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 1-8 of 8 results.

A234095 Primes p such that 2*p + 1 is semiprime.

Original entry on oeis.org

7, 17, 19, 43, 47, 59, 61, 71, 79, 101, 107, 109, 149, 151, 163, 167, 197, 223, 257, 263, 271, 311, 317, 347, 349, 353, 383, 389, 401, 421, 439, 449, 461, 479, 503, 521, 523, 557, 569, 599, 601, 613, 631, 673, 677, 691, 701, 811, 821, 827, 839, 853, 863, 881
Offset: 1

Views

Author

Clark Kimberling, Dec 27 2013

Keywords

Comments

Also primes of the form (p*q - 1)/2, where p and q are distinct primes.

Examples

			7 is in the sequence because it is prime and 7*2 + 1 = 15 = 3*5 is a semiprime.

Links

Crossrefs

Cf. A233561, A234096, A233562.

Programs

  • Magma
    IsSemiprime:=func< p | &+[ k[2]: k in Factorization(p)] eq 2 >; [p: p in PrimesUpTo(1000)| IsSemiprime(2*p+1)]; // Vincenzo Librandi, Feb 21 2014
  • Mathematica
    t = Select[Range[1, 7000, 2], Map[Last, FactorInteger[#]] == Table[1, {2}] &]; Take[(t - 1)/2, 120] (* A234093 *)
v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A233561 *)
(w - 1)/2 (* A234095 *)  (* Peter J. C. Moses, Dec 23 2013 *)
Select[Prime[Range[200]],PrimeOmega[2#+1]==2&] (* Harvey P. Dale, Mar 19 2015 *)
  • PARI
    is(n)=isprime(n) && bigomega(2*n+1)==2 \\ Charles R Greathouse IV, Feb 19 2014

Formula

2*a(n)+1 = A233561(n). - R. J. Mathar, Aug 30 2016

Extensions

New name from Zak Seidov, Feb 19 2014

A234096 Integers of the form (p*q + 1)/2, where p and q are distinct primes.

Original entry on oeis.org

8, 11, 17, 18, 20, 26, 28, 29, 33, 35, 39, 43, 44, 46, 47, 48, 56, 58, 60, 62, 65, 67, 71, 72, 73, 78, 80, 81, 89, 92, 93, 94, 101, 102, 103, 105, 107, 108, 109, 110, 111, 118, 119, 124, 125, 127, 130, 133, 134, 144, 146, 148, 150, 151, 152, 153, 155, 160
Offset: 1

Views

Author

Clark Kimberling, Dec 27 2013

Keywords

Examples

			(3*5 + 1)/2 = 8, (3*7 + 1)/2 = 11.

Crossrefs

Cf. A234093, A233562, A234098.

Programs

  • Mathematica
    t = Select[Range[1, 7000, 2], Map[Last, FactorInteger[#]] == Table[1, {2}] &]; Take[(t + 1)/2, 120] (* A234096 *)
v = Flatten[Position[PrimeQ[(t + 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A233562 *)
(w + 1)/2 (* A234098 *)    (* Peter J. C. Moses, Dec 23 2013 *)

Formula

1 + A234093.

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

A233561 Products p*q of distinct primes such that (p*q - 1)/2 is prime.

Original entry on oeis.org

15, 35, 39, 87, 95, 119, 123, 143, 159, 203, 215, 219, 299, 303, 327, 335, 395, 447, 515, 527, 543, 623, 635, 695, 699, 707, 767, 779, 803, 843, 879, 899, 923, 959, 1007, 1043, 1047, 1115, 1139, 1199, 1203, 1227, 1263, 1347, 1355, 1383, 1403, 1623, 1643
Offset: 1

Views

Author

Clark Kimberling, Dec 14 2013

Keywords

Examples

			15 = 3*5 is the least product of distinct primes p and q for which (p*q - 1)/2 is prime, so a(1) = 15.

Links

Crossrefs

Cf. A234093, A234095, A046388.

Programs

  • Mathematica
    t = Select[Range[1, 7000, 2], Map[Last, FactorInteger[#]] == Table[1, {2}] &]; Take[(t - 1)/2, 120] (* A234093 *)
v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A233561 *)
(w - 1)/2 (* A234095 *)  (* Peter J. C. Moses, Dec 23 2013 *)
With[{upto=2000},Select[Times@@#&/@Select[Subsets[Prime[Range[ PrimePi[ upto/2]]],{2}],PrimeQ[(Times@@#-1)/2]&]//Union,#<=upto&]] (* Harvey P. Dale, Nov 02 2017 *)

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 *)

A234105 Integers of the form (p*q*r*s - 1)/2, where p, q, r, s are distinct primes.

Original entry on oeis.org

577, 682, 892, 997, 1072, 1207, 1402, 1501, 1522, 1567, 1627, 1657, 1852, 1897, 1942, 1963, 2152, 2194, 2242, 2257, 2320, 2392, 2422, 2467, 2502, 2557, 2593, 2656, 2782, 2827, 2932, 3022, 3052, 3097, 3139, 3202, 3272, 3277, 3349, 3382, 3391, 3517, 3547, 3580
Offset: 1

Views

Author

Clark Kimberling, Jan 01 2014

Keywords

Crossrefs

Cf. A234498, A234499, A234500, A234093, A234097.

Programs

  • Mathematica
    t = Select[Range[1, 20000, 2], Map[Last, FactorInteger[#]] == Table[1, {4}] &]; Take[(t - 1)/2, 120] (* A234105 *)
v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234498 *)
(w - 1)/2 (* A234499 *) (* Peter J. C. Moses, Dec 23 2013 *)

Formula

-1 + A234500.

A350101 Numbers k such that 2*k-1 and 2*k+1 are squarefree semiprimes (A046388).

Original entry on oeis.org

17, 28, 43, 46, 47, 71, 72, 80, 92, 93, 101, 102, 107, 108, 109, 110, 118, 124, 133, 150, 151, 152, 160, 161, 164, 170, 196, 197, 206, 207, 208, 223, 226, 235, 236, 258, 259, 267, 268, 272, 276, 290, 291, 295, 317, 334, 335, 340, 343, 344, 348, 349, 361, 377, 390
Offset: 1

Views

Author

Hugo Pfoertner, Dec 14 2021

Keywords

Examples

			a(1) = 17: 2*17 - 1 = 33 = 3*11 and 2*17 + 1 = 35 = 5*7 are both in A046388.

Links

Crossrefs

Cf. A046388.
Intersection of A234093 and A234096.

Programs

  • Maple
    N:= 1000: # for terms <= N
P:= select(isprime,[seq(i,i=3..2*N/3,2)]):
S:= NULL:
for i from 1 to nops(P) do
  for j from 1 to i-1 while P[i]*P[j] <= 2*N+1 do S:= S,P[i]*P[j] od
od:
S:= {S}:
T:= S intersect map(`-`,S,2):
sort(convert(map(t -> (t+1)/2, T),list)); # Robert Israel, Nov 11 2022
  • Mathematica
    semiQ[n_] := FactorInteger[n][[;; , 2]] == {1, 1}; Select[Range[400], AllTrue[2*# + {-1, 1}, semiQ] &] (* Amiram Eldar, Dec 14 2021 *)
  • PARI
    a350101(limit) = {my(sp(k)=omega(k)==2&&bigomega(k)==2);forstep(k=2,2*limit,2, if(sp(k-1)&&sp(k+1),print1(k/2,", ")))};
a350101(390)
Showing 1-8 of 8 results.

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