A005597 -id:A005597 - cp's OEIS frontend

A092816 Number of Sophie Germain primes less than 10^n.

3, 10, 37, 190, 1171, 7746, 56032, 423140, 3308859, 26569515, 218116524, 1822848478, 15462601989, 132822315652

Offset: 1

Eric W. Weisstein, Mar 06 2004

Hardy-Littlewood conjecture: Number of Sophie Germain primes less than n ~ 2*C2*n/(log(n))^2, where C2 = 0.6601618158... is the twin prime constant (see A005597). The truth of the above conjecture would imply that there are an infinite number of Sophie Germain primes (which is also conjectured). - Robert G. Wilson v, Jan 31 2013

			The Sophie Germain primes up to 10 are 2 (since 5 is prime), 3 (since 7 is prime), and 5 (since 11 is prime), so a(1) = 3.

P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, New York, 1991, p. 228.

C. K. Caldwell, An amazing prime heuristic, Table 6.
Eric Weisstein's World of Mathematics, Sophie Germain Prime

Cf. A005384, A156874, A182265.

For 1 < n < 15, a(n) ~ e * (pi(2*10^n) - pi(10^n)) / (5*n - 5) where e is Napier's constant, see A001113 (we use n > 1 to avoid division by zero; whether the formula holds for any n > 14 is unknown). - Sergey Pavlov, Apr 07 2021 [This formula fails under the Hardy-Littlewood conjecture; the leading constant is wrong. - Charles R Greathouse IV, Aug 03 2023]

For any n, a(n) = qcc(x) - (10^n - pi(10^n) - pi(2 * 10^n + 1) + 1) where qcc(x) is the number of "common composite numbers" c <= 10^n such that both c and c' = 2*c + 1 are composite (trivial). - Sergey Pavlov, Apr 08 2021

a(10) computed by Eric W. Weisstein, Nov 02 2005
a(11)-a(12) from Donovan Johnson, Jun 19 2010
a(13)-a(14) from Giovanni Resta, Sep 04 2017

A167864 Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2))).

Original entry on oeis.org

1, 5, 1, 4, 7, 8, 0, 1, 2, 8, 1, 3, 7, 4, 9, 1, 2, 5, 7, 7, 9, 0, 9, 1, 9, 2, 5, 5, 6, 4, 9, 4, 7, 4, 8, 9, 2, 4, 1, 5, 2, 7, 0, 1, 5, 8, 2, 8, 6, 2, 1, 4, 3, 9, 5, 3, 5, 7, 4, 8, 4, 2, 7, 1, 4, 8, 4, 9, 3, 2, 2, 0, 9, 8, 1, 5, 6, 1, 1, 5, 8, 1, 0, 8, 7, 7, 5, 8, 5, 3, 8, 2, 7, 6, 9, 8, 0, 7, 6, 7, 7, 6, 5, 6, 2

Offset: 1

Jonathan Sondow, Nov 13 2009, Nov 17 2009

Coefficient in formulas for the distribution of integers with a fixed number of prime factors.
Reciprocal of the twin prime constant A005597. See A005597 for links and additional references and comments.
Numerators of partial products are A062271. Denominators are A062270.
An analog for primes of Wallis' product pi/2 = Product_{n >=1} (2n)^2/(2n-1)(2n+1), because A167864 = Product_{prime p>2} (p-1)^2/(p-2)p.
Grosswald (see links) proves that Sum_{k<=x} 2^Omega(k) ~ (1/(8*log(2))) * c * x * (log(x))^2 + O(x * log(x)) where c is this constant. - Amiram Eldar, Jun 06 2020
The asymptotic density of numbers m with A046660(m) = Omega(m) - omega(m) = k is asymptotically ~ c/2^(k+2) as k -> oo, where c is this constant (Rényi, 1955). - Amiram Eldar, Aug 08 2020
Named after the Norwegian mathematician Atle Selberg (1917-2007) and the French mathematician Hubert Delange (1914-2003). - Amiram Eldar, Jun 20 2021

			Product_{prime p > 2} (1 + 1/(p(p-2))) = 1.5147801281374912577909192556...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 84-93.
Atle Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc., Vol. 18, No. 1 (1954), pp. 83-87.
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 206.

Michel Balazard, Hubert Delange and Jean-Louis Nicolas, Sur le nombre de facteurs premiers des entiers, C. R. Acad. Sci., Paris, Ser. I, Vol. 306 (1988), pp. 511-514. [From Jonathan Sondow, Nov 17 2009]
Hubert Delange, Sur des formules de Atle Selberg, Acta Arith., Vol. 19 (1971), pp. 105-146.
Steven R. Finch, Mathematical Constants, Errata and Addenda, Sec. 2.1.
Emil Grosswald, The average order of an arithmetic function, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44.
Jean-Louis Nicolas, Sur la distribution des nombres entiers ayant une quantite fixée de facteurs premiers, Acta Arith., Vol. 44 (1984), pp. 191-200.
Alfred Rényi, On the density of certain sequences of integers, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.

Cf. A005597.

Cf. A001222 (Omega), A046660, A061142 (2^Omega), A069205 (partial sums of 2^Omega).

s[n_] := (1/n)* N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], 160]; C2 = (175/256)*Product[(Zeta[ n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[ n]), {n, 2, 160}]; RealDigits[1/C2][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 30 2012, after Pari program in A005597 *)
$MaxExtraPrecision = 300; digits = 105; terms = 600; P[n_] := PrimeZetaP[n] - 1/2^n; LR = Join[{0, 0}, LinearRecurrence[{3, -2}, {2, 6}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n-1]/(n-1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 19 2016 *)

prodeulerrat((1 + 1/(p*(p-2))),,3) \\ Hugo Pfoertner, Aug 08 2020

Equals 1/A005597.

Equals Product_{prime p>2} (p-1)^2/(p-2)p = (2^2/1*3)(4^2/3*5)(6^2/5*7)(10^2/9*11) ....

A274121 The gap prime(n+1) - prime(n) occurs for the a(n)-th time.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 3, 1, 5, 2, 4, 6, 5, 3, 4, 7, 5, 6, 8, 6, 7, 7, 1, 8, 9, 9, 10, 10, 1, 11, 8, 11, 1, 12, 9, 10, 12, 11, 12, 13, 2, 14, 13, 15, 1, 2, 14, 16, 15, 13, 17, 3, 14, 15, 16, 18, 17, 16, 19, 4, 2, 17, 20, 18, 3, 18, 5, 21, 19, 19, 2, 20, 21, 20, 22, 3, 21, 4, 6, 22, 7, 23, 23, 22, 24, 5, 23, 24, 24, 3, 6

Offset: 1

David A. Corneth, Jun 10 2016

nonn

Terms of this sequence grow without bound; any even number occurs in this sequence. Zhang proved that there are infinitely many primes 4680 apart from each other (see link "Bounded gaps between primes").
For a conjectured count of gap n below x, see link Polignac's conjecture.
Polignac's conjecture states that "For any positive even number n, there are infinitely many prime gaps of size n.". By this conjecture, every positive apppears infinitely many times in this sequence (see link "Polignac's conjecture").

			(p, g) denotes a prime p and the gap up to the next prime. So p + g is the next prime after p. These pairs start (2, 1), (3, 2), (5, 2), (7, 4), (11, 2). From here we see that:
- the gap after the first prime, 1 occurs for the first time, so a(1) = 1.
- the gap after the second prime, 2, occurs for the first time, so a(2) = 1.
- the gap after the third prime, 2, occurs for the second time, so a(3) = 2.
- the gap after the fourth prime, 4, occurs for the first time, so a(4) = 1.
- the gap after the fifth prime, 2, occurs for the third time, so a(5) = 3.

David A. Corneth, Table of n, a(n) for n = 1..100021 (all terms up to 1300000)
David A. Corneth, PARI program
PolyMath, Bounded gaps between primes.
Wikipedia, Polignac's conjecture.

Cf. A000230, A005597, A001359, A029710, A274122, A274123.

\\ See link by name "PARI program" for an extended version with comments.
upto(n) = {my(gapcount=List(), freqgap = List([1])); n = max(n, 3); forprime(i=3,n,
g = nextprime(i+1) - i; for(i=#gapcount+1, g\2, listput(gapcount,0));  gapcount[g\2]++; listput(freqgap,gapcount[g\2]));freqgap} \\ David A. Corneth, Jun 28 2016

a(primepi(A000230(n))) = 1.
a(primepi(A001359(n))) = n.
a(primepi(A029710(n))) = n.

A062270 Numerators in partial products of the twin prime constant.

Original entry on oeis.org

3, 45, 175, 693, 11011, 2807805, 302307005, 402243205, 714186915, 42803602439, 11086133031701, 5908908905896633, 1488200914442251997, 3041106216468949733, 16213234917387714257, 21611220383343195817

Offset: 2

Frank Ellermann, Jun 16 2001

For n>1, a(n) is the absolute value of the numerator of the determinant of the n X n matrix with elements M[i,j] = 1/(prime(i)-1)^2 for i=j and 1 otherwise. - Alexander Adamchuk, Jun 02 2006

			a(4) = 175 = 3*1*5*3*7*5 / gcd(3*1*5*3*7*5, 2*2*4*4*6*6).

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, ch. 22.20

Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]

Cf. A062271 (denominators), A005597 (decimal expansion).

Numerator[Abs[Table[ Det[ DiagonalMatrix[ Table[ 1/(Prime[i]-1)^2 - 1, {i, 1, n} ] ] + 1 ], {n, 2, 20} ]]] (* Alexander Adamchuk, Jun 02 2006 *)

a(n) = numerator(prod(k=2, n, 1-1/(prime(k)-1)^2)); \\ Michel Marcus, May 31 2022

a(n) = a(n-1)*(prime(n)*(prime(n)-2)) / gcd(a(n-1)*prime(n)*(prime(n)-2), A062271(n)) for n > 2.

Typo in link corrected by Martin Griffiths, Apr 03 2009

A160910 Decimal expansion of c = sum over twin primes (p, p+2) of (1/p^2 + 1/(p+2)^2).

Original entry on oeis.org

2, 3, 7, 2, 5, 1, 7, 7, 6, 5, 7

Offset: 0

William Royle (seriesandsequences(AT)yahoo.com), May 29 2009

Compare Viggo Brun's constant (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) + (1/29 + 1/31) + ... (see A065421, A005597).
It appears that c = Sum 1/A001359(n)^2 + 1/A006512(n)^2. - R. J. Mathar, May 30 2009
0.237251776574746 < c < 0.237251776947124. - Farideh Firoozbakht, May 31 2009
c < 0.2725177657771. - Hagen von Eitzen, Jun 03 2009
From Farideh Firoozbakht, Jun 01 2009: (Start)
We can show that a(9)=6, a(10)=5, and a(11) is in the set {7, 8, 9}.
Proof: s1 = 0.237251776576249072... is the sum up to prime(499,000,000) and s2 = 0.237251776576250009... is the sum up to prime(500,000,000).
By using the fact that number of twin primes between the first 10^6*n primes and the first 10^6*(n+1) primes is decreasing (up to the first 2*10^9 primes), we conclude that the sum up to prime(2,000,000,000) is less than s2 + 1500*(s2-s1).
But since s2-s1 < 10^(-15), the sum up to prime(2*10^9) is less than s2 + 1.5*10^(-12) = 0.237251776576250009... + 1.5*10^(-12) = 0.237251776577550009... .
Hence the constant c is less than
0.237251776577550009... + lim(sum(1/k^2,{k, prime(2,000,000,001), n}, n -> infinity)
< 0.237251776577550009... + 2.12514*10^(-11)
< 0.237251776598801409.
So we have 0.237251776576250009 < c < 0.237251776598801409, hence a(9)=6, a(10)=5, and a(11) is in the set {7, 8, 9}.
I guess that a(11)=7. (End)
From Jon E. Schoenfield, Jan 02 2019: (Start)
Given that the Hardy-Littlewood approximation to the number of twin prime pairs < y is
     2 * C_2 * Integral_{x=2..y} dx/log(x)^2
  where C_2 = 0.660161815846869573927812110014555778432623 (see A152051), we can estimate the size of the tail of the summation Sum(1/A001359(j)^2) + 1/A006512(j)^2) for twin primes > y as
     t(y) = 2 * C_2 * Integral_{x>y} 2*dx/(x*log(x))^2.
Let s(y) be the sum of the squares of the reciprocals of all the twin primes <= y, and let s'(y) = s(y) + t(y) be the result of adding to the actual value s(y) the estimated tail size t(y). Evaluating s(y), t(y), and s'(y) at y = 2^d for d = 20..33 gives
.
   d        s(2^d)        t(2^d)*10^10    s(2^d) + t(2^d)
  == ==================== ============ ====================
  20 0.237251764919808326 115.34589710 0.237251776454398036
  21 0.237251771317612979  52.59702970 0.237251776577315949
  22 0.237251774173347724  24.08221952 0.237251776581569676
  23 0.237251775469086555  11.06766714 0.237251776575853269
  24 0.237251776066813995   5.10395459 0.237251776577209454
  25 0.237251776340760021   2.36119196 0.237251776576879217
  26 0.237251776467109357   1.09553336 0.237251776576662693
  27 0.237251776525743797   0.50967952 0.237251776576711749
  28 0.237251776552887645   0.23771866 0.237251776576659511
  29 0.237251776565549906   0.11113468 0.237251776576663374
  30 0.237251776571456873   0.05207020 0.237251776576663892
  31 0.237251776574218065   0.02444677 0.237251776576662742
  32 0.237251776575513036   0.01149984 0.237251776576663020
  33 0.237251776576121140   0.00541938 0.237251776576663078
.
which agrees with all the terms in the Data section and suggests likely values for additional terms.
(End)

			(1/9 + 1/25) + (1/25 + 1/49) + (1/121 + 1/169) + (1/289 + 1/361) + (1/841 + 1/961) + ... = 0.237251...

Various authors, On the computation of A160910

Cf. A001359, A005597, A006512, A065421, A152051.

R. J. Mathar pointed out that the value of c as originally submitted was incorrect (see link). - N. J. A. Sloane, May 31 2009
More terms from Farideh Firoozbakht and Hagen von Eitzen, Jun 01 2009
Name changed by Michael B. Porter, Jan 04 2019

A005722 a(n) = (prime(n) - 1)^2.

Original entry on oeis.org

1, 4, 16, 36, 100, 144, 256, 324, 484, 784, 900, 1296, 1600, 1764, 2116, 2704, 3364, 3600, 4356, 4900, 5184, 6084, 6724, 7744, 9216, 10000, 10404, 11236, 11664, 12544, 15876, 16900, 18496, 19044, 21904, 22500, 24336, 26244, 27556, 29584, 31684, 32400, 36100

Offset: 1

Scorpion(AT)aol.com

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Richard K. Guy, Letter to N. J. A. Sloane, 1987.

Cf. A001248, A005597, A006093, A065485, A086242, A095874, A192134.

[(p-1)^2: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 27 2014

A005722:=n->(ithprime(n)-1)^2; seq(A005722(n), n=1..40); # Wesley Ivan Hurt, Mar 27 2014

Table[(Prime[n] - 1)^2, {n, 40}] (* Vincenzo Librandi, Mar 27 2014 *)

a(n) = (prime(n) - 1)^2; \\ Michel Marcus, Nov 09 2020

a(n) = A192134(A095874(A001248(n))) - 1. - Reinhard Zumkeller, Jun 26 2011
a(n) = A006093(n)^2. - Wesley Ivan Hurt, Mar 27 2014
Sum_{n>=1} 1/a(n) = A086242. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065485.
Product_{n>=2} (1 - 1/a(n)) = A005597. (End)

A062271 Denominators in partial products of the twin prime constant.

Original entry on oeis.org

4, 64, 256, 1024, 16384, 4194304, 452984832, 603979776, 1073741824, 64424509440, 16698832846848, 8906044184985600, 2244323134616371200, 4588393964104581120, 24471434475224432640, 32628579300299243520

Offset: 2

Frank Ellermann, Jun 16 2001

			a(4)= 256= 2*2*4*4*6*6 / gcd( 3*1*5*3*7*5, 2*2*4*4*6*6 ).

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, ch. 22.20

Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]

A062270 (numerators), A005597 (decimal expansion).

a(n)= a(n-1)*(p(n)-1)^2 / gcd( A062270(n), a(n-1)*(p(n)-1)^2 ) for n > 2.

A065645 Continued fraction for twin prime constant.

Original entry on oeis.org

0, 1, 1, 1, 16, 2, 2, 2, 2, 1, 18, 2, 2, 11, 1, 1, 2, 4, 1, 16, 3, 2, 4, 21, 2, 405, 2, 1, 33, 1, 2, 8, 2, 29, 1, 4, 4, 4, 4, 1, 9, 3, 1, 4, 1, 1, 2, 26, 1, 8, 2, 6, 1, 4, 1, 3, 9, 46, 1, 6, 1, 1, 4, 2, 1, 12, 1, 1, 7, 35, 1, 1, 2, 1, 4, 1199, 2, 3, 1, 2, 3, 3, 13, 15, 4, 1, 1, 1, 10, 9, 6, 3, 1, 3, 1

Offset: 0

Vladeta Jovovic, Nov 08 2001

			1/(1+1/(1+1/(1+1/(16+1/(2+1/(2+1/(2+1/(2+1/(1+1/(18+...)))))))))).

Harry J. Smith, Table of n, a(n) for n = 0..981

Cf. A005597 (decimal expansion), A065646 (denominators of convergents to twin prime constant), A065647 (numerators of convergents to twin prime constant), A062270, A062271.

{ default(realprecision,1002); c2=\
0.66016181584686957392781211001455577843262336028473341331944842333\
5405642304495277143760031413839867911779005226693304002965847755123\
3662277471657132139869687410976206302141537354348531315960978036699\
3213525529976719930247459059310108297829155383446929750520591665713\
3653611991532464281301172462306379341060056466676584434063501649322\
7235289680109349664756004788123579627894598424336557493755818548141\
7362867809870596949870384124336338658931196907915004057371781437108\
1810615401233104810577794415613125444598860988997585328984038108718\
0355252617198871121363828087823497223742240971426974417644552252655\
4899482977179097778404375789195659064999456706290782860882839599039\
4287082529070521554595671723599449769037800675978761690802426600295\
7110920996337082725592846721298580011486979418554018246398874939417\
1182852838236599705032872570808798066220106863047430520199239428201\
4311102297265141514194258422242375342296879836738796224286600285358\
098482833679152235700192585875285961205994728621007171131607980572; x=contfrac(c2); for (n=1, 982, write("b065645.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 15 2009

A269843 Decimal expansion of Hardy-Littlewood constant C_5 = Product_{p prime > 5} 1/(1-1/p)^5 (1-5/p).

Original entry on oeis.org

4, 0, 9, 8, 7, 4, 8, 8, 5, 0, 8, 8, 2, 3, 6, 4, 7, 4, 4, 7, 8, 7, 8, 1, 2, 1, 2, 3, 3, 7, 9, 5, 5, 2, 7, 7, 8, 9, 6, 3, 5, 8, 0, 1, 3, 2, 5, 4, 9, 4, 5, 4, 6, 9, 8, 2, 6, 3, 3, 6, 3, 9, 8, 8, 2, 2, 6, 4, 8, 2, 3, 6, 1, 7, 3, 9, 6, 5, 9, 6, 5, 1, 5, 4, 6, 0, 8, 4, 5, 4, 4, 9, 9, 6, 2, 0, 2, 8, 1

Offset: 0

Jean-François Alcover, Apr 17 2016

			0.4098748850882364744787812123379552778963580132549454698263363988...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood Constants, p. 86.

B. H. Mayoh, The 2nd Goldbach conjecture revisited, BIT 8 (1968) 128-133 Table 5.

Cf. A005597, A065418, A065419, A001692.

$MaxExtraPrecision = 800; digits = 99; terms = 800; P[n_] := PrimeZetaP[n] - 1/2^n - 1/3^n - 1/5^n; LR = Join[{0, 0}, LinearRecurrence[{6, -5}, {-20, -120}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First

prodeulerrat(1/(1-1/p)^5*(1-5/p), 1, 7) \\ Amiram Eldar, Mar 11 2021

A305444 a(n) = Product_{p is odd and prime and divisor of n} (p - 2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 1, 1, 3, 9, 1, 11, 5, 3, 1, 15, 1, 17, 3, 5, 9, 21, 1, 3, 11, 1, 5, 27, 3, 29, 1, 9, 15, 15, 1, 35, 17, 11, 3, 39, 5, 41, 9, 3, 21, 45, 1, 5, 3, 15, 11, 51, 1, 27, 5, 17, 27, 57, 3, 59, 29, 5, 1, 33, 9, 65, 15, 21, 15, 69, 1, 71, 35, 3, 17

Offset: 1

Markus Sigg, Aug 12 2018

Denominator of c_n = Product_{odd p| n} (p-1)/(p-2). Numerator is A173557. [Yamasaki and Yamasaki]. - N. J. A. Sloane, Jan 19 2020

This ratio, multiplied by the twin prime constant, occurs in the asymptotic behavior of prime gaps of size 2*n as decribed by the Hardy-Littlewood asymptotic conjecture for the number of prime pairs. See A005597 for more information. - Hugo Pfoertner, Dec 25 2024

Markus Sigg, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, Plot of A173557(n)/a(n) vs n, using Plot 2.
Yasuo Yamasaki and Aiichi Yamasaki, On the Gap Distribution of Prime Numbers, Kyoto University Research Information Repository, October 1994. MR1370273 (97a:11141).

Cf. A173557, A000010, A008683, A001221.

A305444 := proc(n) mul(d - 2, d = numtheory[factorset](n) minus {2}) end proc:

a[n_] := If[n == 1, 1, Times @@ (DeleteCases[FactorInteger[n][[All, 1]], 2] - 2)];
Array[a, 100] (* Jean-François Alcover, Apr 08 2020*)

a(n)={my(f=factor(n>>valuation(n,2))[,1]); prod(i=1, #f, f[i]-2)} \\ Andrew Howroyd, Aug 12 2018

from math import prod
from sympy import primefactors
def A305444(n): return prod(p-2 for p in primefactors(n>>(~n&n-1).bit_length())) # Chai Wah Wu, Sep 08 2023

Sum_{k=1..n} a(k) ~ c * n^2, where c = (2/3) * Product_{p prime} (1 - 3/(p*(p+1))) = 0.1950799046... . - Amiram Eldar, Nov 12 2022
a(n) = abs( Sum_{d divides n, d odd} mobius(d) * phi(d) ). - Peter Bala, Feb 01 2024
a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*phi(2*d), where omega = A001221. - Ridouane Oudra, Jul 30 2025

cp's OEIS Frontend

A092816 Number of Sophie Germain primes less than 10^n.

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A167864 Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2))).

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A274121 The gap prime(n+1) - prime(n) occurs for the a(n)-th time.

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PARI

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A062270 Numerators in partial products of the twin prime constant.

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Mathematica

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A160910 Decimal expansion of c = sum over twin primes (p, p+2) of (1/p^2 + 1/(p+2)^2).

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A005722 a(n) = (prime(n) - 1)^2.

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Magma

Maple

Mathematica

PARI

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A062271 Denominators in partial products of the twin prime constant.

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