A184856 -id:A184856 - cp's OEIS frontend

A077545 Primes of the form floor(k*e).

2, 5, 13, 19, 29, 43, 59, 67, 73, 89, 97, 103, 127, 149, 157, 163, 173, 179, 233, 239, 241, 263, 269, 271, 277, 293, 307, 331, 337, 347, 353, 383, 421, 443, 467, 521, 557, 587, 617, 619, 641, 701, 709, 733, 739, 761, 769, 823, 829, 839, 853, 883, 907, 929, 937

Offset: 1

Amarnath Murthy, Nov 09 2002

Primes not in A077545 are in A184856, since {floor(k*e)} and {floor(j*e/(e-1))} are complementary Beatty sequences (A022843 and A054385).

Cf. A062409, A184855, A184856, A184587, A184858.

r=E; s=r/(r-1);
a[n_]:=Floor[n*r];
b[n_]:=Floor[n*s];
Table[a[n], {n, 1, 120}]  (* A022843 *)
t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1, a[n]]], {n, 1, 600}]; t1
t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2, n]], {n, 1, 600}]; t2
t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3, n]], {n, 1, 300}]; t3
t4={}; Do[If[PrimeQ[b[n]], AppendTo[t4, b[n]]], {n, 1, 600}]; t4
t5={}; Do[If[PrimeQ[b[n]], AppendTo[t5, n]], {n, 1, 600}]; t5
t6={}; Do[If[MemberQ[t4, Prime[n]], AppendTo[t6, n]], {n, 1, 300}]; t6
(* List t1 matches A077545; list t2 matches A062409;
lists t3-t6 match A184855-A184858. *)

More terms from Sascha Kurz, Jan 12 2003

Mathematica code and crossreferences by Clark Kimberling, Jan 24 2011

A373464 Largest of a quadruple of primes p[1..4] such that (p[k]+1, k=1..4) is in geometric progression.

Original entry on oeis.org

23, 47, 107, 191, 499, 647, 719, 809, 863, 1249, 1439, 1999, 2591, 2879, 3023, 3779, 4079, 5323, 6911, 7039, 7127, 7559, 8231, 8231, 8747, 9839, 10289, 10289, 10499, 10499, 10529, 10691, 11279, 11519, 12959, 13229, 13309, 13999, 15551, 15551, 15971, 18143, 19207

Offset: 1

M. F. Hasler, Jul 12 2024

nonn

a(10) = 1249 is the first term not in A299171, a(15) = 3023 is the first term not in A293194, a(17) = 4079 is the first term not in A347977 and also the first term not in A374482, and a(21) = 7127 is the first term not in A184856.

			The terms of the sequence are column "p[4]" in the following table which lists the sequences of primes, and ratios of the geometric progression (p[k]+1):
   n  | p[1], p[2], p[3], p[4]  |  r = (p[k+1]+1) / (p[k]+1)
------+-------------------------+---------------------------
   1  |    2,    5,   11,   23  |  2 = 6/3 = 12/6 = 24/12
   2  |    5,   11,   23,   47  |  2 = 12/6 = 24/12 = 48/24
   3  |   31,   47,   71,  107  |  3/2 = 48/32 = 72/48 = 108/72
   4  |    2,   11,   47,  191  |  4 = 12/3 = 48/12 = 192/48
   5  |   31,   79,  199,  499  |  5/2 = 80/32 = 200/80 = 500/200
   6  |    2,   17,  107,  647  |  6 = 18/3 = 108/18 = 648/108
   7  |   89,  179,  359,  719  |  2 = 180/90 = ...
   8  |   29,   89,  269,  809  |  3 = 90/30 = ...
   9  |  499,  599,  719,  863  |  6/5 = 600/500 = ...
  10  |   79,  199,  499, 1249  |  5/2 = 200/80 = ...
  11  |  179,  359,  719, 1439  |  2 = 360/180 = ...
  12  |   53,  179,  599, 1999  |  10/3 = 180/54 = ...

Chai Wah Wu, Table of n, a(n) for n = 1..2372
Doddy Kastanya, Fun Math #241, Number Theory group on LinkedIn.com, Jul 04 2024

Subsequence of A089199 (primes p such that p+1 is divisible by a cube).

A373464_upto(N, show=0, D = 1, LIM=N\2) = { my(L=List()); forprime(p=1, LIM, my(denom = p+D); for(numer=denom+1, sqrtnint((N+D) * denom^2, 3), my(r=numer/denom); for(k=1,3, (type(denom * r^k)=="t_INT" && isprime(denom * r^k - D)) || next(2)); listput(L, denom * r^3 - D); show && printf(" | %4d, %4d, %4d, %4d | %s\n",denom-D, denom*r-D, denom*r^2-D, denom*r^3-D, numer/denom))); vecsort(L)}

from itertools import islice
from fractions import Fraction
from sympy import nextprime
def A373464_gen(): # generator of terms
    p, plist, pset = 1, [], set()
    while True:
        p = nextprime(p)
        for q in plist:
            r = Fraction(q+1,p+1)
            q2 = r*(q+1)-1
            if q2 < 2:
                break
            if q2.denominator == 1:
                q2 = int(q2)
                if q2 in pset:
                    q3 = r*(q2+1)-1
                    if q3 < 2:
                        break
                    if q3.denominator == 1 and int(q3) in pset:
                        yield p
        plist = [p]+plist
        pset.add(p)
A373464_list = list(islice(A373464_gen(),20)) # Chai Wah Wu, Jul 16 2024

a(26)-a(43) from Chai Wah Wu, Jul 16 2024

A184857 Numbers k such that floor(k*e/(e-1)) is prime.

Original entry on oeis.org

2, 5, 7, 11, 15, 20, 24, 26, 30, 34, 39, 45, 50, 53, 64, 68, 69, 72, 83, 87, 88, 96, 106, 115, 121, 122, 125, 126, 134, 141, 144, 145, 159, 163, 178, 179, 197, 198, 201, 221, 227, 232, 236, 240, 246, 251, 254, 259, 265, 273, 274, 278, 284, 289, 292, 293, 303, 308, 311, 316, 318, 322, 331, 342, 346, 356, 360, 361, 365, 375, 379, 380, 384, 388, 399, 407, 409, 413, 417, 418, 426, 428, 432, 437, 455, 460, 470, 475, 479, 489, 498, 504, 512, 513, 519, 523, 542, 543, 546, 555, 557, 561, 576, 581, 595, 599

Offset: 1

Clark Kimberling, Jan 23 2011

nonn

			See A077545.

Cf. A077545, A184856, A184858.

(See A077545.)
Select[Range[600],PrimeQ[Floor[# E/(E-1)]]&]  (* Harvey P. Dale, Jan 25 2011 *)

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A077545 Primes of the form floor(k*e).

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A373464 Largest of a quadruple of primes p[1..4] such that (p[k]+1, k=1..4) is in geometric progression.

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A184857 Numbers k such that floor(k*e/(e-1)) is prime.

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