cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Amiram Eldar, Sep 04 2020

Keywords

Comments

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).

Examples

			0.669740969937071220538922431571764406688370157436482...

References

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

Links

Crossrefs

Cf. A000068, A037896, A088367, A334826.
Similar constants: A005597, A331941, A337606, A337608.

Programs

  • Mathematica
    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 *)

Formula

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.

Extensions

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

Views

Author

Amiram Eldar, Sep 04 2020

Keywords

Comments

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).

Examples

			0.792208238167541668775455566579024101128932250986221...

References

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

Links

Crossrefs

Cf. A000068, A037896, A088367, A210630, A217795, A217796.
Similar constants: A005597, A331941, A337606, A337607.

Programs

  • Mathematica
    $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 *)

Formula

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.

Extensions

More terms from Vaclav Kotesovec, Jan 16 2021

A182347 Primes of the form k^4 - 2.

Original entry on oeis.org

79, 2399, 14639, 28559, 194479, 707279, 2313439, 2825759, 3418799, 5764799, 7890479, 12117359, 28398239, 47458319, 52200623, 57289759, 81450623, 96059599, 104060399, 200533919, 276922879, 395254159, 418161599, 937890623, 1073283119, 1171350623, 1275989839
Offset: 1

Views

Author

Patrick Devlin, Apr 25 2012

Keywords

Examples

			79 = 3^4 - 3; 2399 = 7^4 - 3.

Links

Crossrefs

Cf. A037896.

Programs

  • Maple
    # choose N large, then S is the desired set
f:=n->n^4 - 2:
S:={}:
for n from 0 to N do if(isprime(f(n))) then S:=S union {f(n)}: fi: od:
  • Mathematica
    lst = {}; Do[If[PrimeQ[r=(2k+1)^4-2], AppendTo[lst, r]], {k, 1, 1000}]; lst[[1;;100]]

A217128 Numbers n such that (2n)^4 + 1 is not prime.

Original entry on oeis.org

4, 5, 6, 7, 9, 11, 13, 15, 16, 18, 19, 20, 21, 22, 25, 26, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 42, 43, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 72, 73, 74, 75, 76, 78, 79, 81, 83, 84, 85, 86, 88, 89, 91, 92, 93, 94, 95, 96, 98, 100
Offset: 1

Views

Author

Vincenzo Librandi, Sep 27 2012

Keywords

Links

Crossrefs

Cf. A000059, A037896.

Programs

  • Magma
    [n: n in [1..100] | not IsPrime((2*n)^4+1)];
  • Mathematica
    Select[Range[100], ! PrimeQ[(2 #)^4 + 1] &]

A217129 Numbers n such that n^4 + 1 is not prime.

Original entry on oeis.org

3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83
Offset: 1

Views

Author

Vincenzo Librandi, Sep 27 2012

Keywords

Links

Crossrefs

Cf. A000068, A037896.

Programs

  • Magma
    [n: n in [1..100] |not IsPrime(n^4+1)];
  • Mathematica
    Select[Range[100], ! PrimeQ[#^4 + 1] &]

A272137 Primes of the form k^16 + 1.

Original entry on oeis.org

2, 65537, 197352587024076973231046657, 808551180810136214718004658177, 1238846438084943599707227160577, 37157429083410091685945089785857, 123025056645280288014028950372089857, 150838912030874130174020868290707457
Offset: 1

Views

Author

Jaroslav Krizek, May 08 2016

Keywords

Comments

Corresponding values of k are in A006313.

Links

Crossrefs

Cf. Sequences of numbers n such that n^(2^k)+1 is a prime p for k = 1-13: A005574 (k=1), A000068 (k=2), A006314 (k=3), A006313 (k=4), A006315 (k=5), A006316 (k=6), A056994 (k=7), A056995 (k=8), A057465 (k=9), A057002 (k=10), A088361 (k=11), A088362 (k=12), A226528 (k=13).
Corresponding sequences of primes p of the form n^(2^k)+1 for k = 1-4: A002496 (k=1), A037896 (k=2), A258805 (k=3), A272137 (k=4).

Programs

  • Magma
    [n^16 + 1: n in [1..700] | IsPrime(n^16 + 1)];
  • Maple
    A272137:=n->`if`(isprime(n^16+1), n^16+1, NULL): seq(A272137(n), n=1..200); # Wesley Ivan Hurt, May 11 2016

A140834 Primes that are the sum of at most four nonzero 4th powers.

Original entry on oeis.org

2, 3, 17, 19, 83, 97, 113, 163, 179, 257, 337, 353, 419, 499, 593, 641, 643, 673, 769, 787, 881, 883, 1153, 1297, 1409, 1459, 1553, 1889, 2003, 2083, 2131, 2417, 2579, 2593, 2609, 2657, 2659, 2689, 2819, 3169, 3217, 3697, 3779, 3889, 3907, 4099, 4129, 4177
Offset: 1

Views

Author

Jonathan Vos Post, Jul 18 2008

Keywords

Comments

This sequence was checked by T. D. Noe, who had supplied the b-list for A004833. A037896 is a subset of {Primes that are the sum of at exactly 2 nonzero 4th powers}, itself a subset of A002645 Quartan primes: primes of the form x^4 + y^4, x>0, y>0.

Crossrefs

Cf. A000040, A000583, A002645, A003294, A004833, A037896, A085318, A122540, A126657, A133740.

Formula

A000040 INTERSECTION A004833. {A133740 = Primes that are the sum of at exactly 4 nonzero 4th powers} UNION {A085318 = Primes that are the sum of at exactly 3 nonzero 4th powers} UNION {A002645 = Primes that are the sum of at exactly 2 nonzero 4th powers}.

Extensions

Missing term 353 inserted by Georg Fischer, May 11 2024

A182346 Primes of the form n^4 + 6.

Original entry on oeis.org

7, 631, 279847, 4879687, 12117367, 20151127, 28398247, 62742247, 68574967, 81450631, 174900631, 1330863367, 1698181687, 3373402567, 3722098087, 4499860567, 5719140631, 9354951847, 16610312167, 18141126727
Offset: 1

Views

Author

Patrick Devlin, Apr 25 2012

Keywords

Examples

			7 = 1^4 + 6; 631 = 5^4 + 6.

Crossrefs

Cf. A037896.

Programs

  • Maple
    # choose N large, then S is the desired set
f:=n->n^4 + 6:
S:={}:
for n from 0 to N do if(isprime(f(n))) then S:=S union {f(n)}: fi: od:
  • Mathematica
    Select[Range[1,400,2]^4+6,PrimeQ] (* Harvey P. Dale, Jan 12 2013 *)

A182348 Primes of the form n^4 - 3.

Original entry on oeis.org

13, 4093, 1048573, 1336333, 2085133, 4477453, 16777213, 54700813, 92236813, 116985853, 146409997, 236421373, 285609997, 479785213, 959512573, 1003875853, 1097199373, 1303209997, 1871773693, 2097273613, 2342559997
Offset: 1

Views

Author

Patrick Devlin, Apr 25 2012

Keywords

Examples

			13 = 2^4 - 3; 4093 = 8^4 - 3.

Crossrefs

Cf. A037896.

Programs

  • Maple
    # choose N large, then S is the desired set
f:=n->n^4 - 3:
S:={}:
for n from 0 to N do if(isprime(f(n))) then S:=S union {f(n)}: fi: od:

A182350 Primes of the form n^4 - 5.

Original entry on oeis.org

11, 251, 1291, 4091, 20731, 104971, 1048571, 2085131, 9834491, 11316491, 14776331, 18974731, 29986571, 49787131, 78074891, 168896011, 236421371, 406586891, 429981691, 454371851, 479785211, 959512571, 1146228731
Offset: 1

Views

Author

Patrick Devlin, Apr 25 2012

Keywords

Examples

			11 = 2^4 - 5; 251 = 4^4 - 5.

Crossrefs

Cf. A037896.

Programs

  • Maple
    # choose N large, then S is the desired set
f:=n->n^4 - 5:
S:={}:
for n from 0 to N do if(isprime(f(n))) then S:=S union {f(n)}: fi: od:
  • Mathematica
    Select[Range[200]^4-5,PrimeQ] (* Harvey P. Dale, Apr 10 2017 *)
Previous Showing 21-30 of 32 results. Next

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