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.

Showing 1-6 of 6 results.

A234099 Integers of the form (p*q*r - 1)/2, where p, q, r are distinct primes.

Original entry on oeis.org

52, 82, 97, 115, 127, 136, 142, 172, 178, 192, 199, 214, 217, 227, 232, 241, 277, 280, 297, 304, 307, 313, 322, 325, 331, 332, 352, 357, 370, 379, 388, 397, 402, 430, 442, 448, 451, 457, 467, 478, 484, 493, 500, 502, 507, 511, 522, 532, 542, 547, 552, 556
Offset: 1

Views

Author

Clark Kimberling, Dec 27 2013

Keywords

Examples

			52 = (3*5*7 - 1)/2.

Crossrefs

Cf. A046389, A234100, A234101, A234093, A234102.

Programs

  • Mathematica
    t = Select[Range[1, 10000, 2], Map[Last, FactorInteger[#]] == Table[1, {3}] &]; Take[(t - 1)/2, 120] (* A234099 *)
v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234100 *)
(w - 1)/2 (* A234101 *)    (* Peter J. C. Moses, Dec 23 2013 *)
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A234099(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    return bisection(f,n,n)>>1 # Chai Wah Wu, Oct 18 2024

Formula

-1 + A234102.
a(n) = (A046389(n)-1)/2. - Chai Wah Wu, Oct 18 2024

A234103 Products p*q*r of distinct primes for which (p*q*r + 1)/2 is prime.

Original entry on oeis.org

105, 165, 273, 345, 357, 385, 465, 561, 705, 777, 861, 885, 897, 957, 1005, 1045, 1113, 1173, 1185, 1281, 1353, 1545, 1645, 1653, 1677, 1705, 1905, 1965, 2037, 2065, 2121, 2185, 2193, 2233, 2301, 2373, 2445, 2553, 2613, 2865, 2877, 2905, 2985, 3021, 3157
Offset: 1

Views

Author

Clark Kimberling, Dec 27 2013

Keywords

Examples

			105 = 3*5*7, and (105 + 1)/2 is prime.

Crossrefs

Cf. A234100, A234102, A234104.

Programs

  • Mathematica
    t = Select[Range[1, 10000, 2], Map[Last, FactorInteger[#]] == Table[1, {3}] &]; Take[(t + 1)/2, 120] (* A234102 *)
v = Flatten[Position[PrimeQ[(t + 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234103 *)
(w + 1)/2  (* A234104 *)    (* Peter J. C. Moses, Dec 23 2013 *)
With[{nn=50},Select[Union[Select[Times@@@Subsets[Prime[Range[2,nn]],{3}],PrimeQ[(#+1)/2]&]],#<=15*Prime[nn]&]]  (* Harvey P. Dale, May 12 2025 *)

A234104 Primes of the form (p*q*r + 1)/2, where p, q, r are distinct primes.

Original entry on oeis.org

53, 83, 137, 173, 179, 193, 233, 281, 353, 389, 431, 443, 449, 479, 503, 523, 557, 587, 593, 641, 677, 773, 823, 827, 839, 853, 953, 983, 1019, 1033, 1061, 1093, 1097, 1117, 1151, 1187, 1223, 1277, 1307, 1433, 1439, 1453, 1493, 1511, 1579, 1583, 1601, 1619
Offset: 1

Views

Author

Clark Kimberling, Dec 27 2013

Keywords

Examples

			(3*5*7 + 1)/2 = 53.

Links

Crossrefs

Cf. A234102, A234103, A234101.

Programs

  • Maple
    filter:= proc(n) local s;
  if not isprime(n) then return false fi;
  s:= ifactors(2*n-1)[2];
  nops(s)=3 and map(t -> t[2],s)=[1,1,1]
end proc:
select(filter, [seq(i,i=3..1619,2)]); # Robert Israel, May 11 2020
  • Mathematica
    t = Select[Range[1, 10000, 2], Map[Last, FactorInteger[#]] == Table[1, {3}] &]; Take[(t + 1)/2, 120] (* A234102 *)
v = Flatten[Position[PrimeQ[(t + 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234103 *)
(w + 1)/2  (* A234104 *)    (* Peter J. C. Moses, Dec 23 2013 *)
Module[{nn=100},Select[(Times@@#+1)/2&/@Subsets[Prime[Range[nn]],{3}],PrimeQ[ #] && #<=5*Prime[nn]&]]//Union (* Harvey P. Dale, Jan 29 2023 *)

A234100 Products p*q*r of distinct primes for which (p*q*r - 1)/2 is prime.

Original entry on oeis.org

195, 255, 399, 455, 483, 555, 615, 627, 663, 759, 795, 915, 935, 1095, 1235, 1239, 1295, 1419, 1455, 1479, 1515, 1547, 1595, 1659, 1767, 1955, 2067, 2139, 2235, 2247, 2343, 2387, 2555, 2595, 2607, 2639, 2847, 2895, 2919, 2967, 3219, 3243, 3335, 3395, 3399
Offset: 1

Views

Author

Clark Kimberling, Dec 27 2013

Keywords

Examples

			97 = (3*5*13 - 1)/2, and 3*5*13 is the least product p*q*r of 3 distinct primes for which (p*q*r - 1)/2 is prime, so a(1) = 3*5*13.

Links

Crossrefs

Cf. A234099, A234101, A234093, A234102.

Programs

  • Mathematica
    t = Select[Range[1, 10000, 2], Map[Last, FactorInteger[#]] == Table[1, {3}] &]; Take[(t - 1)/2, 120] (* A234099 *)
v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* this sequence *)
(w - 1)/2 (* A234101 *)    (* Peter J. C. Moses, Dec 23 2013 *)
With[{upto=4000},Select[Union[Times@@@Select[Subsets[Prime[ Range[ PrimePi[ upto/ 6]]],{3}],PrimeQ[(Times@@#-1)/2]&]],#<=upto&]] (* Harvey P. Dale, May 12 2017 *)

A234101 Primes of the form (p*q*r - 1)/2, where p, q, r are distinct primes.

Original entry on oeis.org

97, 127, 199, 227, 241, 277, 307, 313, 331, 379, 397, 457, 467, 547, 617, 619, 647, 709, 727, 739, 757, 773, 797, 829, 883, 977, 1033, 1069, 1117, 1123, 1171, 1193, 1277, 1297, 1303, 1319, 1423, 1447, 1459, 1483, 1609, 1621, 1667, 1697, 1699, 1747, 1753
Offset: 1

Views

Author

Clark Kimberling, Dec 27 2013

Keywords

Examples

			(See A234100.)

Crossrefs

Cf. A234099, A234100, A234093, A234102.

Programs

  • Mathematica
    t = Select[Range[1, 10000, 2], Map[Last, FactorInteger[#]] == Table[1, {3}] &]; Take[(t - 1)/2, 120] (* A234099 *)
v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234100 *)
(w - 1)/2 (* A234101 *)    (* Peter J. C. Moses, Dec 23 2013 *)

A376734 Numbers k such that 2*k-1 and 2*k+1 are products of exactly three distinct odd primes (A046389).

Original entry on oeis.org

332, 655, 773, 943, 1007, 1018, 1033, 1046, 1117, 1172, 1277, 1333, 1358, 1369, 1424, 1622, 1667, 1783, 1810, 1828, 1865, 1907, 1928, 2008, 2216, 2252, 2293, 2348, 2404, 2447, 2473, 2518, 2567, 2608, 2645, 2698, 2711, 2726, 2797, 2898, 2942, 2972, 2978, 3031, 3048, 3049
Offset: 1

Views

Author

Hugo Pfoertner, Oct 18 2024

Keywords

Links

Crossrefs

Cf. A046389, A234099, A234102.

Programs

  • Maple
    q:= k-> andmap(x-> map(i-> i[2], ifactors(x)[2])=[1$3], [2*k-1, 2*k+1]):
select(q, [$1..4000])[];  # Alois P. Heinz, Oct 18 2024
  • PARI
    is_a046389(k) = k%2 && omega(k)==3 && bigomega(k)==3;
is_a376734(n) = is_a046389(2*n-1) && is_a046389(2*n+1)
Showing 1-6 of 6 results.

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