A108570 -id:A108570 - cp's OEIS frontend

A108604 Squares of the form prime(k)prime(k+1) + 2prime(k+1).

25, 49, 169, 361, 961, 1849, 3721, 5329, 10609, 11881, 19321, 22801, 32761, 37249, 39601, 52441, 58081, 73441, 80089, 97969, 121801, 177241, 187489, 214369, 273529, 326041, 361201, 383161, 413449, 436921, 657721, 677329, 687241, 737881, 779689

Offset: 1

Giovanni Teofilatto, Jul 06 2005

Squares of greater of twin primes.

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A006512, A108570.

ZL:=[]:for p from 1 to 950 do if (isprime(p) and isprime(p+2)) then ZL:=[op(ZL),((p+2)^2)]; fi; od; print(ZL); # Zerinvary Lajos, Mar 08 2007

Select[Sqrt/@Table[Prime[k]*Prime[k+1]+2Prime[k+1],{k,160}],IntegerQ]^2 (* James C. McMahon, Jan 27 2024 *)

a(n) = A006512(n)^2.

Corrected and extended by Ray Chandler, Jul 10 2005

A111046 Difference between squares of twin prime pairs.

Original entry on oeis.org

16, 24, 48, 72, 120, 168, 240, 288, 408, 432, 552, 600, 720, 768, 792, 912, 960, 1080, 1128, 1248, 1392, 1680, 1728, 1848, 2088, 2280, 2400, 2472, 2568, 2640, 3240, 3288, 3312, 3432, 3528, 4080, 4128, 4200, 4248, 4368, 4608, 4920, 5112, 5160, 5208, 5280

Offset: 1

Giovanni Teofilatto, Oct 06 2005

Except for the first term 16 = 4^2, a(n) is never a square.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001359, A006512, A014574, A040040, A054735, A108570, A108604.

a111046 = (* 2) . a054735  -- Reinhard Zumkeller, Feb 10 2015

ZL:=[]:for p from 1 to 1400 do if (isprime(p) and isprime(p+2)) then ZL:=[op(ZL),(((p+2)^2)-p^2)]; fi; od; print(ZL); # Zerinvary Lajos, Mar 08 2007

Select[Table[Prime[n] + 1, {n, 220}], PrimeQ[ # + 1] &] *4 (* Ray Chandler, Oct 12 2005 *)
4+4#&/@Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==2&] [[All,1]] (* Harvey P. Dale, Apr 12 2018 *)

a(n) = A006512(n)^2 - A001359(n)^2 = A108604(n) - A108570(n) = 2*A054735(n) = 4*A014574(n) = 8*A040040(n).

Edited and extended by Ray Chandler, Oct 12 2005

A171727 The number of twin prime pairs in the interval (p^2,p*q), where (p,q) runs over the twin prime pairs (A001359(n),A006512(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 1, 3, 2, 2, 4, 7, 3, 3, 5, 7, 4, 4, 7, 6, 11, 9, 5, 11, 9, 9, 11, 10, 11, 9, 11, 11, 12, 11, 12, 18, 12, 12, 16, 11, 16, 20, 14, 16, 15, 20, 16, 22, 13, 22, 16, 17, 21, 20, 20, 23, 22, 23, 20, 21, 21, 26, 20, 28, 24, 24, 23, 24, 25, 21, 24, 37, 27, 21, 28, 24, 31

Offset: 1

Jaspal Singh Cheema, Dec 16 2009

nonn

If you graph the order of the twin primes along the x-axis (i.e., first twin, second, third, ...) and the number of twins in the sequence given above along the y-axis, a clear pattern emerges. As you go farther along the x-axis, the number of twin primes, on average, within the interval increases. The pattern appears to be nonlinear. If one could prove that there's at least one twin prime within each interval, the twin prime conjecture would be proved since the n-th twin produces larger intervals with more twin primes. The evidence seems overwhelming.

			The first twin prime pair (3,5) corresponds to the interval (9,15), which contains one twin prime pair (11,13), so a(1) = 1.
The fifth twin prime pair (29,31) corresponds to the interval (841,899), which contains the twin prime pairs (857,859) and (881,883), so a(5) = 2.

C. C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Perseus Books, 1999.
J. Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Penguin Books Canada Ltd., 2004.
M. du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins Publishers Inc., 2004.

J. S. Cheema, Table of n, a(n) for n = 1..1044

Cf. A001359, A006512, A108570, A037074. - Charlie Neder, Feb 12 2019

{for(k=1, 300, if(prime(k+1)-prime(k)==2, my(c=0); forprime(m=prime(k)^2, prime(k)*prime(k+1), c+=isprime(m+2)); print1(c, ", ")))} \\ Zhandos Mambetaliyev, Mar 28 2021

Partially edited by Michel Marcus, Mar 19 2013

Edited by Charlie Neder, Feb 12 2019

A330477 Semiprimes (A001358) pq such that pq+p+q is also a semiprime.

Original entry on oeis.org

9, 22, 25, 39, 62, 69, 77, 87, 91, 94, 95, 106, 115, 119, 121, 122, 133, 134, 142, 146, 159, 183, 187, 202, 213, 214, 218, 219, 226, 235, 237, 249, 253, 259, 262, 265, 274, 287, 289, 291, 299, 303, 305, 309, 314, 335, 362, 381, 386, 393, 403, 411, 417, 422, 446, 458, 469, 473, 489, 501, 502, 505

Offset: 1

J. M. Bergot and Robert Israel, Dec 15 2019

nonn

			a(3) = 25 is a member because 25 = 5*5 and 25+5+5 = 5*7 is also a semiprime.

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

Cf. A001358.

Contains A108570.

N:= 1000:
Primes:= select(isprime, [2,seq(i,i=3..N)]):
SP:= sort([seq(seq([p,q],q=select(t -> t >= p and p*t<=N, Primes)),p=Primes)],(a,b) -> a[1]*a[2] t[1]*t[2], select(t -> numtheory:-bigomega(t[1]*t[2]+t[1]+t[2])=2, SP));

Select[Union@ Apply[Join, Table[Flatten@{p #, Sort[{p, #}]} & /@ Prime@ Range@ PrimePi@ Floor[Max[#]/p], {p, #}]] &@ Prime@ Range@ 97, PrimeOmega[Total@ #] == 2 &][[All, 1]] (* Michael De Vlieger, Dec 15 2019 *)

issemi(n)=bigomega(n)==2
list(lim)=my(v=List()); forprime(p=2, sqrtint(lim\=1), forprime(q=p, lim\p, if(issemi(p*q+p+q), listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Dec 16 2019

from sympy import factorint
def is_semiprime(n): return sum(e for e in factorint(n).values()) == 2
def ok(n):
    f = factorint(n, multiple=True)
    if len(f) != 2: return False
    p, q = f
    return len(factorint(p*q + p + q, multiple=True)) == 2
print(list(filter(ok, range(506)))) # Michael S. Branicky, Sep 22 2021

A110487 Squares of the form p*q - p - q + 2, where p and q are primes.

Original entry on oeis.org

9, 25, 49, 81, 121, 169, 225, 289, 361, 529, 625, 841, 961, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3481, 3721, 4225, 4489, 4761, 5041, 5329, 5929, 6241, 6889, 7225, 7921, 8281, 9025, 9409, 10201, 10609, 11449, 11881, 12769

Offset: 1

Giovanni Teofilatto, Sep 11 2005

nonn

Includes A108570 squares of lesser twin primes and A110284 squares of form 4p-3 where p is prime (q=5).

Cf. A108570, A110284, A110484.

Lim=13000;f[{p_,q_}]:=p*q-p-q+2;Union[Select[f/@Subsets[Prime[Range[PrimePi[Lim]]], {2}],#James C. McMahon, Apr 20 2024 *)

Corrected and extended by Ray Chandler, Sep 13 2005

cp's OEIS Frontend

A108604 Squares of the form prime(k)*prime(k+1) + 2*prime(k+1).

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A111046 Difference between squares of twin prime pairs.

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A171727 The number of twin prime pairs in the interval (p^2,p*q), where (p,q) runs over the twin prime pairs (A001359(n),A006512(n)).

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A330477 Semiprimes (A001358) p*q such that p*q+p+q is also a semiprime.

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A110487 Squares of the form p*q - p - q + 2, where p and q are primes.

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Mathematica

Extensions

A108604 Squares of the form prime(k)prime(k+1) + 2prime(k+1).

A330477 Semiprimes (A001358) pq such that pq+p+q is also a semiprime.