, 85202, 86210, 86330, 88010

Offset: 1

Henk Koppelaar, Jun 11 2013

nonn

Suggestion: difference between successive terms is always at least 3. (With the known 115885 terms <10^9, the smallest difference is 24.) - Zak Seidov, Jun 12 2013

			2210 = 19^2 + 43^2 = 23^2 + 41^2 = 29^2 + 37^2;

Stan Wagon, Mathematica in Action, Springer, 2000 (2nd ed.), Ch. 17.5, pp. 375-378.

Zak Seidov, Table of n, a(n) for n = 1..2464 (all terms up to 10^7).

Cf. A054735 (restricted to twin primes), A037073, A069496.
Cf. A045636 (sum of two squared primes), A226539.
Cf. A214511 (least number having n representations).
Cf. A226539 (restricted to sums decomposed in exactly three ways).

Prime2PairsSum := s -> select( x -> `if`(andmap(isprime, x), true, false), numtheory:-sum2sqr(s)):
for n from 2 to 10 do
if nops(Prime2PairsSum(n)) = 3 then print(n,Prime2PairsSum(n)) fi
od;

Select[Range@20000, Length[Select[ PowersRepresentations[#, 2, 2], And @@ PrimeQ[#] &]] == 3 &] (* Giovanni Resta, Jun 11 2013 *)

a(22)-a(37) from Giovanni Resta, Jun 11 2013

A154773 Numbers n such that 252n^2 is the average of a twin prime pair.

Original entry on oeis.org

3, 5, 11, 14, 18, 20, 26, 27, 28, 29, 31, 38, 42, 52, 58, 64, 73, 82, 85, 90, 110, 125, 138, 156, 167, 180, 212, 234, 248, 297, 299, 303, 305, 308, 312, 314, 317, 319, 334, 336, 348, 361, 365, 371, 372, 377, 379, 414, 451, 465, 478, 499, 508, 509, 535, 554, 564

Offset: 1

M. F. Hasler, Jan 15 2009

nonn

Inspired by Zak Seidov's post to the SeqFan list, cf. link: This yields A154673 as 252 a(n)^2. Indeed, if N/7 is a square, then N=7m^2 and this can't be the average of a twin prime pair unless m=6a.

Zak Seidov, "A154676", Jan 15 2009

Cf. A037073, A154331, A154772.

Select[Range[600],And@@PrimeQ[252#^2+{1,-1}]&] (* Harvey P. Dale, Dec 13 2012 *)

for(i=1,999, isprime(252*i^2+1) & isprime(252*i^2-1) & print1(i","))

a(n) = sqrt(A154673(n)/252).

A154774 Numbers n such that 9900n^2 is the average of a twin prime pair.

Original entry on oeis.org

10, 14, 15, 25, 60, 74, 76, 87, 127, 129, 130, 140, 171, 196, 207, 224, 259, 263, 302, 314, 315, 319, 333, 337, 377, 389, 451, 454, 470, 491, 508, 518, 549, 568, 574, 589, 592, 624, 629, 636, 690, 696, 729, 748, 753, 769, 770, 771, 781, 791, 802, 833, 844

Offset: 1

M. F. Hasler, Jan 15 2009

nonn

Inspired by Zak Seidov's post to the SeqFan list, cf. link: This yields A154674 as 9900 a(n)^2. Indeed, if N/11 is a square, then N=11 m^2 and this can't be the average of a twin prime pair unless m=30a (considering N+1 mod 2,3,5 and N-1 mod 5).

Zak Seidov, "A154676", Jan 15 2009

Cf. A037073, A154331, A154772-A154773.

Select[Range[1000],AllTrue[9900#^2+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 19 2019 *)

for(i=1,999, isprime(9900*i^2+1) & isprime(9900*i^2-1) & print1(i","))

a(n) = sqrt(A154674(n)/9900).

cp's OEIS Frontend

A173167 Values n such that n and n+1 are both in A037073.

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A154670 Averages of twin prime pairs k such that k*2 and k/2 are squares.

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A152786 Integers k such that (k^2)/2 is the arithmetic mean of a pair of twin primes.

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A154331 Numbers m such that 12 m^2 is the average of a twin prime pair.

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A154772 Numbers m such that 180 m^2 is the average of a twin prime pair.

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

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A226539 Numbers which are the sum of two squared primes in exactly two ways (ignoring order).

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Maple

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A226562 Numbers which are the sum of two squared primes in exactly three ways (ignoring order).