A001359 -id:A001359 - cp's OEIS frontend

A126193 Lesser of twin primes (A001359) of the form p = k^2+s such that q = k^4+s is also a lesser of twin primes, q > p and s >= 0.

5, 17, 29, 41, 59, 71, 107, 137, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607, 1619, 1667, 1697, 1721

Offset: 1

Tomas Xordan, Mar 07 2007

p = q-k^4+k^2 where p and q are lesser of twin primes and p < q.

May be connected with the twin prime conjecture (see link).

			5 = 2^2+1 and 17 = 2^4+1; 5 and 17 are lesser of twin primes;
41 = 4^2+25 and 281 = 4^4+25; 41 and 281 are lesser of twin primes.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Prime Conjecture

Cf. A001359, A126769, A126194.

filter:= proc(p) local s,k;
  if not(isprime(p) and isprime(p+2)) then return false fi;
  for k from 2 do
    s:= p - k^2;
    if s < 0 then return false fi;
    if isprime(s+k^4) and isprime(s+k^4+2) then return true fi;
  od
end proc:
select(filter, [seq(i,i=5..2000, 6)]); # Robert Israel, Sep 15 2024

{m=42; v=[]; for(k=2, m, for(s=1, (m+1)^2-1, if((p=k^2+s)p&&isprime(q)&&isprime(q+2), v=concat(v,p)))); v=listsort(List(v), 1); for(j=1, #v, print1(v[j], ","))} /* Klaus Brockhaus, Mar 09 2007 */

Edited and checked by Klaus Brockhaus, Mar 09 2007

Definition corrected by Robert Israel, Sep 15 2024

A126251 a(n) = (p+2)!/p! where p is the n-th lesser twin prime, A001359(n).

Original entry on oeis.org

20, 42, 156, 342, 930, 1806, 3660, 5256, 10506, 11772, 19182, 22650, 32580, 37056, 39402, 52212, 57840, 73170, 79806, 97656, 121452, 176820, 187056, 213906, 273006, 325470, 360600, 382542, 412806, 436260, 656910, 676506, 686412, 737022, 778806

Offset: 1

Zerinvary Lajos, Mar 08 2007

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

Cf. A001359, A082669.

Subsequence of A002378.

ZL:=[]:for p from 1 to 1653 do if (isprime(p) and isprime(p+2)) then ZL:=[op(ZL),((p+2)!/p!)]; fi; od; print(ZL);

#[[2]]!/#[[1]]!&/@Select[Partition[Prime[Range[200]],2,1],#[[2]]-#[[1]] == 2&] (* Harvey P. Dale, Feb 16 2020 *)

a(n) = 2*A082669(n).

Name edited by Michel Marcus, Apr 29 2023

A139187 Smallest twin prime member A001359 of the form k!/n-1.

Original entry on oeis.org

5, 11, 239, 5, 7983359, 3

Offset: 1

Artur Jasinski, Apr 11 2008

"Factorial" twin primes are a pair (k!/n-1, k!/n+1) = (A001359(j), A006512(j)).
Given n, the sequence shows the smallest a(n)=A001359(j) solving this pair equation.
The associated upper twin prime is A139188(n) = A006512(j) = A001359(j)+2 = a(n)+2, and the associated factorial index is k(n) = A139186(n).
The twin prime indices j(n) are 2, 3, 17, 2, 48525, 1.
a(7) is unknown, with k(7) > 25000. A continuation of the sequence, with unknown terms indicated by 0, is a(7)..a(50): 0, 453599, 0, 11, 0, 59, 0, 0, 2687, 0, 0, 0, 2688996956405759999, 5, 239, 0, 0, 29, 44960029111104307199, 0, 134399, 179, 0, 3, 0, 0, 0, 0, 1151, 100799, 0, 0, 536481791999, 17, 0, 0, 0, 141523199, 0, 1313375283986387731246850697141608641462271999999999, 0, 7559, 0, 8065829222532112711679999. - Hugo Pfoertner, Mar 30 2020

			For n=1, the smallest k is 3, where (3!/1-1,3!/1+1) = (5,7) = (A001359(2),A006512(2)).
For n=3, the smallest k is 6, where (6!/3-1,6!/3+1) = (239,241) = (A001359(17),A006512(17)).

Cf. A139186, A139188.

a = {}; Do[k = 1; While[ ! (PrimeQ[(k! - n)/n] && PrimeQ[(k! + n)/n]), k++ ]; AppendTo[a, (k! - n)/n], {n, 1, 6}]; a

A182513 a(n) is the solution of equation A001359(x) = A182482(n).

Original entry on oeis.org

2, 3, 4, 8, 5, 8, 6, 14, 10, 7, 15, 8, 20, 22, 13, 14, 9, 10, 16, 17, 35, 30, 11, 23, 12, 20, 31, 76, 21, 13, 45, 14, 15, 36, 22, 23, 61, 16, 57, 17, 42, 69, 37, 46, 18, 33, 19, 41, 35, 27, 67, 20, 149, 52, 30, 76, 123, 21, 39, 171, 282, 50, 69, 41, 60, 84, 51, 98, 33, 22, 43

Offset: 1

Vladimir Shevelev, May 03 2012

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A182481, A182482, A182483, A001359.

More terms from Ray Chandler, Sep 18 2019

A242918 Positions of smaller of twin primes in A001359 with index 5.

Original entry on oeis.org

11, 18, 21, 27, 43, 48, 62, 77, 107, 109, 110, 118, 131, 145, 172, 201, 216, 258, 260, 265, 289, 294, 301, 307, 315, 319, 340, 350, 363, 365, 367, 381, 442, 449, 451, 453, 491, 496, 500, 515, 522, 528, 540, 559, 569, 581, 603, 613, 616, 620, 623, 630, 659, 689

Offset: 1

Vladimir Shevelev, May 26 2014

nonn

For the definition of the index of a twin prime pair, see the comment in A242767.

Peter J. C. Moses, Table of n, a(n) for n = 1..5000

Cf. A001359, A242758, A242767, A242768, A242807, A242881, A242913, A242917.

A242767=Join[{1},Differences[Flatten[Position[Differences[Prime[Range[10^4]]],2]]]];
A242918=Flatten[Position[A242767,5]] (* Peter J. C. Moses, May 27 2014 *)

More terms from Peter J. C. Moses, May 26 2014

A242919 Positions of smaller of twin primes in A001359 with index 6.

Original entry on oeis.org

9, 13, 24, 26, 53, 59, 61, 63, 76, 88, 94, 100, 104, 115, 126, 146, 156, 160, 184, 203, 206, 210, 224, 229, 240, 266, 276, 279, 298, 309, 333, 338, 352, 386, 406, 415, 431, 435, 444, 450, 469, 473, 508, 525, 529, 535, 537, 546, 550, 576, 580, 615, 633, 634

Offset: 1

Vladimir Shevelev, May 26 2014

nonn

For the definition of the index of a twin prime pair, see the comment in A242767.

Peter J. C. Moses, Table of n, a(n) for n = 1..5000

Cf. A001359, A242758, A242767, A242768, A242807, A242881, A242913, A242917, A242918.

A242767=Join[{1},Differences[Flatten[Position[Differences[Prime[Range[10^4]]],2]]]];
A242919=Flatten[Position[A242767,6]] (* Peter J. C. Moses, May 27 2014 *)

More terms from Peter J. C. Moses, May 26 2014

A243096 Lesser of twin primes (A001359) such that both are full reptend primes (A001913).

Original entry on oeis.org

17, 59, 179, 821, 1019, 1301, 1619, 2141, 2339, 3257, 3299, 3461, 4217, 4259, 4337, 4421, 5417, 5501, 5657, 5741, 6659, 6701, 7457, 8819, 8861, 9341, 10139, 10457, 10859, 10937, 11057, 11699, 11939, 12377, 12821, 13337, 13901, 15137, 15581, 15737, 16979, 17417, 17579, 18059, 19139, 19541, 19697

Offset: 1

Robert G. Wilson v, Aug 18 2014

A proper subset of both A001359 and A001913.

Number of terms < 10^k: 0, 2, 4, 26, 152, 1015, 7618, 56282, 436385, …, .

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A001359 & A001913.

Select[ Prime@ Range@ 2300, MultiplicativeOrder[10, #] == # - 1 && MultiplicativeOrder[10, # + 2] == # + 1 &]
Select[Partition[Prime[Range[2500]],2,1],#[[2]]-#[[1]]==2&& PrimitiveRoot[ #,10]=={10,10}&][[All,1]] (* Harvey P. Dale, Dec 02 2017 *)

Intersection of A001359 and A001913.

A270535 Integers k such that A001359(k) + A001359(k+3) = A001359(k+1) + A001359(k+2).

Original entry on oeis.org

5, 8, 10, 11, 15, 16, 17, 27, 36, 68, 69, 71, 111, 132, 189, 200, 212, 214, 234, 252, 262, 279, 317, 332, 343, 344, 364, 424, 426, 500, 506, 518, 520, 543, 563, 577, 606, 620, 658, 672, 696, 697, 737, 766, 882, 907, 982, 1009, 1064, 1087, 1089, 1091, 1162, 1164, 1172, 1226, 1256, 1268

Offset: 1

Altug Alkan, Mar 18 2016

nonn

Integers k such that A006512(k) + A006512(k+3) = A006512(k+1) + A006512(k+2).

Integers k such that A014574(k) + A014574(k+3) = A014574(k+1) + A014574(k+2).

			5 is a term because A001359(5) = 29, A001359(6) = 41, A001359(7) = 59, A001359(8) = 71 and 29 + 71 = 41 + 59.

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

Cf. A001359, A006512, A014574, A053319.

s = Select[Prime@ Range[10^6], PrimeQ[# + 2] &]; Select[Range@ 1300, s[[#]] + s[[# + 3]] == s[[# + 1]] + s[[# + 2]] &] (* after Robert G. Wilson v at A001359 *)

t(n, p=3) = { while( p+2 < (p=nextprime( p+1 )) || n-->0, ); p-2}
b(n) = t(n) + t(n+3) - t(n+1) - t(n+2);
for(n=1, 2000, if(b(n) == 0, print1(n, ", ")));

list(lim) = {my(k = 0, p1 = 2, t = [0, 0, 0, 0]); forprime(p2 = 3, lim, if(p2 - p1 == 2, k++; t = concat(t[2..4], p1); if(t[1] + t[4] == t[2] + t[3], print1(k-3, ", "))); p1 = p2);} \\ Amiram Eldar, Feb 22 2025

A276831 For a lesser p=A001359(n-1), n>=2, of twin primes, let B_k be the sequence defined as A159559 but with initial term k; a(n) is the smallest m such that B_(p+2)(m)-B_p(m) = max_{t>=2} (B_(p+2)(t)-B_p(t)).

Original entry on oeis.org

5, 17, 11, 5, 3, 17, 3, 11, 11, 5, 31, 107, 13, 333, 17, 5, 3, 3, 281, 5, 997, 3, 487, 659, 5178, 5, 15, 3, 23, 53, 13, 1567, 13, 13, 181, 3, 5, 443, 37, 21, 19, 11, 5, 3, 5, 5, 7, 20786, 13, 7, 5, 21, 3, 5, 17, 61, 31, 23, 7, 3, 11, 5, 11, 5, 3, 3, 157, 37

Offset: 2

Vladimir Shevelev, Sep 20 2016

nonn

			Let n=2, p=A001359(1)=3. Then B_3(2)=3, B_3(3)=5, B_3(4)=6, B_3(5)=7, B_3(6)=8, B_3(7)=11, B_3(8)=12, B_3(9)=14, B_3(10)=15, B_3(11)=17;
Further, B_5(2)=5, B_5(3)=7, B_5(4)=8, B_5(5)=11, B_5(6)=12, B_5(7)=13, B_5(8)=14, B_5(9)=15, B_5(10)=16, B_5(11)=17 and, beginning with t=11,
B_3 merges with B_5. So, max(B_5(t)-B_3(t))=4 reaching at t=5 and t=6.
Thus a(2)=min(5,6)=5.

Vladimir Shevelev, Peter J. C. Moses, Constellations of primes generated by twin primes, arXiv:1610.03385 [math.NT], 2016.

Cf. A001359, A159559, A229019, A276676, A276703, A276767, A276826.

B_(p+2)(a(n)) - B_p(a(n)) = A276826(n).

More terms from Peter J. C. Moses, Sep 20 2016

A299196 Number of partitions of n into distinct parts that are lesser of twin primes (A001359).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 0, 2, 1, 0, 2, 1, 1, 3, 2, 1, 3, 2, 2, 2, 0, 2, 2, 0, 1, 2, 2, 2, 2, 3, 3, 3

Offset: 0

Ilya Gutkovskiy, Feb 04 2018

nonn

For n > 0 let b(n) be the inverse Euler transform of a(n). It appears that, if p is the lesser of twin primes, then b(p) = 1 and b(2*p) = -1; otherwise b(n) = 0. - Georg Fischer, Aug 15 2020

			a(46) = 2 because we have [41, 5] and [29, 17].

Robert Israel, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Scatter plot of a(n) - A299197(n)
Eric Weisstein's World of Mathematics, Twin Primes
Index entries for related partition-counting sequences

Cf. A000586, A000607, A001097, A001359, A077608, A129363, A283875, A283876, A299197.

P:= select(isprime,{seq(i,i=3..201,2)}):
TP:= P intersect map(`-`,P,2):
G:= mul(1+x^p,p=TP):
seq(coeff(G,x,i),i=0..200); # Robert Israel, Dec 15 2024

nmax = 105; CoefficientList[Series[Product[1 + Boole[PrimeQ[k] && PrimeQ[k + 2]] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A001359(k)).

cp's OEIS Frontend

A126193 Lesser of twin primes (A001359) of the form p = k^2+s such that q = k^4+s is also a lesser of twin primes, q > p and s >= 0.

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A126251 a(n) = (p+2)!/p! where p is the n-th lesser twin prime, A001359(n).

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A139187 Smallest twin prime member A001359 of the form k!/n-1.

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A182513 a(n) is the solution of equation A001359(x) = A182482(n).

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A242918 Positions of smaller of twin primes in A001359 with index 5.

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A242919 Positions of smaller of twin primes in A001359 with index 6.

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A243096 Lesser of twin primes (A001359) such that both are full reptend primes (A001913).

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A270535 Integers k such that A001359(k) + A001359(k+3) = A001359(k+1) + A001359(k+2).

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