A054735 -id:A054735 - cp's OEIS frontend

A118071 Primes which are the sum of a twin prime pair + 1.

13, 37, 61, 277, 397, 457, 541, 1201, 1237, 1321, 1621, 1657, 2557, 2857, 3217, 4057, 4177, 4261, 4621, 5101, 5581, 6337, 6661, 6781, 7057, 7537, 8101, 8317, 8461, 8521, 8677, 9277, 9601, 10837, 10957, 11317, 11701, 12541, 12601, 12721, 13381, 13921

Offset: 1

Jonathan Vos Post, May 11 2006

			a(1) = 13 = 5 + 7 + 1 where (5,7) is a twin prime pair.
a(2) = 37 = 17 + 19 + 1.
a(3) = 61 = 29 + 31 + 1.
a(4) = 277 = 137 + 139 + 1.
a(5) = 397 = 197 + 199 + 1.

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

Cf. A000040, A001359, A006512, A054735, A038869.

Select[Total/@Select[Partition[Prime[Range[1000]],2,1],#[[2]]-#[[1]] == 2&]+1,PrimeQ] (* Harvey P. Dale, Jul 25 2019 *)

is(n)=n%12==1 && isprime(n) && isprime(n\2-1) && isprime(n\2+1) \\ Charles R Greathouse IV, Jan 21 2015

{A001359(k) + A006512(k) + 1} INTERSECT {A000040}.
{A054735(k) + 1} INTERSECT {A000040}.
{2*A001359(k) + 3} INTERSECT {A000040}.
{2*A006512(k) - 1} INTERSECT {A000040}. - Juri-Stepan Gerasimov, Apr 26 2010

More terms added by Vladimir Joseph Stephan Orlovsky, Mar 10 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

A309152 Numbers that can be written as the sum of two primes whose difference is also prime.

Original entry on oeis.org

7, 8, 9, 12, 15, 21, 24, 33, 36, 45, 60, 63, 75, 84, 105, 111, 120, 141, 144, 153, 183, 195, 201, 204, 216, 231, 243, 273, 276, 285, 300, 315, 351, 360, 384, 396, 423, 435, 456, 465, 480, 525, 540, 564, 573, 603, 621, 624, 645, 663, 696, 813, 825, 831, 840

Offset: 1

Wesley Ivan Hurt, Jul 14 2019

nonn

Numbers k such that k = p + q where p < q and p, q, and q - p are all prime.
Union of A054735 and (A006512 + 2). - Robert Israel, Jul 15 2019
From Bernard Schott, Jul 15 2019: (Start)
If k is even, then k is in A054735 with q - p = 2.
If k is odd, then k is in (A006512 + 2) with p = 2. (End)

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

Cf. A006512, A054735.

P:= select(isprime, {seq(i,i=3..10000,2)}):
T:= P intersect map(`+`,P,2):
A1:= map(`+`,T, 2):
A2:= select(`<`, map(t -> 2*t-2, T), max(A1)):
sort(convert(A1 union A2,list); # Robert Israel, Jul 15 2019

is(n) = my(x=n-1, y=1); while(x >= y, if(ispseudoprime(x) && ispseudoprime(y), if(ispseudoprime(x-y), return(1))); x--; y++); 0 \\ Felix Fröhlich, Jul 14 2019

A118072 Primes which are the sum of a twin prime pair - 1.

Original entry on oeis.org

7, 11, 23, 59, 83, 359, 383, 479, 563, 839, 863, 1283, 1319, 1619, 2039, 2063, 2099, 2459, 2579, 2903, 2963, 3863, 4259, 4283, 4679, 5099, 5939, 6599, 6659, 6719, 6779, 7079, 7643, 7703, 8039, 8543, 8963, 10463, 10559, 10883, 11003, 11279, 11483, 11699, 12263

Offset: 1

Jonathan Vos Post, May 11 2006

Dickson's conjecture implies this sequence is infinite. - Charles R Greathouse IV, Apr 18 2013

			a(1) = 7 = 3 + 5 - 1 where (3,5) is a twin prime pair.
a(2) = 11 = 5 + 7 - 1 where (5,7) is a twin prime pair.

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

Cf. A000040, A001359, A005384, A006512, A054735.

[p: p in PrimesUpTo(13000)|IsPrime((p-1) div 2) and IsPrime((p+3) div 2)]; // Marius A. Burtea, Jan 01 2020

Select[(Total[#]-1)&/@Select[Partition[Prime[Range[500]],2,1], Last[#]- First[#]== 2&],PrimeQ]  (* Harvey P. Dale, Apr 04 2011 *)

is(p)=isprime((p-1)\2)&&isprime((p+3)\2)&&isprime(p) \\ Charles R Greathouse IV, Apr 18 2013

{A001359(k) + A006512(k) - 1} INTERSECT {A000040}. {A054735(k) - 1} INTERSECT {A000040}. {2*A001359(k) + 1} INTERSECT {A000040}.

More terms from Harvey P. Dale, Apr 04 2011

More terms from Amiram Eldar, Jan 01 2020

A158870 Sums of the form (twin primes + 1) which are also an upper twin prime.

Original entry on oeis.org

13, 61, 1321, 1621, 4261, 5101, 6661, 6781, 11701, 12541, 21061, 66361, 83221, 88261, 107101, 110881, 114661, 127681, 130201, 140761, 141961, 144541, 148201, 149521, 157561, 161341, 163861, 175081, 186481, 204601, 230941, 249541, 267961

Offset: 1

Cino Hilliard, Mar 28 2009

nonn

If the sum is a member of a twin prime pair, it always is the upper member, shown in A158866.
Moreover, except the first term, these numbers are of the form 10k+1. [We prove this by exhausting the possibilities when calculating the upper, summing and inspecting the lower of the sum. Here are the possible outcomes.
p1(k), p2(k)  p2(m) = p1(k)+p2(k)+1
------------  ---------------------------------
10k+1 10k+3   20k+4+1 not prime
10k+3 10k+5   p2(k) not prime
10k+5 10k+7   p1(k) not prime
10k+7 10k+9   20k+16+1 upper => p1(m) not prime
10k+9 10k+11  20k+20+1 = 10(2k+2)+1
So the only form that was not eliminated, is 10k+1. 13 defies this scheme because 10k+5 is prime for k=0, Q.E.D.]

			The 30th lower twin prime is 659. 659+661+1 = 1321, prime and 1319 is too.
Then 1319 is the lower member of the twin prime pair (1319,1321). So 1321 is in the sequence.

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

Cf. A158866.

With[{tws=Total/@Select[Partition[Prime[Range[25000]],2,1],#[[2]]-#[[1]] == 2&]+1},Select[tws,And@@PrimeQ[#+{0,-2}]&]] (* Harvey P. Dale, Apr 30 2014 *)

gp > g(n)=for(x=1,n,y=2*twinl(x)+3;if(isprime(y)&&isprime(y-2), print1(y",")))

{A054735(k)+1: A054735(k)+1 = A006512(j), any j,k} - R. J. Mathar, Apr 06 2009

Edited by R. J. Mathar, Apr 06 2009

A135283 Sum of staircase twin primes according to the rule: top + bottom + next top.

Original entry on oeis.org

13, 23, 41, 65, 101, 143, 191, 245, 311, 353, 425, 479, 551, 581, 623, 695, 749, 821, 875, 971, 1115, 1271, 1325, 1445, 1613, 1739, 1817, 1877, 1943, 2129, 2441, 2471, 2513, 2597, 2783, 3071, 3113, 3161, 3215, 3335, 3533, 3737, 3845, 3881, 3923, 4067

Offset: 1

Cino Hilliard, Dec 02 2007

nonn

We list the twin primes in staircase fashion as follows.
3
5_5
__7_11
____13_17
_______19_29
__________31_41
___________.._..
________________tu(n)_tl(n)
______________________tu(n+1)
...
where tl(n) = n-th lower twin prime, tu(n) = n-th upper twin prime. Then a(n) = tl(n) + tu(n) + tl(n+1).

g(n) = for(x=1,n,y=twinu(x)+twinl(x) + twinl(x+1);print1(y",")) twinl(n) = / *The n-th lower twin prime. */ { local(c,x); c=0; x=1; while(c

a(n) = A054735(n)+A001359(n+1). - R. J. Mathar, Sep 10 2016

A158866 Indices of twin primes if the sum of these twin primes+1 is an upper twin prime.

Original entry on oeis.org

2, 5, 30, 31, 66, 73, 88, 91, 141, 147, 217, 513, 607, 637, 743, 760, 784, 845, 856, 911, 920, 938, 949, 958, 994, 1015, 1031, 1092, 1150, 1246, 1373, 1470, 1553, 1586, 1768, 1814, 1871, 2017, 2029, 2129, 2261, 2271, 2331, 2370, 2458, 2488, 2510, 2545, 2579

Offset: 1

Cino Hilliard, Mar 28 2009

nonn

If the sum is a member of a twin prime pair, it always is the upper twin prime member. [Proof: Twin primes are sequentially of the form 6n-1 and 6n+1. Then adding according to the condition, we get 6n-1+6n+1+1 = 12n+1. This implies 12n+1 is an upper member since if it were a lower member, 12n+1+2 would be the upper member but 12n+3 is not prime contradicting the definition of a twin prime. Therefore 12n+1 must be an upper twin prime member as stated.]

			The 30th lower twin prime is 659. 659+661+1 = 1321, prime and 1319 is too.
Then 1319 is the lower member of the twin prime pair (1319,1321). So 30 is in the sequence.

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

Cf. A158870.

count:= 0: Res:= NULL:
k:= 1:
for j from 1 while count < 100 do
  if isprime(6*j-1) and isprime(6*j+1) then
    k:= k+1;
    if isprime(12*j-1) and isprime(12*j+1) then
       count:= count+1;
       Res:= Res,k;
    fi
  fi
od:
Res; # Robert Israel, Mar 06 2018

utpQ[{a_, b_}]:=And@@PrimeQ[a + b + {1, -1}]; Flatten[Position[Select[ Partition[Prime[Range[25000]],2,1],#[[2]]-#[[1]]==2&],?utpQ]] (* _Harvey P. Dale, Sep 16 2013 *)

twinl(n) = { local(c,x); c=0; x=1; while(c

{k: A054735(k)+1 = A006512(j), any j} - R. J. Mathar, Apr 06 2009

Edited by R. J. Mathar, Apr 06 2009

A165966 Triangular numbers that are sums of twin prime pairs.

Original entry on oeis.org

36, 120, 276, 300, 2556, 3240, 5460, 8256, 12720, 23436, 26796, 34980, 41616, 46056, 56616, 59340, 103740, 122760, 139656, 147696, 157080, 185136, 195000, 231540, 277140, 333336, 353220, 386760, 401856, 516636, 609960, 860016, 1001820

Offset: 1

Zak Seidov, Oct 01 2009

nonn

All terms are multiples of 12.

			36 = 8*9/2 is a term since it is triangular and the sum of the twin primes 17 and 19.
120 = 15*16/2 is a term since it is triangular and the sum of the twin primes 59 and 61.

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

Subsequence of A111163 and of A054735.

Cf. A000217, A001359, A105174.

tri[n_] := n*(n+1)/2; Select[tri /@ Range[10^3], And @@ PrimeQ[#/2 + {-1, 1}] &] (* Amiram Eldar, Dec 27 2019 *)

lista(nn) = {for (n = 1, nn, trg = n*(n+1)/2; if (!(trg % 2) && isprime(trg/2-1) && isprime(trg/2+1), print1(trg, ", ")););} \\ Michel Marcus, Oct 16 2013

a(n) = A105174(n)*(A105174(n) + 1)/2.

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

cp's OEIS Frontend

A118071 Primes which are the sum of a twin prime pair + 1.

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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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A309152 Numbers that can be written as the sum of two primes whose difference is also prime.

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A118072 Primes which are the sum of a twin prime pair - 1.

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Magma

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A158870 Sums of the form (twin primes + 1) which are also an upper twin prime.

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A135283 Sum of staircase twin primes according to the rule: top + bottom + next top.

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PARI

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A158866 Indices of twin primes if the sum of these twin primes+1 is an upper twin prime.

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