A073273 -id:A073273 - cp's OEIS frontend

A073274 A000040(n+1) - A073273(n).

0, 1, 0, 2, 0, 2, 0, 0, 3, -1, 2, 2, 0, 0, 1, 3, -1, 2, 2, -1, 2, 0, 0, 3, 2, 0, 2, 0, -4, 6, 0, 3, -3, 5, -1, 1, 2, 0, 1, 3, -3, 5, 0, 2, -4, 1, 5, 2, 0, 0, 3, -3, 3, 1, 1, 3, -1, 2, 2, -3, -1, 6, 2, 0, -4, 5, -1, 5, 0, 0, 0, 2, 1, 2, 0, 0, 3, -1, 0, 5, -3, 5, -1, 2, 0, 0, 3, 2, 0, -3, 3, 3, -1, 3

Offset: 1

Reinhard Zumkeller, Jul 22 2002

sign

			For n=11, A000040(11)*A000040(13) = 31*41 = 1271 = 35*35+46, therefore A073273(11)=35; a(11) = A000040(12)-A073273(11) = 37-35 = +2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A073272.

vector(80, n, prime(n+1) - floor(sqrt(prime(n)*prime(n+2)))) \\ Michel Marcus, Jun 02 2015

a(n,p=prime(n))=my(q=nextprime(p+1),r=nextprime(q+1)); q - sqrtint(p*r) \\ Charles R Greathouse IV, Jun 02 2015

A126990 Largest prime preceding geometric mean of prime(n) and prime(n+2).

Original entry on oeis.org

3, 3, 7, 7, 13, 13, 19, 23, 23, 31, 31, 37, 43, 47, 47, 53, 61, 61, 67, 73, 73, 83, 89, 89, 97, 103, 103, 109, 113, 113, 131, 131, 139, 139, 151, 151, 157, 167, 167, 173, 181, 181, 193, 193, 199, 199, 211, 223, 229, 233, 233, 241, 241, 251, 257, 263, 271, 271, 277

Offset: 1

Artur Jasinski, Jan 01 2007

nonn

With duplicates removed, seems to be a subsequence of A105399 and A105792. - M. F. Hasler, Jun 14 2007

P. Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007917, A073273.

<< NumberTheory`NumberTheoryFunctions` a = {}; Do[AppendTo[a,PreviousPrime[Sqrt[(Prime[x])*(Prime[x + 2])]]], {x, 1, 100}]; a

A126990(n)={ n=sqrtint(prime(n)*prime(n+2)); if( 0==n%2, n--); while(!isprime(n), n-=2); n } /* then vector(50,n,A126990(n)) displays a list of values, M. F. Hasler, Jun 14 2007 */

a(n)= precprime(sqrtint(prime(n)*prime(n+2))); \\ Michel Marcus, Nov 07 2013

a(n) = A007917(A073273(n)). - Michel Marcus, Nov 07 2013

Edited by M. F. Hasler, Jun 14 2007

Definition changed so that offset is now 1 by Michel Marcus, Nov 07 2013

A073271 a(n) = floor( prime(n)*prime(n+2) / prime(n+1) ).

Original entry on oeis.org

3, 4, 7, 8, 14, 14, 20, 23, 24, 34, 34, 38, 44, 48, 52, 54, 64, 64, 68, 76, 76, 84, 90, 92, 98, 104, 104, 110, 122, 116, 132, 132, 146, 140, 154, 156, 160, 168, 172, 174, 188, 182, 194, 194, 208, 210, 214, 224, 230, 234, 234, 248, 246, 256, 262, 264, 274, 274

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

A000040(n) < a(n) < A000040(n+2);
Conjecture: a(n) is even except for a(1)=3, a(3)=7 and a(8)=23 (A073272(n)=0).
Conjecture: when a(n) = a(n+1) then A001223(n+1) = A001223(n) + A001223(n+2). - Gionata Neri, May 31 2015

			A000040(10)*A000040(12)/A000040(11) = 29*37/31 = 1073/31 = (34*31+19)/31, therefore a(10)=34; A073272(10) = A000040(11)-a(10) = 31-34 = -3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A022462, A073272, A073273.

[Floor(NthPrime(n)*NthPrime(n+2) / NthPrime(n+1)): n in [1..60]]; // Vincenzo Librandi, May 31 2015

Table[Floor[Prime[n] Prime[n + 2] / Prime[n + 1]], {n, 60}] (* Vincenzo Librandi, May 31 2015 *)
Floor[(#[[1]]#[[3]])/#[[2]]]&/@Partition[Prime[Range[60]],3,1] (* Harvey P. Dale, Jun 07 2021 *)

vector(100, n, (prime(n)*prime(n+2))\prime(n+1)) \\ Michel Marcus, May 31 2015

A118469 Triangle read by rows: a(n,m) = If(n = 1, then 1, else Prime(n) - 1 + Sum_{k=n..m} (Prime(k + 1) - Prime(k))/2 ).

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 6, 7, 8, 1, 7, 8, 9, 11, 1, 9, 10, 11, 13, 14, 1, 10, 11, 12, 14, 15, 17, 1, 12, 13, 14, 16, 17, 19, 20, 1, 15, 16, 17, 19, 20, 22, 23, 25, 1, 16, 17, 18, 20, 21, 23, 24, 26, 29, 1, 19, 20, 21, 23, 24, 26, 27, 29, 32, 33, 1, 21, 22, 23, 25, 26, 28, 29, 31, 34, 35

Offset: 1

Roger L. Bagula, May 04 2006

An improved triangular Goldbach sequence in which the gap sum is taken from a start at n.

			1
1, 3
1, 4, 5
1, 6, 7, 8
1, 7, 8, 9, 11
1, 9, 10, 11, 13, 14
1, 10, 11, 12, 14, 15, 17
1, 12, 13, 14, 16, 17, 19, 20
1, 15, 16, 17, 19, 20, 22, 23, 25
1, 16, 17, 18, 20, 21, 23, 24, 26, 29

Main diagonal: A078444, 2nd diagonal: A073273.

Columns 1-8: A000012, A006254, A098090, A089253, A097069, A097338, A097480, A098605.

t[n_, m_] := If[n == 1, 1, Prime[n] + Sum[(Prime[k + 1] - Prime[k])/2, {k, n, m}] - 1]; Table[ t[n, m], {m, 11}, {n, m}] // Flatten

cp's OEIS Frontend

A073274 A000040(n+1) - A073273(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

A126990 Largest prime preceding geometric mean of prime(n) and prime(n+2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A073271 a(n) = floor( prime(n)*prime(n+2) / prime(n+1) ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A118469 Triangle read by rows: a(n,m) = If(n = 1, then 1, else Prime(n) - 1 + Sum_{k=n..m} (Prime(k + 1) - Prime(k))/2 ).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica