A062737 -id:A062737 - cp's OEIS frontend

A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

3, 2, 5, 13, 11, 19, 17, 31, 29, 43, 41, 61, 59, 73, 71, 103, 101, 109, 107, 139, 137, 151, 149, 181, 179, 193, 191, 199, 197, 229, 227, 241, 239, 271, 269, 283, 281, 313, 311, 349, 347, 421, 419, 433, 431, 463, 461, 523, 521, 571, 569, 601, 599, 619, 617

Offset: 1

Clark Kimberling, Dec 15 2015

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...).
See A265772 and A265774 for definitions of lower POBA and upper POBA. In the following guide, for example, A001359/A006512 represents (conjecturally in some cases) the Lower POBAs p(n)/q(n) to 1, where p = A001359 and q = A006512 except for first terms in some cases. Every POBA is either a lower POBA or an upper POBA.
x        Lower POBA       Upper POBA       POBA
1        A001359/A006512  A006512/A001359  A265759/A265760
3/2      A104163/A158708  A162336/A158709  A265761/A222565
2        A005383/A005382  A005385/A005384  A079149/A120628
3        A091180/A088878  A094525/A023208  A265763/A265764
4        A162857/A062737  A090866/A023212  A265765/A120639
5        A265766/A158318  A265767/A023217  A265768/A265769
6        A227756/A158015  A051644/A007693  A265770/A265771
sqrt(2)  A265772/A265773  A265774/A265775  A265776/A265777
sqrt(3)  A265778/A265779  A265780/A265781  A265782/A265783
sqrt(5)  A265784/A265785  A265786/A265787  A265788/A265789
sqrt(8)  A265790/A265791  A265792/A265793  A265794/A265795
tau      A265796/A265797  A265798/A265799  A265800/A265801
1/tau    A265799/A265798  A265797/A265796  A265806/A265807
pi       A265808/A265809  A265810/A265811  A265812/A265813
e        A265814/A265815  A265816/A265817  A265818/A265819

			The POBAs for 1 start with 3/2, 2/3, 5/7, 13/11, 11/13, 19/17, 17/19, 31/29, 29/31, 43/41, 41/43, 61/59, 59/61. For example, if p and q are primes and q > 13, then 11/13 is closer to 1 than p/q is.

Cf. A000040, A001359, A006512, A265759, A265760.

x = 1; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265759/A265760 *)
Numerator[tL]   (* A001359 *)
Denominator[tL] (* A006512 *)
Numerator[tU]   (* A006512 *)
Denominator[tU] (* A001359 *)
Numerator[y]    (* A265759 *)
Denominator[y]  (* A265760 *)

A158015 Primes p such that 6*p-1 is also prime.

Original entry on oeis.org

2, 3, 5, 7, 17, 19, 23, 29, 43, 47, 53, 59, 67, 103, 107, 109, 113, 127, 137, 157, 163, 197, 199, 227, 229, 239, 269, 283, 313, 317, 347, 359, 373, 379, 383, 389, 397, 439, 443, 449, 457, 463, 467, 523, 569, 577, 593, 599, 613, 617, 647, 653, 709, 733, 743, 773

Offset: 1

Roger L. Bagula, Mar 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005382 for the type 2p-1, A062737 for 4p-1, A158016 for 8p-1, A158017 for 10p-1.

Primes in A024898, i.e., intersection of A024898 with A000040.

[p: p in PrimesUpTo(800) | IsPrime(6*p-1)]; // Vincenzo Librandi, Apr 14 2013

Select[Prime[Range[200]], PrimeQ[(6 # - 1)]&] (* Vincenzo Librandi, Apr 14 2013 *)

Edited by the Associate Editors of the OEIS, Apr 22 2009

A158017 Primes p such that 10*p-1 is also prime.

Original entry on oeis.org

2, 3, 11, 23, 41, 71, 83, 101, 107, 113, 149, 167, 179, 227, 239, 269, 311, 317, 347, 353, 389, 479, 491, 521, 557, 569, 587, 647, 653, 683, 809, 821, 827, 839, 863, 911, 977, 983, 1091, 1229, 1259, 1283, 1289, 1301, 1367, 1373, 1439, 1487, 1493, 1607, 1619

Offset: 1

Roger L. Bagula, Mar 11 2009

The family of prime sequences that generate primes k*p-1 for k = 2, 4, 6, 8, ... also comprises A005382 (k=2), A062737 (k=4), A158015 (k=6), and A158016 (k=8).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005382, A062737.

[p: p in PrimesUpTo(3000)|IsPrime(10*p-1)] // Vincenzo Librandi, Jan 29 2011

Flatten[Table[If[PrimeQ[n] && PrimeQ[10*n - 1], n, {}], {n, 1, 10000}]]
Select[Prime[Range[600]], PrimeQ[(10 # - 1)]&] (* Vincenzo Librandi, Apr 14 2013 *)

A158016 Primes p such that 8*p-1 is also prime.

Original entry on oeis.org

3, 13, 19, 61, 79, 103, 163, 181, 193, 223, 229, 313, 331, 349, 409, 433, 439, 541, 571, 613, 619, 691, 751, 769, 853, 859, 919, 991, 1021, 1033, 1039, 1321, 1381, 1423, 1483, 1543, 1549, 1621, 1699, 1759, 1801, 1861, 1873, 1879, 1951, 1999, 2011, 2029, 2113

Offset: 1

Roger L. Bagula, Mar 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005382 for the type 2p-1, A062737 for 4p-1, A158015 for 6p-1, A158017 for 10p-1.

[p: p in PrimesUpTo(2200) | IsPrime(8*p - 1)]; // Vincenzo Librandi, Apr 14 2013

Select[Prime[Range[600]], PrimeQ[(8 # - 1)]&] (* Vincenzo Librandi, Apr 14 2013 *)

Edited by the Associate Editors of the OEIS, Apr 22 2009

A162857 Primes of the form 4p - 1, p a prime.

Original entry on oeis.org

7, 11, 19, 43, 67, 163, 211, 283, 331, 523, 547, 691, 787, 907, 1051, 1123, 1171, 1531, 1723, 1867, 2011, 2083, 2251, 2347, 2371, 2467, 2707, 2731, 2803, 2971, 3187, 3307, 3547, 3643, 3907, 3931, 4051, 4243, 4363, 4603, 4651, 4723, 5107, 5227, 5443, 5923

Offset: 1

Daniel Tisdale, Jul 14 2009

If 4p - 1 is prime then n^2 + n + p = p(4p - 1) for some n = 1, 2, 3, ... [Proof. Let n + 1 = 2p, etc.]
From Alonso del Arte, Jan 14 2024: (Start)
The first six terms correspond to rings of algebraic integers of Q(sqrt(-a(n))) which are unique factorization domains.
In the ring of algebraic integers of Q(sqrt(-a(n))), the corresponding prime p = (a(n) + 1)/4 is divisible by 1/2 - sqrt(-a(n))/2 and 1/2 + sqrt(-a(n))/2, both of those being algebraic integers with minimal polynomial x^2 - x + p. For example, in Q(sqrt(-163)), we see that (1/2 - sqrt(-163)/2)(1/2 + sqrt(-163)/2) = 1/4 + 163/4 = 41, with both of the divisors having the minimal polynomial x^2 - x + 41. (End)

Cf. A062737 for the corresponding primes p.

Overlaps with A003173, the Heegner numbers (last six terms of that one match the first six of this one).

Select[Prime[Range[1000]], PrimeQ[(# + 1)/4] &] (* Alonso del Arte, Jan 14 2024 *)

is(n)=isprime(n) && isprime(4*n-1) \\ Charles R Greathouse IV, Jun 14 2022

def isPrime(num: Int): Boolean = Math.abs(num) match {
  case 0 => false; case 1 => false; case n => (2 to Math.floor(Math.sqrt(n)).toInt) forall (p => n % p != 0)
}
((1 to 2500).map(4 *  - 1)).filter(n => isPrime(n) && isPrime((n + 1)/4)) // _Alonso del Arte, Jan 14 2024

a(n) >> n log^2 n. - Charles R Greathouse IV, Jun 14 2022

More terms from N. J. A. Sloane, Jul 19 2009

A265765 Numerators of primes-only best approximates (POBAs) to 4; see Comments.

Original entry on oeis.org

11, 7, 13, 11, 19, 29, 43, 53, 67, 149, 163, 173, 211, 269, 283, 293, 317, 331, 389, 509, 523, 547, 557, 653, 691, 773, 787, 797, 907, 1051, 1109, 1123, 1171, 1229, 1493, 1531, 1637, 1723, 1733, 1867, 1949, 1997, 2011, 2083, 2251, 2309, 2347, 2371, 2467

Offset: 1

Clark Kimberling, Dec 18 2015

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences.

			The POBAs for 4 start with  11/2, 7/2, 13/3, 11/3, 19/5, 29/7, 43/11, 53/13, 67/17. For example, if p and q are primes and q > 13, then 53/13 is closer to 3 than p/q is.

Cf. A000040, A023212, A062737, A090866, A120639, A162857.

x = 4; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265765/A120639 *)
Numerator[tL]   (* A162857 *)
Denominator[tL] (* A062737 *)
Numerator[tU]   (* A090866 *)
Denominator[tU] (* A023212 *)
Numerator[y]    (* A265765 *)
Denominator[y]  (* A120639 *)

A292024 a(n) is the smallest k such that n divides psi(k!) (k > 0).

Original entry on oeis.org

1, 3, 2, 3, 10, 3, 13, 4, 5, 10, 22, 3, 26, 13, 10, 4, 34, 5, 37, 10, 13, 22, 46, 4, 15, 26, 6, 13, 58, 10, 61, 5, 22, 34, 13, 5, 73, 37, 26, 10, 82, 13, 86, 22, 10, 46, 94, 4, 14, 15, 34, 26, 106, 6, 22, 13, 37, 58, 118, 10, 122, 61, 13, 6, 26, 22, 134, 34, 46, 13, 142, 5, 146, 73, 15, 37, 22, 26, 157

Offset: 1

Altug Alkan, Sep 07 2017

From Robert Israel, Sep 14 2017: (Start)
If m and n are coprime then a(m*n) = max(a(m),a(n)).
a(n) <= 2n.
Suppose p is a prime >= 5.  Then
  a(p) = 2p-1 if p is in A005382, otherwise 2p.
  a(p^2) = 2p if p is in A005382, otherwise 3p.
  a(p^3) = 3p if p is in A005382, 4p-1 if p is in A062737, otherwise 4p.
(End)

			a(4) = 3 because 4 divides psi(3!) = 12 and 3 is the least number with this property.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001615, A005382, A062737, A275985.

A:= proc(n) option remember;
    local F, p, e, t, k;
    F:= ifactors(n)[2];
    if nops(F)=1 then
      p:= F[1][1];
      e:= F[1][2];
      if p = 3 then
        t:= 1; if e =1 then return 2 fi
      else t:= 0:
      fi;
      for k from 2*p by p do
        if isprime(k-1) then
          t:= t+padic:-ordp(k, p);
          if t >= e then return(k-1) fi;
        fi;
        t:= t + padic:-ordp(k, p);
        if t >= e then return k fi;
      od
    else
      max(seq(procname(t[1]^t[2]), t=F))
    fi
end proc:
A(1):= 1:
map(A, [$1..100]); # Robert Israel, Sep 14 2017

psi[n_] := Module[{p, e}, Product[{p, e} = pe; p^e + p^(e-1), {pe, FactorInteger[n]}]];
a[n_] := Module[{k = 1}, While[!Divisible[psi[k!], n], k++]; k]; a[2] = 3;
Array[a, 100] (* Jean-François Alcover, Oct 15 2020, after PARI *)

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
a(n) = {my(k=1); while(a001615(k!) % n, k++); k; } \\ after Charles R Greathouse IV at A001615

A337055 Numbers that are k+A000010(k) for at least two different k.

Original entry on oeis.org

24, 32, 33, 38, 45, 48, 56, 64, 76, 86, 93, 96, 112, 113, 128, 140, 150, 152, 153, 172, 182, 192, 200, 203, 213, 216, 224, 225, 231, 256, 258, 263, 280, 293, 297, 300, 304, 320, 325, 326, 333, 342, 344, 345, 352, 364, 374, 380, 383, 384, 393, 397, 400, 402, 410, 413, 429, 432, 448, 459, 470, 473

Offset: 1

J. M. Bergot and Robert Israel, Aug 12 2020

nonn

If p>2 and 4*p-1 are prime, then 12*p-4 is in the sequence.

If p>3 and (5*p-1)/2 are prime, then 5*p-2 is in the sequence.

			a(3)=33 is in the sequence because 33 = 17 + A000010(17) = 21 + A000010(21).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A062737.

N:= 20000: # for terms <= N
V:= Vector(N):
for n from 1 to N do
  v:= n + numtheory:-phi(n);
  if v <= N then V[v]:= V[v]+1 fi
od:
select(t -> V[t]>=2, [$1..N]);

A200997 Terms of A135506 sorted (after removing the 1's).

Original entry on oeis.org

2, 2, 5, 5, 5, 5, 7, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 17, 17, 17, 19, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 31, 37, 41, 41, 41, 43, 43, 43, 43, 43, 47, 47, 47, 47, 47, 53, 53, 53, 59, 59, 59, 59, 59, 61, 61, 61, 61, 61, 61, 61, 67, 67, 67, 67, 67, 71, 71, 71, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 79, 83, 83, 83, 89

Offset: 1

Benoit Cloitre, Jan 08 2013

nonn

Terms of the sequence are primes and all primes except 3 appear finitely many times. p>=5 occurs 1 time iff 2p-1 is prime (cf. A005382). p>=11 occurs 3 times iff 4p-1 is prime (cf. A062737). In general p occurs 2m-1 times iff 2mp-1 is prime and 2ip-1 is composite for i=1,2,3,...,m-1.

v=[];u1=1;for(n=2,1000,u2=u1+lcm(n,u1);r=u1;u1=u2;if(u2/r-1>1,v=concat(v,[u2/r-1]);));w=vecsort(v);a(n)=w[n];

cp's OEIS Frontend

A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

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A158015 Primes p such that 6*p-1 is also prime.

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Magma

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A158017 Primes p such that 10*p-1 is also prime.

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Magma

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A158016 Primes p such that 8*p-1 is also prime.

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Magma

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A162857 Primes of the form 4p - 1, p a prime.

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A265765 Numerators of primes-only best approximates (POBAs) to 4; see Comments.

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A292024 a(n) is the smallest k such that n divides psi(k!) (k > 0).

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PARI

A337055 Numbers that are k+A000010(k) for at least two different k.

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