A053182 -id:A053182 - cp's OEIS frontend

A188596 Decimal expansion of Product_{primes p} (1-1/p)^(-2)*(1-(2+A102283(p))/p).

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

Offset: 1

R. J. Mathar, Apr 05 2011

This is the principal scale factor in an estimate of the number of primes p not exceeding N such that p^2+p+1 is also prime [Bateman-Horn].
A102283 in the definition plays the role of the Dirichlet character modulo 3.
After splitting the product into the three modulo-3 classes of primes, this constant turns out to be the product of four factors.
One factor as mentioned by Bateman and Horn is the inverse of A073010.
The second factor is 3/4 arising from the prime 3 which is the sole prime in the class == 0 (mod 3).
The third factor is product_{p == 1 (mod 3)} (1-(3p-1)/(p-1)^3) = 0.8675121817.. which is the constant C(m=3,n=1,s=3) of the arXiv preprint, basically the C(3) variant of A065418 reduced to the modulo class.
The final factor is product_{p == 2 (mod 3)} (1+1/(p^2-1)) = 1/product_{p == 2 (mod 3)} (1-1/p^2) = 1.41406439089214763.. which is the constant zeta(m=3,n=2,s=2) of the preprint and mentioned in A175646.

			Equals 1.5217315350757058188419... = 0.92003856361849186... / A073010 .

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 86.

Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363-367, constant C.
H. Davenport and A. Schinzel, A note on certain arithmetical constants, Illinois Math. J. 10 (2) (1966), 181-185.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A053182.

a073010 := evalf(Pi/3/sqrt(3)) ;
Cm3n0s2 := 1-1/(3-1)^2 ;
Cm3n1s3 := 0.867512181712394919089076584762888869720269526863 ;
Zm3n2s2 := 1.4140643908921476375655018190798293799076950693931 ;
Cm3n0s2*Cm3n1s3*Zm3n2s2/a073010 ;

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[3^(5/2)*Zs[3, 1, 3]*Z[3, 2, 2]/(4*Pi), digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)

More terms from Vaclav Kotesovec, Jan 16 2021

A216061 Primes p such that p^3 + p + 1 is prime.

Original entry on oeis.org

2, 3, 5, 17, 29, 41, 53, 71, 83, 131, 179, 191, 239, 263, 311, 389, 491, 509, 557, 569, 593, 653, 701, 719, 743, 797, 821, 863, 887, 953, 971, 977, 1019, 1049, 1097, 1109, 1277, 1301, 1319, 1373, 1427, 1481, 1523, 1559, 1601, 1607, 1613, 1667, 1787, 1823

Offset: 1

César Eliud Lozada, Aug 31 2012

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A053182.

Subsequence of A045309.

[p: p in PrimesUpTo(2000) | IsPrime(p^3+p+1)]; // Bruno Berselli, Sep 01 2012

A := {}; for n to 1000 do p := ithprime(n); if isprime(p^3+p+1) then A := `union`(A, {p}) end if end do; A := A

Select[Prime[Range[400]], PrimeQ[#^3 + # + 1] &] (* Bruno Berselli, Sep 01 2012 *)

A238400 Primes in A238399.

Original entry on oeis.org

2, 3, 7, 1237, 66067, 525593, 974167, 1412473, 2675759, 4471237, 5264333, 8107961, 8308271, 12615151, 20145407, 34926433, 43167569, 94772749, 104612297, 115103327, 144450221, 153124973, 165108557, 196634723, 211696049, 213507241, 255963131, 263979101

Offset: 1

Torlach Rush, Feb 26 2014

nonn

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A238399, A053182, A053183.

Select[(PrimePi[#^2 + #] - PrimePi[#]) & /@ Select[Prime@Range[3000], PrimeQ[#^2 + # + 1] &], PrimeQ] (* Giovanni Resta, Feb 27 2014 *)

Corrected by Torlach Rush, Feb 26 2014

a(16)-a(28) from Giovanni Resta, Feb 27 2014

A243544 Primes p such that p^2 - p + 1 is semiprime.

Original entry on oeis.org

5, 11, 29, 37, 41, 43, 53, 61, 71, 73, 83, 97, 109, 113, 127, 137, 149, 157, 167, 181, 191, 211, 223, 229, 241, 271, 277, 281, 307, 317, 331, 359, 389, 421, 433, 443, 461, 463, 487, 499, 547, 557, 571, 587, 601, 617, 631, 659, 661, 683, 691, 701, 709, 733, 757

Offset: 1

K. D. Bajpai, Jun 06 2014

nonn

Intersection of A000040 and A180748.

			11 is in the sequence because 11 is prime and 11^2 - 11 + 1 = 111 = 3 * 37 is semiprime.
29 is in the sequence because 29 is prime and 29^2 - 29 + 1 = 813 = 3 * 271 is semiprime.
17 is not in the sequence though 17 is prime, because 17^2 - 17 + 1 = 273 = 3 * 7 * 13, has more than two prime factors.

K. D. Bajpai, Table of n, a(n) for n = 1..7090

Cf. A000040, A001358, A053182, A053184, A065508, A091567, A180748.

with(numtheory): A243544 := proc() local a; a:=ithprime(n);  if bigomega(a^2-a+1)=2 then RETURN (a); fi; end: seq(A243544 (), n=1..200);

c = 0; Do[k = Prime[n]; If[PrimeOmega[k^2 - k + 1] == 2, c++; Print[c, " ", k]], {n, 1, 30000}];
Select[Prime[Range[150]],PrimeOmega[#^2-#+1]==2&] (* Harvey P. Dale, Oct 22 2024 *)

s=[]; forprime(p=2, 800, if(bigomega(p^2-p+1)==2, s=concat(s, p))); s \\ Colin Barker, Jun 06 2014

A290767 Primes p such that p^2 +/- p +/- 1 are all nonprimes.

Original entry on oeis.org

23, 37, 43, 73, 107, 109, 113, 137, 157, 179, 211, 223, 227, 229, 239, 251, 257, 271, 277, 283, 311, 313, 317, 347, 353, 367, 389, 439, 443, 467, 503, 509, 521, 523, 547, 557, 563, 577, 587, 593, 601, 631, 653, 661, 719, 733, 757, 797, 811, 821, 823, 829, 853, 859, 877, 883

Offset: 1

Ralf Steiner, Aug 10 2017

nonn

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

Cf. A000040, A053184, A053182, A065508, A091567, A236056.

select(p -> isprime(p) and not ormap(isprime, [p^2+p+1,p^2+p-1,p^2-p+1,p^2-p-1]), [2,seq(i,i=3..1000,2)]); # Robert Israel, Aug 10 2017

Select[Prime[Range[1000]], ! (PrimeQ[#^2 + # + 1] || PrimeQ[#^2 + # - 1] ||PrimeQ[#^2 - # + 1] || PrimeQ[#^2 - # - 1]) &]
Select[Prime[Range[200]],NoneTrue[{#^2+#+1,#^2+#-1,#^2-#+1,#^2-#-1},PrimeQ]&] (* Harvey P. Dale, Oct 13 2024 *)

is(n) = my(v=[n^2+n+1, n^2+n-1, n^2-n+1, n^2-n-1]); for(k=1, #v, if(ispseudoprime(v[k]), return(0))); 1
forprime(p=1, 900, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Aug 10 2017

Intersection of the complements of A053184, A053182, A065508, and A091567 within the primes A000040.

A325435 Numbers m such that m! / sigma(m) is an integer.

Original entry on oeis.org

1, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Jaroslav Krizek, Apr 26 2019

nonn

Complement of A325436.
Corresponding integers are 1, 20, 60, 630, 2688, 201600, 3326400, 17107200, ...
Equals A163162 without number 3.

			6 is in the sequence because 6! / sigma(6) = 720 / 12 = 60 (integer).

Cf. A000142, A000203, A053182, A163162, A325436.

[n: n in [1..1000] | IsIntegral(Factorial(n)/&+[d: d in Divisors(n)])]

Select[Range[80],IntegerQ[#!/DivisorSigma[1,#]]&] (* Harvey P. Dale, Jul 20 2022 *)

isok(n) = ((n! % sigma(n)) == 0); \\ Michel Marcus, Apr 26 2019

A325436 Numbers m such that m! / sigma(m) is not an integer.

Original entry on oeis.org

2, 3, 4, 9, 16, 25, 64, 289, 729, 1681, 2401, 3481, 4096, 5041, 7921, 10201, 15625, 17161, 27889, 28561, 29929, 65536, 83521, 85849, 146689, 262144, 279841, 458329, 491401, 531441, 552049, 579121, 597529, 683929, 703921, 707281, 734449, 829921, 1190281

Offset: 1

Jaroslav Krizek, Apr 26 2019

nonn

Numbers A053182(n)^2 are terms for n >= 1.
Complement of A325435.
Union of the number 3 and numbers m from A023194 (sigma(m) = prime).

Cf. A000142, A000203, A023194, A053182, A325435.

[n: n in [1..10000] | not IsIntegral(Factorial(n)/&+[d: d in Divisors(n)])]

isok(n) = ((n! % sigma(n)) != 0); \\ Michel Marcus, Apr 26 2019

A224781 Primes p such that both 2*p + 1 and p^2 + p + 1 are primes.

Original entry on oeis.org

2, 3, 5, 41, 89, 131, 173, 293, 743, 761, 911, 1559, 1583, 1811, 1931, 1973, 2129, 2273, 2339, 2969, 3449, 3491, 4409, 4733, 5003, 5039, 5501, 6173, 6551, 6761, 7883, 7901, 8093, 8741, 9059, 9689, 10589, 10781, 11171, 11549, 13229, 13553, 14939, 15569

Offset: 1

Jayanta Basu, Apr 17 2013

nonn

Intersection of A005384 and A053182.

Note that 2p+1 is the derivative of p^2+p+1 with respect to p. - T. D. Noe, Apr 18 2013

			5 is a member since 5+6=11 and 5*6+1=31 are both primes.

Cf. A005384, A053182.

Select[Prime[Range[1850]], PrimeQ[2*# + 1] && PrimeQ[#^2 + # + 1] &]

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A188596 Decimal expansion of Product_{primes p} (1-1/p)^(-2)*(1-(2+A102283(p))/p).

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A216061 Primes p such that p^3 + p + 1 is prime.

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A238400 Primes in A238399.

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A243544 Primes p such that p^2 - p + 1 is semiprime.

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A290767 Primes p such that p^2 +/- p +/- 1 are all nonprimes.

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A325435 Numbers m such that m! / sigma(m) is an integer.

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A325436 Numbers m such that m! / sigma(m) is not an integer.

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A224781 Primes p such that both 2*p + 1 and p^2 + p + 1 are primes.

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