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.

A364411 a(n) = prime(n) + 2*prime(n+1).

Original entry on oeis.org

8, 13, 19, 29, 37, 47, 55, 65, 81, 91, 105, 119, 127, 137, 153, 171, 181, 195, 209, 217, 231, 245, 261, 283, 299, 307, 317, 325, 335, 367, 389, 405, 415, 437, 451, 465, 483, 497, 513, 531, 541, 563, 577, 587, 595, 621, 657, 677, 685, 695, 711, 721, 743, 765, 783
Offset: 1

Views

Author

Paul Curtz, Jul 23 2023

Keywords

Comments

All terms > 8 are odd.

Links

Crossrefs

Cf. A000040, A001043, A062234, A094105, A100484, A191472 (first differences), A210497.

Programs

  • Mathematica
    ListConvolve[{2,1},Prime[Range[100]]] (* Paolo Xausa, Nov 02 2023 *)

Formula

a(n) = a(n-1) + A191472(n-1).
a(n) = A000040(n) + A100484(n+1).
a(n) = A000040(n+1) + A001043(n).

A211280 Numerator of prime(n+1) - prime(n)/2.

Original entry on oeis.org

2, 7, 9, 15, 15, 21, 21, 27, 35, 33, 43, 45, 45, 51, 59, 65, 63, 73, 75, 75, 85, 87, 95, 105, 105, 105, 111, 111, 117, 141, 135, 143, 141, 159, 153, 163, 169, 171, 179, 185, 183, 201, 195, 201, 201, 223, 235, 231, 231, 237, 245, 243, 261, 263, 269, 275, 273, 283, 285, 285, 303, 321, 315, 315, 321, 345, 343, 357, 351, 357, 365, 375, 379
Offset: 1

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Author

Paul Curtz, Jul 05 2012

Keywords

Comments

Second row of the inverse semi-binomial transform of A000040(n+1) as introduced in A213268.
The list of denominators is 1, 2, 2, ... (2 repeated), so a(n) = A210497(n) for n>1.
a(n) - prime(n) = 2*prime(n+1)-prime(n)-prime(n) are prime differences (A001223) multiplied by 2, and therefore multiples of 4.

Crossrefs

Denominators are A040000.

Programs

  • Maple
    A211280 := proc(n)
        ithprime(n+1)-ithprime(n)/2 ;
        numer(%) ;
end proc: # R. J. Mathar, Jul 10 2012
  • Mathematica
    Numerator[#[[2]]-#[[1]]/2]&/@Partition[Prime[Range[80]],2,1] (* Harvey P. Dale, Mar 05 2023 *)

Formula

a(n) ~ n log n. Apart from the first term, a(n) = 2*prime(n+1) - prime(n). - Charles R Greathouse IV, Jul 10 2012
a(n) = prime(n+2) - A036263(n), n>1. - R. J. Mathar, Jul 10 2012

A376762 Number of composite numbers c in the range prime(n) < c <= 2*prime(n+1).

Original entry on oeis.org

2, 5, 6, 11, 11, 16, 16, 21, 28, 25, 33, 35, 35, 41, 47, 51, 50, 59, 60, 61, 69, 71, 78, 85, 84, 85, 91, 92, 98, 117, 111, 117, 115, 131, 126, 134, 140, 142, 150, 154, 152, 168, 162, 168, 168, 187, 196, 192, 192, 197, 205, 203, 219, 220, 225, 232, 230, 240, 242, 242, 258, 271, 264, 265, 271, 290, 288, 300, 295, 301, 309, 317, 320, 325, 327, 334, 344, 344, 355, 364, 358
Offset: 1

Views

Author

N. J. A. Sloane, Oct 29 2024

Keywords

Examples

			a(2) = 5 because there are 5 composite numbers c in the range 3 < c <= 10, namely 4, 6, 8, 9, and 10.

Links

Crossrefs

Cf. A000720, A020900, A065855, A210497, A376759.

Programs

  • Mathematica
    A376762[n_] := n - Prime[n] + 2*Prime[n+1] - PrimePi[2*Prime[n+1]];
Array[A376762, 100] (* Paolo Xausa, Oct 29 2024 *)
  • Python
    from sympy import prime, nextprime, primepi
def A376762(n): return int(n-(p:=prime(n))+(q:=nextprime(p)<<1)-primepi(q)) # Chai Wah Wu, Oct 29 2024

Formula

a(n) = 2*q - pi(2*q) - p + n, where p = prime(n), q = prime(n+1), and pi() = A000720().
a(n) = A210497(n) - A020900(n+1) + n. - Paolo Xausa, Oct 29 2024

A224895 Let p = prime(n). Smallest odd number m > p such that m + p is semiprime.

Original entry on oeis.org

7, 7, 9, 15, 15, 21, 21, 27, 35, 33, 43, 45, 45, 51, 59, 65, 63, 73, 75, 75, 85, 87, 95, 105, 105, 105, 111, 111, 117, 141, 135, 143, 141, 159, 153, 163, 169, 171, 179, 185, 183, 201, 195, 201, 201, 223, 235, 231, 231, 237, 245, 243, 261, 263, 269, 275, 273
Offset: 1

Views

Author

Zak Seidov, Jul 24 2013

Keywords

Comments

Apparently a(n) = A210497(n) for n>1, which basically indicates that the search for the smallest even semiprime larger than 2*prime(n) produces 2*prime(n+1). - R. J. Mathar, Jul 27 2013
a(n) <= A165138(n); a(n) = A165138(n) when a(n) is prime, corresponding n's: 1, 2, 11, 15, 18, 36, 39, 46, 54, 55, 58, 73, 91,.. .
Also of interest is that sequence in not monotonic: e.g., a(10) - a(9) = 33 - 35 = -2, a(31) - a(30) = 135 - 141 = -6.

Examples

			2 + 7 = 9 = 3*3, 3 + 7 = 10 = 2*5, 5 + 9 = 14 = 2*7.

Crossrefs

Cf. A000040, A001358, A084704, A165138, A211280, A210497.

Programs

  • Maple
    A224895 := proc(n)
    local p,m ;
    p := ithprime(n) ;
    for m from p+1 do
        if type(m,'odd') and numtheory[bigomega](m+p) = 2 then
            return m ;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 28 2013
  • Mathematica
    Reap[Sow[7];Do[p=Prime[n];k=p+2;While[!PrimeQ[r=(p+k)/2],k=k+2];Sow[k],{n,2,100}]][[2,1]]
son[n_]:=Module[{m=If[EvenQ[n],n+1,n+2]},While[PrimeOmega[n+m]!=2,m = m+2]; m]; Table[son[n],{n,Prime[Range[60]]}] (* Harvey P. Dale, Apr 24 2017 *)
Showing 1-4 of 4 results.

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

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