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.

Previous Showing 11-18 of 18 results.

A118747 a(n) = product[k=1..n] P(k), where P(k) is the largest prime <= 2*k. a(n) = product[k=1..n] A060308(k).

Original entry on oeis.org

2, 6, 30, 210, 1470, 16170, 210210, 2732730, 46456410, 882671790, 16770764010, 385727572230, 8871734161290, 204049885709670, 5917446685580430, 183440847252993330, 5686666264842793230, 176286654210126590130
Offset: 1

Views

Author

Jonathan Vos Post, Apr 29 2006

Keywords

Crossrefs

Cf. A020482, A049653, A060266-A060268, A060264, A060265, A060308, A118455, A118456, A118748.

A118752 a(n) = product[k=0..n] P(k), where P(k) is the smallest prime > 3*n. a(n) = product[k=0..n] A118751(k).

Original entry on oeis.org

2, 10, 70, 770, 10010, 170170, 3233230, 74364290, 2156564410, 62540367890, 1938751404590, 71733801969830, 2654150672883710, 108820177588232110, 4679267636293980730, 219925578905817094310, 11656055682008305998430
Offset: 0

Views

Author

Jonathan Vos Post, Apr 29 2006

Keywords

Comments

Analogous to A118456 a(n) = product{k=1..n} P(k), where P(k) is the smallest prime >= k.

Links

Crossrefs

Cf. A020482, A049653, A060266-A060268, A060264, A060265, A060308, A118455, A118456, A118748-51.

Programs

  • Mathematica
    Rest[FoldList[Times,1,Table[NextPrime[3n],{n,0,20}]]] (* Harvey P. Dale, Mar 09 2014 *)

Extensions

Definition corrected by Harvey P. Dale, Mar 09 2014

A118751 Smallest prime >= 3*n.

Original entry on oeis.org

2, 5, 7, 11, 13, 17, 19, 23, 29, 29, 31, 37, 37, 41, 43, 47, 53, 53, 59, 59, 61, 67, 67, 71, 73, 79, 79, 83, 89, 89, 97, 97, 97, 101, 103, 107, 109, 113, 127, 127, 127, 127, 127, 131, 137, 137, 139, 149, 149, 149, 151, 157, 157, 163, 163, 167, 173, 173, 179, 179, 181
Offset: 0

Views

Author

Jonathan Vos Post, Apr 29 2006

Keywords

Crossrefs

Analogous to A060264 = first prime after 2n.
Cf. A060308, A060264, A118750, A118752.

A118754 Smallest prime >= 5*n.

Original entry on oeis.org

2, 5, 11, 17, 23, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 97, 101, 107, 113, 127, 127, 127, 131, 137, 149, 149, 151, 157, 163, 167, 173, 179, 181, 191, 191, 197, 211, 211, 211, 223, 223, 227, 233, 239, 241, 251, 251, 257, 263, 269, 271, 277, 281
Offset: 0

Views

Author

Jonathan Vos Post, Apr 29 2006

Keywords

Links

Crossrefs

Cf. A007918, A008587, A060308, A060264, A118747-A118755.

Programs

  • Mathematica
    Table[If[PrimeQ[5n],5n,NextPrime[5n]],{n,0,60}] (* Harvey P. Dale, Nov 29 2024 *)
  • PARI
    a(n) = nextprime(5*n); \\ Michel Marcus, Feb 13 2021

Formula

a(n) = A007918(A008587(n)). - Michel Marcus, Feb 13 2021

A118755 Smallest prime >= 6*n.

Original entry on oeis.org

2, 7, 13, 19, 29, 31, 37, 43, 53, 59, 61, 67, 73, 79, 89, 97, 97, 103, 109, 127, 127, 127, 137, 139, 149, 151, 157, 163, 173, 179, 181, 191, 193, 199, 211, 211, 223, 223, 229, 239, 241, 251, 257, 263, 269, 271, 277, 283, 293, 307, 307
Offset: 0

Views

Author

Jonathan Vos Post, Apr 29 2006

Keywords

Crossrefs

Cf. A007918, A008588, A060308, A060264, A118747-A118754.
A002476 is a subsequence.

Programs

  • Mathematica
    Prime[1+PrimePi[6Range[0,50]]] (* T. D. Noe, Nov 15 2006 *)
NextPrime[6*Range[0,50]] (* Harvey P. Dale, Sep 05 2015 *)
  • PARI
    a(n) = nextprime(6*n); \\ Michel Marcus, Feb 13 2021

Formula

a(n) = A007918(A008588(n)). - Michel Marcus, Feb 13 2021

Extensions

Corrected by T. D. Noe, Nov 15 2006

A206770 Smallest number k such that sigma(k-2*n)=sigma(k)-2*n.

Original entry on oeis.org

5, 7, 11, 11, 13, 17, 17, 19, 23, 23, 29, 29, 21, 31, 37, 37, 37, 41, 28, 33, 47, 47, 53, 53, 53, 59, 59, 44, 61, 67, 67, 67, 71, 57, 73, 79, 79, 79, 83, 83, 69, 89, 74, 101, 68, 97, 97, 85, 101, 103, 107, 107, 109, 113, 93, 131, 127, 127, 131, 127, 127, 127
Offset: 1

Views

Author

Paolo P. Lava, Jan 10 2013

Keywords

Comments

Note all k>=1 are considered, even if k-2n<0. If the search space is k>=2n, variants of A020484 and A060264 appear. - R. J. Mathar, Jan 12 2013

Examples

			a(15)=37 because 37 is the minimum number for which sigma(37-2*15)=sigma(7)=8 and sigma(37)-2*15=38-30=8.

Crossrefs

Cf. A054906

Programs

  • Maple
    A206770:=proc(q)
local k,n;
for n from 1 to q do
for k from 1 to q do
  if sigma(k-2*n)=sigma(k)-2*n then print(k); break; fi;
od; od; end:
A206770(1000000000);
A206770 := proc(n)
    local k ;
    for k from 1 do
        if numtheory[sigma](k-2*n) = numtheory[sigma](k)-2*n then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Jan 12 2013

A107257 Smallest prime p such that for each j <= n there are primes a < b <= p whose difference b - a is 2*j.

Original entry on oeis.org

5, 7, 11, 11, 13, 17, 17, 19, 23, 23, 29, 29, 29, 31, 37, 37, 37, 41, 41, 43, 47, 47, 53, 53, 53, 59, 59, 59, 61, 67, 67, 67, 71, 71, 73, 79, 79, 79, 83, 83, 89, 89, 89, 101, 101, 101, 101, 101, 101, 103, 107, 107, 109, 113, 113, 131, 131, 131, 131, 131, 131, 131, 131
Offset: 1

Views

Author

Klaus Brockhaus, May 15 2005

Keywords

Comments

Every positive even number <= 2*n is the difference of two suitable primes <= a(n).
Sequence is nondecreasing, whereas the related sequence A020484 is not; first divergence is at 45: a(45) = 101, A020484(45) = 97.

Examples

			Consider n = 45: 89, 97, 101 are consecutive primes, 2*45 = 97 - 7, but 2*44 = 101 - 13 cannot be written as b - a where a and b are primes <=97, hence a(45) = 101.

Crossrefs

Cf. A060264, A020484.

A365241 a(n) is the n-th prime of the form 2*n + k where k > 0.

Original entry on oeis.org

3, 7, 13, 19, 23, 31, 41, 43, 53, 61, 67, 73, 79, 83, 97, 103, 107, 109, 127, 131, 139, 151, 157, 167, 173, 179, 191, 193, 197, 211, 227, 229, 233, 241, 251, 263, 271, 277, 281, 293, 307, 313, 317, 331, 347, 349, 353, 359, 373, 379, 389, 401, 409, 421, 433, 439
Offset: 1

Views

Author

Tamas Sandor Nagy, Aug 28 2023

Keywords

Examples

			a(3) = 13 because at k = 1, 2 * 3 + 1 = 7, this being the first prime result of the sum. At k = 2, the sum is 8, which is not a prime, so trying incremental k's, the second prime 11 is found with k = 5. The third prime at n = 3 and k = 7 is found to be 13, therefore a(3) = 13.

Crossrefs

Cf. A000040, A060264.

Programs

  • Maple
    f:= proc(n)
      ithprime(numtheory:-pi(2*n)+n)
end proc:
map(f, [$1..100]); # Robert Israel, Aug 28 2023
  • Mathematica
    a[n_] := NextPrime[2*n, n]; Array[a, 60] (* Amiram Eldar, Aug 28 2023 *)
  • PARI
    a(n) = prime(primepi(2*n) + n) \\ David A. Corneth, Aug 28 2023 after Jon E. Schoenfield
  • PARI
    first(n) = {my(res = vector(n), ind = 1, nextpp = 2, pp = 2); forprime(p = 3, oo, if(pp >= nextpp, res[ind] = p;if(isprime(2*ind+1), nextpp+=2,nextpp+=1);ind++;if(ind > n,return(res))); pp++;)} \\ David A. Corneth, Aug 28 2023
Previous Showing 11-18 of 18 results.

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