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 1-4 of 4 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

A128283 Numbers of the form m = p1 * p2 where for each d|m we have (d+m/d)/2 prime and p1 < p2 both prime.

Original entry on oeis.org

21, 33, 57, 85, 93, 133, 145, 177, 205, 213, 217, 253, 393, 445, 553, 565, 633, 697, 793, 817, 865, 913, 933, 973, 1137, 1285, 1345, 1417, 1437, 1465, 1477, 1513, 1537, 1717, 1765, 1837, 1857, 1893, 2101, 2173, 2245, 2305, 2517, 2577, 2581, 2605, 2641, 2653, 2733, 2761
Offset: 1

Views

Author

Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007

Keywords

Comments

The symmetric representation of sigma (A237593) for p1*p2, SRS(p1*p2), consists of either 4 or 3 regions. Let p1 < p2. Then 2*p1 < p2 implies that SRS(p1*p2), consists of 2 pairs of regions of widths 1 having respective sizes (p1*p2 + 1)/2 and (p1 + p2)/2; and p2 < 2*p1 implies that SRS(p1*p2) consists of 2 outer regions of width 1 and size (p1*p2 + 1)/2 and a central region of maximum width 2 of size p1 + p2 . Therefore, if SRS(p1*p2) has four regions, the area of each is a prime number (see A233562) and if it has three regions, the central area is an even semiprime (A100484). - Hartmut F. W. Hoft, Jan 09 2021
Old name was: "a(n) is the n-th smallest product of two distinct odd primes m=p1*p2 with the property that (d+m/d)/2 are all primes for each d dividing m.". - David A. Corneth, Jan 09 2021

Examples

			85=5 * 17, (5 * 17+1)/2=43, (5+17)/2=11 are both primes and 85 is in the sequence.
From _Hartmut F. W. Hoft_, Jan 09 2021: (Start)
9=3*3 is not in the sequence even though (1+9)/2 and (3+3)/2 are primes, see also A340482.
a(33) = 1537 = 29*53 is the first number for which the symmetric representation of sigma consists of three regions ( 769, 82, 769 ) with 5 units of width 2 straddling the diagonal in the central region; (1537+1)/2 = 769 and (29+53)/2 = 41 are primes. (End)

Links

Crossrefs

Cf. A128281, A005383, A128284, A128285, A128286.
Cf. A100484, A233562, A237591, A237593, A249223, A262045, A340482.
Subsequence of A046388.

Programs

  • Mathematica
    ppQ[s_, k_] := Last[Transpose[FactorInteger[s]]]==Table[1, k]
dQ[s_] := Module[{d=Divisors[s]}, AllTrue[Map[(d[[#]]+d[[-#]])/2&, Range[Length[d]/2]], PrimeQ]]
goodL[{m_, n_}, k_] := Module[{i=m, list={}}, While[i<=n, If[ppQ[i, k] && dQ[i], AppendTo[list, i]]; i+=2]; list]/;OddQ[m]
a128283[n_] := goodL[{1, n}, 2]
a128283[2761] (* Hartmut F. W. Hoft, Jan 09 2021 *)

Extensions

Added "distinct" for clarification since 9 satisfies the divisor property. See also A340482. - Hartmut F. W. Hoft, Jan 09 2021
New name from David A. Corneth, Jan 09 2021

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.

A234098 Primes of the form (p*q + 1)/2, where p and q are distinct primes.

Original entry on oeis.org

11, 17, 29, 43, 47, 67, 71, 73, 89, 101, 103, 107, 109, 127, 151, 191, 197, 223, 227, 241, 251, 269, 277, 283, 317, 349, 359, 373, 397, 409, 433, 457, 461, 467, 487, 521, 541, 569, 571, 631, 643, 647, 659, 673, 701, 709, 719, 733, 739, 751, 757, 769, 821
Offset: 1

Views

Author

Clark Kimberling, Dec 27 2013

Keywords

Examples

			11 = (3*7 + 1)/2, 17 = (5*7 + 1)/2.

Links

Crossrefs

Cf. A234096, A233562, A234095.
Cf. A010051, A046388.

Programs

  • Haskell
    a234098 n = a234098_list !! (n-1)
a234098_list = filter ((== 1) . a010051') $
                      map ((flip div 2) . (+ 1)) a046388_list
-- Reinhard Zumkeller, Jan 02 2014
  • 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 *)
Take[Select[(Times@@#+1)/2&/@Subsets[Prime[Range[200]],{2}],PrimeQ]//Union,60] (* Harvey P. Dale, Jun 24 2025 *)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.