Jani Melik - cp's OEIS frontend

A337704 Pythagorean primes that are not congruent numbers.

17, 73, 89, 97, 113, 193, 233, 241, 281, 337, 401, 409, 433, 449, 521, 569, 577, 593, 601, 617, 641, 673, 769, 809, 857, 881, 929, 937, 953, 977, 1009, 1033, 1049, 1097, 1129, 1153, 1193, 1289, 1297, 1361, 1409, 1433, 1481, 1489, 1553, 1601, 1609, 1657, 1697, 1721, 1753

Offset: 1

Jani Melik, Sep 16 2020

nonn

			17 is a Pythagorean prime and is not congruent because it is not the area of a right triangle with rational sides. So 17 is a term.

Intersection of A002144 and A165564.

Cf. A003273, A269727, A165815.

A269430 Decimal expansion of (1 + Pi)/2.

Original entry on oeis.org

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

Offset: 1

Jani Melik, Feb 26 2016

			2.0707963267948966192313216916397514420985846996875529...

Index entries for transcendental numbers

Cf. A000796, A002193, A002194, A002163, A174968, A001622, A002161, A010767, A011002, A011003, A191502, A053004.

PARI
```
(1 + Pi)/2 \\ Altug Alkan, Apr 07 2016
```
Sage
```
N((1+pi)/2, digits=110)
```

Equals A096444 + 1 or A019669 + 1/2.

Equals Sum_{k>=0} 2^k/binomial(2*k+2,k). - Amiram Eldar, Jun 30 2020

A258619 Decimal expansion of Sum_{k>=0} (1/A055209(k)).

Original entry on oeis.org

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

Offset: 1

Jani Melik, Jun 06 2015

			2.2569565016088514950284576370776548515676635143755759249....

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A055209.

SetDefaultRealField(RealField(100)); b:=[1] cat [(&*[(Factorial(k))^2: k in [1..n]]): n in [1..60]];  (&+[1.0/b[k]: k in [1..50]]); // G. C. Greubel, Nov 28 2018

RealDigits[Sum[1/BarnesG[k + 2]^2, {k, 0, 80}], 10, 100][[1]] (* G. C. Greubel, Nov 28 2018 *)

default(realprecision, 100); sum(n=0,50, 1.0/prod(i=0, n, i!)^2) \\ G. C. Greubel, Nov 28 2018

def A055209(n) :
   return prod(factorial(i)^(2) for i in (0..n))
N(sum(1/A055209(n) for n in (0..12)), digits=105)

Name corrected by Amiram Eldar, Nov 17 2020

A258621 Decimal expansion of sum(1/A030450).

Original entry on oeis.org

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

Offset: 1

Jani Melik, Jun 06 2015

			2.2569474585265979554627367244423422105592365088936959533....

Cf. A030450.

#include 
#include 
#define dBIT   512
#define dLIST  n, t, u
int main (void) {
mpfr_t dLIST;
mpfr_rnd_t trnd;
unsigned int i;
trnd = MPFR_RNDU;
mpfr_inits2 (dBIT, dLIST, (mpfr_ptr) 0);
mpfr_set_d (n, 1.0, trnd);
mpfr_set_d (t, 1.0, trnd);
for (i = 1; i <= 9; i++) {
    mpfr_mul_ui (t, t, i, trnd);
    mpfr_mul (t, t, t, trnd);
    mpfr_set_d (u, 1.0, trnd);
    mpfr_div (u, u, t, trnd);
    mpfr_add (n, n, u, trnd);
}
mpfr_printf ("%.106Rg", n);
mpfr_clears (dLIST, (mpfr_ptr) 0);
return 0;
}

def A030450(n) :
   return prod((n-i+1)^(2^i) for i in (1..n))
N(sum(1/A030450(n) for n in (0..9)), digits=106)

A244623 Odd prime powers that are not primes.

Original entry on oeis.org

1, 9, 25, 27, 49, 81, 121, 125, 169, 243, 289, 343, 361, 529, 625, 729, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 14641, 15625, 16129, 16807, 17161, 18769, 19321, 19683

Offset: 1

Jani Melik, Jul 02 2014

nonn

Intersection of A061345 and A014076.

A014076 set minus A061346.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Intersection of A005408 and A025475.

Cf. A061345 (odd prime powers), A061346 (odd neither prime nor prime power), A062739 (odd powerful), A075109 (perfect powers), A136141.

Join[{1},Select[Range[1,20001,2],PrimePowerQ[#]&&(!PrimeQ[#])&]] (* Harvey P. Dale, Dec 11 2018 *)

isok(p) = ((p%2) && !isprime(p) && isprimepower(p)) || (p==1); \\ Michel Marcus, Jul 06 2021

def isA244623(n) :
   return(n % 2 == 1 and is_prime_power(n) == 1 and is_prime(n) == 0)
[n for n in (1..20000) if isA244623(n)]

a(n) = A079290(n) at least in the range n=3..94, and perhaps beyond. - R. J. Mathar, Aug 20 2014

Sum_{n>=1} 1/a(n) = 1/2 + Sum_{p prime} 1/(p*(p-1)) = 1/2 + A136141. - Amiram Eldar, Dec 21 2020

A242459 Maximal differences of A029707.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 20, 24, 27, 29, 42, 54, 72, 75, 103, 128, 131, 151, 153, 162, 164, 181, 204, 208, 209, 211, 237, 265, 285, 286, 326, 335, 340, 356, 368, 392, 409, 432, 439, 441, 444, 446

Offset: 1

Jani Melik, May 15 2014

Cf. A000720, A001359, A027833, A029707, A054691.

nextLesserTwinPrime[p_Integer] := Block[{q = p + 2}, While[ NextPrime@ q - q > 2, q = NextPrime@ q]; q]; p = 2; q = 3; px = 1; qx = 2; mxd = 0; tpx = 0; lst = {}; While[p <
5090000001, d = qx - px; If[ d > mxd, mxd = d; AppendTo[ lst, d]; Print@ d]; p = q; px = qx; q = nextLesserTwinPrime@ q; qx = PrimePi@ q; tpx++]; lst (* Robert G. Wilson v, May 21 2014 *)

def A242459_list(n) :
   a = [ 1 ]
   st = 3
   for i in (4..n) :
      if (nth_prime(i+1)-nth_prime(i) == 2) :
         if i-st > a[len(a)-1] :
            a.append(i-st)
         st = i
   return(a)
A242459_list(10^(5))

a(n) = primepi(next(A054691(n-1))) - primepi(A054691(n-1)) + 1 for n >= 2, where primepi = A000720 and next(k) is the least lesser of twin primes that is larger than k. - Amiram Eldar, May 19 2024

a(20)-a(28) from Robert G. Wilson v, May 21 2014

a(29)-a(42) from Amiram Eldar, May 19 2024

A237704 Numbers n for which the fundamental solution of Pell's equation x^2 - n*y^2 = 1 has both x and y prime.

Original entry on oeis.org

2, 6, 12, 30, 32, 40, 42, 72, 90, 132, 152, 192, 210, 240, 312, 342, 408, 420, 462, 480, 552, 560, 592, 672, 702, 792, 870, 880, 888, 912, 930, 1122, 1152, 1260, 1272, 1320, 1332, 1560, 1584, 1722, 1752, 1792, 1980, 2352, 2520, 2550, 2652, 2712, 2862, 2952, 2970, 3192, 3560, 3640, 4032

Offset: 1

Jani Melik, Feb 11 2014

nonn

			Pell's equation x^2 - 2*y^2 = 1 and its fundamental solution is (x,y) = (3,2) which are both primes, so a(1) = 2.
(x,y) = (5,2) satisfies x^2 - 6*y^2 = 1, so a(2) = 6.
(x,y) = (7,2) satisfies x^2 - 12*y^2 = 1, so a(3) = 12.
Pell's equation x^2 - 2088*y^2 = 1 and (x,y) = (19603, 429), 19603 is prime, 429 = 3 * 11 * 13 is not, so 2088 is not included.
Pell's equation x^2 - 2000*y^2 = 1 and (x,y) = (930249, 20801), 930249 = 3^2 * 41 * 2521 and 20801 = 11 * 31 * 61 are not primes, so 2000 is not included.

Ray Chandler, Table of n, a(n) for n = 1..716
Wikipedia, Pell's equation

Cf. A000037, A033313, A033317.

420 inserted into the sequence by Colin Barker, Feb 12 2014

A236250 Period of the n-th convergent to the continued fraction expansion of Pi.

Original entry on oeis.org

1, 6, 13, 112, 51, 24, 15088, 12284, 88460, 1204, 459, 31824, 93210, 1864254, 531648, 456036, 8299090, 28574910, 1813560, 32552820, 33166008, 133585180, 2503410, 214098720, 3183870690, 7411133309730, 4852769490690, 2294509753536, 175964053944, 3336533898768

Offset: 1

Jani Melik, Jan 21 2014

			The 2nd convergent is 22/7 = 3.142857 142857 ..., whose period is 6, so a(2) = 6.
The 3rd convergent is 333/106 = 3.1 4150943396226 4150943396226 ..., whose period is 13, so a(3) = 13.

Cf. A001203, A007732, A084407.

st_clenov = 30
def A236250(n) :
   vu = continued_fraction_list(pi, nterms=st_clenov);
   p = []
   for i in (0..n) :
      p.append(convergents(vu)[i].period())
   return(p)
A236250(st_clenov-1);

a(n) = A007732(A002486(n+2)). - Michel Marcus, Jan 21 2014

A217458 Decimal expansion of 2^(Pi*sqrt(2)).

Original entry on oeis.org

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

Offset: 2

Jani Melik, Oct 03 2012

			21.7490870543774583340992082660113952338588408897418426759...

G. C. Greubel, Table of n, a(n) for n = 2..10000

Cf. A063448 (Pi * sqrt(2)).

SetDefaultRealField(RealField(100)); R:=RealField(); 2^(Pi(R)*Sqrt(2)); // G. C. Greubel, Oct 05 2018

RealDigits[2^(Pi*Sqrt[2]), 10, 100][[1]] (* G. C. Greubel, Oct 05 2018 *)

fpprec : 100; ev(bfloat(2^(%pi*sqrt(2)))); /* Martin Ettl, Oct 04 2012 */

default(realprecision, 100); 2^(Pi*sqrt(2)) \\ G. C. Greubel, Oct 05 2018

Sage
```
2^(pi*sqrt(2)).n(digits=100)
```

A217459 Decimal expansion of 2^Pi.

Original entry on oeis.org

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

Offset: 1

Jani Melik, Oct 03 2012

			8.8249778270762876238564296042080015817044108152714849266....

"SackVideo", What is 2^π?, 2022 video.

Cf. A000796, A086054.

RealDigits[2^Pi, 10, 100][[1]] (* Amiram Eldar, Nov 24 2020 *)

fpprec : 100; ev(bfloat(2^(%pi)));  /* Martin Ettl, Oct 04 2012 */

default(realprecision,2000);2^Pi \\ Anders Hellström, Nov 11 2015

Sage
```
2^(pi).n(digits=100)
```

Equals exp(A086054). - Amiram Eldar, Nov 24 2020

cp's OEIS Frontend

User: Jani Melik

A337704 Pythagorean primes that are not congruent numbers.

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Examples

Crossrefs

A269430 Decimal expansion of (1 + Pi)/2.

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Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Sage

Formula

A258619 Decimal expansion of Sum_{k>=0} (1/A055209(k)).

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Magma

Mathematica

PARI

Sage

Extensions

A258621 Decimal expansion of sum(1/A030450).

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Examples

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Programs

C

Sage

A244623 Odd prime powers that are not primes.

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Comments

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Mathematica

PARI

Sage

Formula

A242459 Maximal differences of A029707.

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Author

Keywords

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A237704 Numbers n for which the fundamental solution of Pell's equation x^2 - n*y^2 = 1 has both x and y prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A236250 Period of the n-th convergent to the continued fraction expansion of Pi.

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Author

Keywords

Examples

Crossrefs

Programs

Sage

Formula