A005597 -id:A005597 - cp's OEIS frontend

A271742 Decimal expansion of Hardy-Littlewood constant C_7 = Product_{p prime > 7} 1/(1-1/p)^7 (1-7/p).

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

Offset: 0

Jean-François Alcover, Apr 17 2016

			0.3694375103864986893231907498767507770553729138930318252910123...

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, A269843, A269846.

$MaxExtraPrecision = 1100; digits = 99; terms = 1000; P[n_] := PrimeZetaP[ n] - 1/2^n - 1/3^n - 1/5^n - 1/7^n; LR = Join[{0, 0}, LinearRecurrence[ {8, -7}, {-42, -336}, 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)^7*(1-7/p), 1, 11) \\ Amiram Eldar, Mar 11 2021

A337606 Decimal expansion of the Gaussian twin prime constant: the Hardy-Littlewood constant for A096012.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 04 2020

The name of this constant was suggested by Finch (2003).
Gaussian twin primes on the line x + i in the complex plane are Gaussian primes pair of the form (m - 1 + i, m + 1 + i). The numbers m are numbers such that (m-1)^2 + 1 and (m+1)^2 + 1 are both primes (A096012 plus 1).
Shanks (1960) conjectured that the number of these pairs with m <= x is asymptotic to c * li_2(x), where li_2(x) = Integral_{t=2..n} (1/log(t)^2) dt, and c is this constant. He defined c as in the formula section and evaluated it by 0.4876.
The first 100 digits of 4*c were calculated by Ettahri et al. (2019).

			0.487622778111571768611646391452388423131677124429735...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 90.

Keith Conrad, Hardy-Littlewood constants in: Mathematical properties of sequences and other combinatorial structures, Jong-Seon No et al. (eds.), Kluwer, Boston/Dordrecht/London, 2003, pp. 133-154, alternative link.
Salma Ettahri, Olivier Ramaré, Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 (Section 8).
Daniel Shanks, , A note on Gaussian twin primes, Mathematics of Computation, Vol. 14, No. 70 (1960), pp. 201-203.

Cf. A002496, A005574, A096012, A206328.

Similar constants: A005597, A331941, A337607, A337608.

S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
Zs[m_, n_, s_] := (w = 2; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = (s^w - s) * P[m, n, w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Pi^2/8 * Zs[4, 1, 4]/Z[4, 1, 2]^2, digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

Equals (Pi^2/8) * Product_{primes p == 1 (mod 4)} (1 - 4/p)*((p + 1)/(p - 1))^2.

More digits from Vaclav Kotesovec, Jan 15 2021

A337607 Decimal expansion of Shanks's constant: the Hardy-Littlewood constant for A000068.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 04 2020

Named by Finch (2003) after the American mathematician Daniel Shanks (1917 - 1996).
Shanks (1961) conjectured that the number of primes of the form m^4 + 1 (A037896) with m <= x is asymptotic to c * li(x), where li(x) is the logarithmic integral function and c is this constant. He defined c as in the formula section and evaluated it by 0.66974.
The first 100 digits of this constant were calculated by Ettahri et al. (2019).

			0.669740969937071220538922431571764406688370157436482...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 90.

Keith Conrad, Hardy-Littlewood constants in: Mathematical properties of sequences and other combinatorial structures, Jong-Seon No et al. (eds.), Kluwer, Boston/Dordrecht/London, 2003, pp. 133-154, alternative link.
Salma Ettahri, Olivier Ramaré, Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 (Corollary 1.8).
Mohan Lal, Primes of the form n^4 + 1, Mathematics of Computation, Vol. 21, No. 98 (1967), pp. 245-247.
Daniel Shanks, On Numbers of the form n^4 + 1, Mathematics of Computation, Vol. 15, No. 74 (1961), pp. 186-189.
Daniel Shanks, Lal's constant and generalizations, Mathematics of Computation, Vol. 21, No. 100 (1967), pp. 705-707.
Eric Weisstein's World of Mathematics, Lal's Constant.

Cf. A000068, A037896, A088367, A334826.

Similar constants: A005597, A331941, A337606, A337608.

S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
Zs[m_, n_, s_] := (w = 2; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = (s^w - s) * P[m, n, w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Pi^2/(16*Log[1+Sqrt[2]]) * Zs[8, 1, 4]/Z[8, 1, 2]^2, digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

Equals (Pi^2/(16*log(1+sqrt(2)))) * Product_{primes p == 1 (mod 8)} (1 - 4/p)*((p + 1)/(p - 1))^2 = (Pi/8) * A088367 * A334826.

More digits from Vaclav Kotesovec, Jan 15 2021

A337608 Decimal expansion of Lal's constant: the Hardy-Littlewood constant for A217795.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 04 2020

Shanks (1967) conjectured that the number of primes of the form (m + 1)^4 + 1 such that (m - 1)^4 + 1 is also a prime (A217795 plus 1), with m <= x, is asymptotic to c * li_2(x), where li_2(x) = Integral_{t=2..n} (1/log(t)^2) dt, and c is this constant. He defined c as in the formula section, evaluated it by 0.79220 and named it after the mathematician Mohan Lal, who conjectured the asymptotic formula without evaluating this constant.

The first 100 digits of this constant were calculated by Ettahri et al. (2019).

			0.792208238167541668775455566579024101128932250986221...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 90-91.

Keith Conrad, Hardy-Littlewood constants in: Mathematical properties of sequences and other combinatorial structures, Jong-Seon No et al. (eds.), Kluwer, Boston/Dordrecht/London, 2003, pp. 133-154, alternative link.
Salma Ettahri, Olivier Ramaré, Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 (Corollary 1.9).
Mohan Lal, Primes of the form n^4 + 1, Mathematics of Computation, Vol. 21, No. 98 (1967), pp. 245-247.
Daniel Shanks, Lal's constant and generalizations, Mathematics of Computation, Vol. 21, No. 100 (1967), pp. 705-707.
Eric Weisstein's World of Mathematics, Lal's Constant.

Cf. A000068, A037896, A088367, A210630, A217795, A217796.

Similar constants: A005597, A331941, A337606, A337607.

$MaxExtraPrecision = 1000; digits = 121;
f[p_] := (p-8)*(p+1)^4/((p-1)^4*p);
coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
m = 2; sump = 0; difp = 1; While[Abs[difp] > 10^(-digits - 5) || difp == 0, difp = coefs[[m]]*(P[8, 1, m] - 1/17^m); sump = sump + difp; m++];
RealDigits[Chop[N[f[17] * Pi^4/(2^7 * Log[1+Sqrt[2]]^2) * Exp[sump], digits]], 10, digits - 1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)

Equals (Pi^4/(2^7 * log(1+sqrt(2))^2)) * Product_{primes p == 1 (mod 8)} (1 - 4/p)^2 * ((p + 1)/(p - 1))^4 * p*(p-8)/(p-4)^2 = (Pi^2/32) * A088367^2 * A334826^2 * A210630 = 2 * A337607^2 * A210630.

More terms from Vaclav Kotesovec, Jan 16 2021

A387362 Cyclic numbers k such that k+2 is also a cyclic number.

Original entry on oeis.org

1, 3, 5, 11, 13, 15, 17, 29, 31, 33, 35, 41, 51, 59, 65, 67, 69, 71, 77, 83, 85, 87, 89, 95, 101, 107, 113, 131, 137, 139, 141, 143, 149, 157, 159, 161, 177, 179, 185, 191, 197, 209, 211, 213, 215, 221, 227, 233, 239, 247, 249, 255, 257, 263, 265, 267, 269, 281, 293

Offset: 1

Amiram Eldar, Aug 27 2025

All the lesser members of twin primes (A001359) are terms since every prime is a cyclic number (A003277).

Cohen (2025) conjectured and Pomerance (2025) proved that this sequence is infinite.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025.
Carl Pomerance, Patterns for cyclic numbers, 2025.

Cf. A001620, A005597, A073004, A387363.
Subsequence of A003277.
A001359 is a subsequence.

cyclicQ[n_] := cyclicQ[n] = CoprimeQ[n, EulerPhi[n]]; Select[Range[1, 300, 2], And @@ cyclicQ[{#, # + 2}] &]

iscyclic(k) = gcd(k, eulerphi(k)) == 1;
isok(k) = k % 2 && iscyclic(k) && iscyclic(k+2);

The number of terms <= x is ~ 2 * C_2 * x / (exp(gamma) * log(log(log(x))))^2, where C_2 = A005597, and gamma = A001620 (Pomerance, 2025).

A387363 The number of decompositions of 2*n into ordered sums of two cyclic numbers.

Original entry on oeis.org

1, 3, 3, 4, 3, 4, 5, 6, 8, 8, 7, 8, 7, 6, 9, 8, 11, 12, 11, 10, 12, 12, 13, 16, 12, 14, 16, 12, 13, 14, 13, 16, 19, 14, 19, 20, 19, 20, 20, 20, 21, 26, 19, 24, 26, 22, 25, 26, 24, 26, 33, 26, 27, 30, 26, 28, 32, 26, 29, 38, 25, 30, 34, 26, 33, 34, 29, 30, 41, 28

Offset: 1

Amiram Eldar, Aug 27 2025

Analogous to A002372 with cyclic numbers (A003277) instead of odd primes.

Pomerance (2025) proved that a(n) > 0 for every sufficiently large n.

			a(1) = 1 since 2*1 = 1 + 1.
a(2) = 3 since 2*2 = 1 + 3 = 2 + 2 = 3 + 1.
a(3) = 3 since 2*3 = 1 + 5 = 3 + 3 = 5 + 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025.
Carl Pomerance, Patterns for cyclic numbers, 2025.

Cf. A002372, A003277, A387362.

Cf. A001620, A005597, A073004.

cyclicQ[n_] := cyclicQ[n] = CoprimeQ[n, EulerPhi[n]]; a[n_] := Count[Range[2*n], _?(And @@ cyclicQ[{#, 2*n-#}] &)]; Array[a, 100]

iscyclic(k) = gcd(k, eulerphi(k)) == 1;
a(n) = sum(k = 1, 2*n, iscyclic(k) * iscyclic(2*n-k));

a(n) ~ C_2 * n / (exp(gamma) * log(log(log(n))))^2 * Product_{p | n, p odd prime < log(log(n/2))} (p-1)/(p-2), where C_2 = A005597, and gamma = A001620 (Pomerance, 2025).

A246061 Decimal expansion of lim_{n->infinity} ((1/log(n)^2)*Product_{2 < p < n, p prime} p/(p-2)).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 11 2014

			1.201303559967362241247555959207383482453838449427113...

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

Eric Weisstein's MathWorld, Twin Primes Constant.

Cf. A005597, A091724.

digits = 101; s[n_] := (1/n)* N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], digits + 60]; C2 = (175/256)*Product[(Zeta[ n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[ n]), {n, 2, digits + 60}]; RealDigits[Exp[2*EulerGamma]/(4*C2), 10, digits] // First

exp(2*Euler)/(4*prodeulerrat(1-1/(p-1)^2, 1, 3)) \\ Amiram Eldar, Apr 27 2025

Equals exp(2*EulerGamma)/(4*C_2), where C_2 is the twin primes constant A005597.

A271886 Decimal expansion of the constant D related to the conjectured asymptotic expression of the counting function of prime triples as D*n/log(n)^3.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 16 2016

			2.8582485957192204324301346607263508780392955929956760290488...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood Constants, p. 85.

Cf. A005597.

$MaxExtraPrecision = 800; digits = 96; terms = 1000; P[n_] := PrimeZetaP[n] - 1/2^n - 1/3^n; LR = Join[{0, 0}, LinearRecurrence[{4, -3}, {-6, -24}, terms + 10]]; r[n_Integer] := LR[[n]]; (9/2)*Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First

(9/2) * prodeulerrat(p^2*(p-3)/(p-1)^3, 1, 5) \\ Amiram Eldar, Mar 11 2021

D = (9/2) Product_{p prime > 3} p^2(p-3)/(p-1)^3.

A271951 Decimal expansion of (1/2) Product_{p prime} 1+1/(p-1)^3, a constant related to I. M. Vinogradov's proof of the "ternary" Goldbach conjecture.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 17 2016

			1.150480772356618527278488074374698090630393298511083680688193059...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood Constants, p. 88.

Eric Weisstein's MathWorld, Goldbach Conjecture.
Eric Weisstein's MathWorld, Vinogradov's theorem.
Wikipedia, Goldbach's conjecture.
Wikipedia, Vinogradov's theorem.

Cf. A005597.

$MaxExtraPrecision = 1600; digits = 99; terms = 1600; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0, 0}, LinearRecurrence[{4, -6, 3}, {3, 12, 30}, terms + 10]]; r[n_Integer] := LR[[n]]; (1/2) Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First

(1/2) * prodeulerrat(1+1/(p-1)^3) \\ Amiram Eldar, Mar 14 2021

A374831 Decimal expansion of Product_{p prime} (1 - (1/(p(p - 1)))p^2/(p^2 + 1)).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jul 21 2024

			0.4589374985054359613063426181...

Rafael Jakimczuk, Numbers of the form p-1 where p is prime, July 2024. See p. 23.

Cf. A005596, A005597, A065414, A065418, A065419, A374830 (lower bound).

PARI
```
prodeulerrat(1-p^2/(p*(p-1)*(p^2+1)))
```

cp's OEIS Frontend

A271742 Decimal expansion of Hardy-Littlewood constant C_7 = Product_{p prime > 7} 1/(1-1/p)^7 (1-7/p).

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A337606 Decimal expansion of the Gaussian twin prime constant: the Hardy-Littlewood constant for A096012.

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A337607 Decimal expansion of Shanks's constant: the Hardy-Littlewood constant for A000068.

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A337608 Decimal expansion of Lal's constant: the Hardy-Littlewood constant for A217795.

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A387362 Cyclic numbers k such that k+2 is also a cyclic number.

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A387363 The number of decompositions of 2*n into ordered sums of two cyclic numbers.

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A246061 Decimal expansion of lim_{n->infinity} ((1/log(n)^2)*Product_{2 < p < n, p prime} p/(p-2)).

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A374831 Decimal expansion of Product_{p prime} (1 - (1/(p(p - 1)))p^2/(p^2 + 1)).