A054735 -id:A054735 - cp's OEIS frontend

A232878 Twin prime pairs which sum to perfect squares.

17, 19, 71, 73, 881, 883, 1151, 1153, 2591, 2593, 3527, 3529, 4049, 4051, 15137, 15139, 20807, 20809, 34847, 34849, 46817, 46819, 69191, 69193, 83231, 83233, 103967, 103969, 112337, 112339, 149057, 149059, 176417, 176419, 179999, 180001, 206081, 206083

Offset: 1

Gary Croft, Dec 01 2013

nonn

All square roots of twin prime sums in this sequence (see A152786) are multiples of 6.
Digital roots of all pairs in this sequence are {8,1}.
Twin primes of the form 18n^2 +- 1. - Charles R Greathouse IV, Aug 26 2014

			17+19 = 36, square root of 36 = 6; 71+73 = 144, square root of 144 = 12.

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

Cf. A001097, A054735, A069496, A077800.

t = {}; Do[ps = {2 n^2 - 1, 2 n^2 + 1}; If[PrimeQ[ps[[1]]] && PrimeQ[ps[[2]]], AppendTo[t, ps]], {n, 1000}]; Flatten[t] (* T. D. Noe, Dec 03 2013 *)

for(n=1,1e3, if(isprime(t=18*n^2-1) && isprime(t+2), print1(t", "t+2", "))) \\ Charles R Greathouse IV, Aug 26 2014

a(2*n) = a(2*n-1) + 2, a(2*n+1) = A069496(n).

A195336 Smallest number k such that k^n is the sum of numbers in a twin prime pair.

Original entry on oeis.org

8, 6, 2, 150, 96, 324, 6, 1518, 174, 168, 21384, 18, 20754, 2988, 2424, 8196, 3786, 14952, 34056, 48, 1620, 8256, 31344, 1176, 123360, 147456, 28650, 132, 90, 12834, 81126, 11790, 2340, 9702, 11496, 33000, 10716, 66954, 6816, 234, 109956, 3012, 6744, 117654, 19950, 26550, 8226, 40584, 23640, 30660

Offset: 1

Kausthub Gudipati, Sep 16 2011

nonn

Schinzel's hypothesis H implies that a(n) exists for every n. [Charles R Greathouse IV, Sep 18 2011]

Cf. A054735.

isA054735 := proc(n)
        if type(n,'odd') then
                false;
        else
                isprime(n/2-1) and isprime(n/2+1) ;
        end if;
end proc:
A195336 := proc(n)
        for k from 1 do
                if isA054735(k^n) then
                        return k;
                end if;
        end do:
end proc:
for n from 1  do print(A195336(n)) ; end do: # R. J. Mathar, Sep 20 2011

a(n)=my(k=2);while(!ispseudoprime(k^n/2-1)||!ispseudoprime(k^n/2+1),k+=2);k \\ Charles R Greathouse IV, Sep 18 2011

from sympy import isprime
def cond(k, n): m = (k**n)//2; return isprime(m-1) and isprime(m+1)
def a(n):
    k = 2
    while not cond(k, n): k += 2
    return k
print([a(n) for n in range(1, 25)]) # Michael S. Branicky, Aug 06 2021

a(n) is the least k such that (1/2)*k^n - 1 and (1/2)*k^n + 1 are prime.

a(11)-a(50) from Charles R Greathouse IV, Sep 18 2011

A037076 Palindromes which are the sum of a twin prime pair.

Original entry on oeis.org

8, 696, 4224, 6336, 42024, 44544, 61116, 67176, 69696, 405504, 423324, 480084, 4050504, 4075704, 4078704, 4258524, 4435344, 4607064, 4656564, 4809084, 4844484, 4863684, 4869684, 4885884, 6161616, 6175716, 6371736, 6527256

Offset: 1

G. L. Honaker, Jr.

			8 is a term since it is a palindrome and the sum of the twin primes (3, 5).

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. De Geest, Palindromic sums of powers of consecutive primes: Sums of Twin Prime Pairs

Intersection of A002113 and A054735.

Cf. A001359, A006512.

Select[2*Range[250000], PalindromeQ[#] && And @@ PrimeQ[#/2 + {-1, 1}] &] (* Amiram Eldar, Dec 27 2019 *)

More terms from Carlos Rivera

A082496 Numbers of the form 2p+1, where p and p+2 are a pair of twin primes.

Original entry on oeis.org

7, 11, 23, 35, 59, 83, 119, 143, 203, 215, 275, 299, 359, 383, 395, 455, 479, 539, 563, 623, 695, 839, 863, 923, 1043, 1139, 1199, 1235, 1283, 1319, 1619, 1643, 1655, 1715, 1763, 2039, 2063, 2099, 2123, 2183, 2303, 2459, 2555, 2579, 2603, 2639, 2855, 2903

Offset: 1

Vincenzo Origlio (vincenzo.origlio(AT)itc.cnr.it), Apr 29 2003

nonn

Eric Weisstein's World of Mathematics, Twin Primes

Cf. A001359, A006512.

Equals A054735 - 1.

2Select[ Prime/@Range[ 250 ], PrimeQ[ #+2 ]& ]+1

a(n) = 2*A001359(n) + 1 = 2*A006512(n) - 3 = A001359(n) + A006512(n) - 1

Edited by Dean Hickerson, Jun 20 2003

A226599 Numbers which are the sum of two squared primes in exactly four ways (ignoring order).

Original entry on oeis.org

10370, 10730, 11570, 12410, 13130, 19610, 22490, 25010, 31610, 38090, 38930, 39338, 39962, 40970, 41810, 55250, 55970, 59330, 59930, 69530, 70850, 73730, 76850, 77090, 89570, 98090, 98930, 103298, 118898, 125450, 126290, 130730, 135218, 139490

Offset: 1

Henk Koppelaar, Jun 13 2013

nonn

It appears that all first differences are divisible by 24. - Zak Seidov, Jun 14 2013

			10370 = 13^2 + 101^2 = 31^2 + 97^2 = 59^2 + 83^2 = 71^2 + 73^2.
10730 = 11^2 + 103^2 = 23^2 + 101^2 = 53^2 + 89^2 = 67^2 + 79^2.

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 is a superset).
Cf. A214511 (least number having n representations).
Cf. A225104 (numbers having at least three representations is a superset).
Cf. A226539, A226562 (sums decomposed in exactly two and three ways).

Prime2PairsSum := s -> select(x ->`if`(andmap(isprime, x), true, false),
   numtheory:-sum2sqr(s)):
for n from 2 to 10^6 do
  if nops(Prime2PairsSum(n)) = 4 then print(n, Prime2PairsSum(n)) fi;
od;

(* Assuming mod(a(n),24) = 2 *) Reap[ For[ k = 2, k <= 2 + 240000, k = k + 24, pr = Select[ PowersRepresentations[k, 2, 2], PrimeQ[#[[1]]] && PrimeQ[#[[2]]] &]; If[Length[pr] == 4 , Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jun 14 2013 *)

a(n) = p^2 + q^2; p, q are (not necessarily different) primes

A239309 a(n) is the smallest k such that prime(n) divides Sum_{i=1..k} A086169(i), or 0 if no such k exists, where A086169(i) is the sum of the first i twin prime pairs.

Original entry on oeis.org

1, 0, 2, 5, 3, 37, 21, 29, 67, 71, 23, 11, 15, 7, 58, 12, 41, 8, 66, 25, 35, 370, 35, 17, 75, 159, 198, 30, 37, 153, 232, 333, 170, 507, 108, 279, 41, 61, 486, 9, 194, 211, 29, 73, 173, 575, 152, 214, 10, 147, 126, 672, 388, 77, 358, 1048, 528, 291, 322, 1491

Offset: 1

Michel Lagneau, Mar 15 2014

nonn

a(2) = 0. Proof

It is easy to see that A054735(1)= 8 ==2 (mod 3) and A054735(n)==0 mod 3 for n > 1 where A054735 is the sum of twin pairs. Hence A086169(n)==2 (mod 3) and the prime 3 is never a divisor of A086169(n).

			a(1)=1 because A086169(1)=(3+5)=8 and prime(1)= 2 divides 8;
a(2)=0 because prime(2)=3 is never a divisor of A086169(n);
a(3)=2 because A086169(2)=(3+5)+(5+7)=20 and prime(3)= 5 divides 20.

Michel Lagneau, Table of n, a(n) for n = 1..1000

Cf. A000040, A054735, A086169.

Transpose[With[{aprs=Thread[{Range[5000],Accumulate[Select[Table[Prime[n]+1,{n,45900}],PrimeQ[#+1]&]*2]}]},Flatten[Table[Select[aprs,Divisible[Last[#],Prime[m]]&,1],{m,1,60}],1]]][[1]]

A243914 Even numbers which are twice the sum of a twin prime pair, but cannot be expressed as the sum of 2 distinct twin prime pairs.

Original entry on oeis.org

16, 24, 792, 1392

Offset: 1

Lear Young, Jun 14 2014

nonn

Subsequence of A111046 (twice A054735).

It seems that this sequence is probably finite (there are no further terms below 10^7).

			a(1) = 16 = 2*(3+5).
16 is in the sequence since it is twice the sum of twin primes 3 and 5, but cannot be expressed as the sum of 2 distinct twin pairs.
36 is not in the sequence because although it is the sum of twin primes 17 and 19, it can also be expressed as the sum of pairs (5, 7) and (11, 13).

Cf. A054735, A111046.

with(SignalProcessing): # requires at least Maple 17
N:= 10^6; # to check primes up to N
Primes:= select(isprime,{seq(2*i+1,i=1..N)}):
Twins:= Primes intersect map(t-> t-2,Primes):
nT:= nops(Twins);
T:= Array(1..(Twins[nT]+1)/2, datatype=float[8]);
for i from 1 to nT do T[(Twins[i]+1)/2]:= 1 od:
TTwins:= Convolution(T,T);
map(t -> 4*(t+1), select(n -> round(TTwins[n])=1,[$1..(nT+1)/2])); # Robert Israel, Jun 15 2014

isok(isum1, vsum2) = {for (k=1, #vsum2, ksum2 = vsum2[k]; if (ksum2 > one, break;); if (isum1 - ksum2 != ksum2, if (vecsearch(vsum2, isum1 - ksum2), return (0)););); return (1);}
lista() = {v = readvec("b014574.txt"); vsum1 = 4*v; vsum2 = 2*v; maxs2 = vecmax(vsum2); for (i=1, #v, isum1 = vsum1[i]; if (isum1 < maxs2, if (isok(isum1, vsum2), print1(isum1, ", "));););} \\ Michel Marcus, Jun 15 2014

l1=l2=List();a=select(p->isprime(p+2),primes(1000));for(i=1,#a-1,if(i<#a/4,listput(l1,4*a[i]+4));for(j=i+1,#a,listput(l2,2*(a[i]+a[j])+4)));print(setminus(Set(l1),Set(l2))) \\ Lear Young, Jun 15 2014

A261889 Primes that are the square of the sum of a twin prime pair plus 1.

Original entry on oeis.org

577, 1297, 7057, 14401, 41617, 90001, 147457, 156817, 484417, 746497, 1299601, 1742401, 2702737, 2944657, 4260097, 5308417, 6051601, 6780817, 8785297, 10497601, 14107537, 15210001, 16451137, 17438977, 18147601, 29419777, 38937601, 45968401, 51322897, 56791297

Offset: 1

K. D. Bajpai, Sep 05 2015

nonn

Alternatively: Primes of the form (p + q)^2 + 1 where p and q are twin primes.

All the terms are congruent to 1 (mod 3).

			577 appears in the sequence because it is a prime resulting from twin prime pair (11,13): (11 + 13)^2 + 1 = 577.
7057 appears in the sequence because it is a prime resulting from twin prime pair (41,43): (41 + 43)^2 + 1 = 7057.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A051779, A054735.

[k : p in PrimesUpTo (10000) | IsPrime(p+2) and IsPrime(k) where k is ((p + p + 2)^2 + 1)];

A261889:= proc() local a, b, d; a:= ithprime(n); b:=a+2; d:=(a+b)^2+1; if isprime(b)and isprime(d) then return (d): fi; end: seq(A261889 (), n=1..10000);

A261889 = {}; Do[p1 = Prime[n]; p2 = p1 + 2; p = (p1 + p2)^2 + 1; If[PrimeQ[p2] &&  PrimeQ[p], AppendTo[A261889, p]], {n, 1, 10000}]; A261889

forprime(p = 1,10000, if(isprime(p+2) && isprime((p + p + 2)^2 + 1), print1(( (p + p + 2)^2 + 1), ", ")));

list(lim)=my(v=List(),t,p=2); forprime(q=3,sqrtint(lim\1-1)\2+1, if(q-p==2 && isprime(t=(p+q)^2+1), listput(v,t)); p=q); Vec(v) \\ Charles R Greathouse IV, Sep 06 2015

A269662 Semiprimes which are the sum of a twin prime pair plus one.

Original entry on oeis.org

9, 25, 85, 121, 145, 205, 217, 301, 361, 481, 565, 697, 841, 865, 1141, 1285, 1717, 1765, 2041, 2101, 2305, 2461, 2581, 2605, 2641, 2965, 2977, 3241, 3337, 3397, 3865, 3901, 3997, 4285, 4537, 4681, 4765, 5317, 5377, 5461, 5941, 6001, 6241, 6505, 6937, 7081, 7117

Offset: 1

K. D. Bajpai, Mar 02 2016

nonn

All the terms, except a(1), are congruent to 1 (mod 3).

			a(2) = 25 = 5 * 5 that is semiprime. Also, 25 = 11 + 13 + 1 where {11, 13} is a twin prime pair.
a(3) = 85 = 5 * 17 that is semiprime. Also, 55 = 41 + 43 + 1 where {41, 43} is a twin prime pair.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A001097, A001358, A001359, A006512, A054735, A118071.

IsP2:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ s: n in [1..1000] | IsPrime(n) and IsPrime(n+2) and IsP2(s) where s is (n + n+2 + 1)];

A269662 = {}; Do[a = Prime[n]; b = a + 2; c = a + b + 1; If[PrimeQ[b] && PrimeOmega[c] == 2, AppendTo[A269662, c]], {n, 1000}]; A269662
Select[Range[1, 7200, 2], And[PrimeOmega@ # == 2, And[PrimeQ@ #, NextPrime[#] - 2] == # &[(# - 1)/2 - 1]] &] (* Michael De Vlieger, Apr 01 2016 *)
Select[1+Total[#]&/@Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]] == 2&],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 10 2016 *)

for(n = 1, 1000, p=prime(n); q=p+2; s=p+q+1; if(isprime(q) && bigomega(s)==2, print1(s,", ")));

A270244 Lesser of twin primes such that the sum of the twin prime pair is the sum of 2 positive cubes.

Original entry on oeis.org

1871, 8819, 74609, 77237, 81647, 93131, 98927, 102059, 108107, 110501, 152837, 180287, 220859, 241919, 256031, 275939, 309851, 319679, 422099, 457001, 459647, 462419, 490247, 530711, 568151, 635291, 660851, 667547, 721619, 729269, 761669, 843677, 859679, 909971, 948401, 1037087, 1041119

Offset: 1

Altug Alkan, Mar 13 2016

nonn

			1871 is a term because 1871 + 1873 = 10^3 + 14^3.
8819 is a term because 8819 + 8821 = 4^3 + 26^3.
74609 is a term because 74609 + 74611 = 7^3 + 53^3.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001359, A003325, A054735.

Select[Select[Prime@ Range[10^5], PrimeQ[# + 2] &], Length[PowersRepresentations[2 # + 2, 2, 3] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Mar 15 2016 *)

isA003325(n) = for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1));
t(n,p=3) = {while( p+2 < (p=nextprime( p+1 )) || n-->0, ); p-2}
for(n=1, 1e4, if(isA003325(2*t(n)+2), print1(t(n), ", ")));

cp's OEIS Frontend

A232878 Twin prime pairs which sum to perfect squares.

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A195336 Smallest number k such that k^n is the sum of numbers in a twin prime pair.

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A037076 Palindromes which are the sum of a twin prime pair.

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A082496 Numbers of the form 2p+1, where p and p+2 are a pair of twin primes.

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A226599 Numbers which are the sum of two squared primes in exactly four ways (ignoring order).

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A239309 a(n) is the smallest k such that prime(n) divides Sum_{i=1..k} A086169(i), or 0 if no such k exists, where A086169(i) is the sum of the first i twin prime pairs.

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A243914 Even numbers which are twice the sum of a twin prime pair, but cannot be expressed as the sum of 2 distinct twin prime pairs.

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A261889 Primes that are the square of the sum of a twin prime pair plus 1.