A075527 -id:A075527 - cp's OEIS frontend

A075526 a(n) = A008578(n+2) - A008578(n+1).

1, 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6

Offset: 0

Reinhard Zumkeller, Sep 22 2002

nonn

n appears this number of times in A000720. - Lekraj Beedassy, Jun 19 2006
a(0) = 1, for n >= 1: a(n) = differences between consecutive primes (A001223(n)) = A158611(n+2) - A158611(n+1). Partial sums give A006093 (shifted). - Jaroslav Krizek, Aug 04 2009
First differences of noncomposite numbers. - Juri-Stepan Gerasimov, Feb 17 2010
This is 1 together with A001223. A054541 is 2 together with A001223. A125266 is 3 together with A001223. - Omar E. Pol, Nov 01 2013

Cf. A000040, A001223, A075527, A158611.

Prime[Range[100]] // Differences // Prepend[#, 1]& (* Jean-François Alcover, Dec 11 2021 *)

a(n) = A001223(n) for n>0.

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010

A031131 Difference between n-th prime and (n+2)-nd prime.

Original entry on oeis.org

3, 4, 6, 6, 6, 6, 6, 10, 8, 8, 10, 6, 6, 10, 12, 8, 8, 10, 6, 8, 10, 10, 14, 12, 6, 6, 6, 6, 18, 18, 10, 8, 12, 12, 8, 12, 10, 10, 12, 8, 12, 12, 6, 6, 14, 24, 16, 6, 6, 10, 8, 12, 16, 12, 12, 8, 8, 10, 6, 12, 24, 18, 6, 6, 18, 20, 16, 12, 6, 10, 14, 14, 12, 10, 10, 14, 12, 12, 18, 12, 12, 12

Offset: 1

Jeff Burch

nonn

Distance between the pair of primes adjacent to the (n+1)-st prime. - Lekraj Beedassy, Oct 01 2004 [Typo corrected by Zak Seidov, Feb 22 2009]
A031131(A261525(n)) = 2*n and A031131(m) != 2*n for m < A261525(n). - Reinhard Zumkeller, Aug 23 2015
The Polymath project 8b proved that a(n) <= 395106 infinitely often (their published paper contains the slightly weaker bound a(n) <= 398130 infinitely often). - Charles R Greathouse IV, Jul 22 2016

			a(10)=8 because the 10th prime=29 is followed by primes 31 and 37, and 37 - 29 = 8.

T. D. Noe, Table of n, a(n) for n = 1..10000
D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Research in the Mathematical Sciences 1:12 (2014).
Polymath project, Bounded gaps between primes

Sum of consecutive terms of A001223.
Cf. A031132, A031133, A031134, A122412, A122413,A046931, A000040, A031165, A031166, A031167, A031168, A031169, A031170, A031171, A031172, A261525, A052288.
Cf. A075527 (allowing 1 to be prime).
First differences of A001043.

a031131 n = a031131_list !! (n-1)
a031131_list = zipWith (-) (drop 2 a000040_list) a000040_list
-- Reinhard Zumkeller, Dec 19 2013

[NthPrime(n+2)-NthPrime(n): n in [1..100] ]; // Vincenzo Librandi, Apr 11 2011

P:= select(isprime, [2,seq(2*i+1,i=1..1000)]):
P[3..-1] - P[1..-3]; # Robert Israel, Jan 25 2015

Differences[lst_]:=Drop[lst,2]-Drop[lst,-2]; Differences[Prime[Range[123]]] (* Vladimir Joseph Stephan Orlovsky, Aug 13 2009 *)
Map[#3 - #1 & @@ # &, Partition[Prime@ Range[84], 3, 1]] (* Michael De Vlieger, Dec 17 2017 *)

ithprime(i+2)-ithprime(i) $ i = 1..65 // Zerinvary Lajos, Feb 26 2007

a(n)=my(p=prime(n));nextprime(nextprime(p+1)+1)-p \\ Charles R Greathouse IV, Jul 01 2013

BB = primes_first_n(67)
L = []
for i in range(65):
    L.append(BB[2+i]-BB[i])
L
# Zerinvary Lajos, May 14 2007

a(n) = A001223(n) + A001223(n-1). - Lior Manor, Jan 19 2005
a(n) = A000040(n+2) - A000040(n).
a(n) = 2*A052288(n-1) for n>1. - Hugo Pfoertner, Apr 16 2025

Corrected by T. D. Noe, Sep 11 2008

Edited by N. J. A. Sloane, Sep 18 2008, at the suggestion of T. D. Noe

A076272 Largest prime factor of A076271(n): A006530(A076271(n)).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37, 37, 37, 37

Offset: 1

Reinhard Zumkeller, Oct 04 2002

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

See A180101 for a variant.

Differences[NestList[#+FactorInteger[#][[-1,1]]&,1,100]] (* Paolo Xausa, Dec 09 2023 *)

a(n) = A076271(n+1) - A076271(n) for all n;

a(A076273(k)+j) = A008578(k) for k>0 and 0 <= j < A075527(k-1).

A076273 a(0) = 1, a(1) = 2; for n>1, a(n) = prime(n)+prime(n-1)-1.

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 23, 29, 35, 41, 51, 59, 67, 77, 83, 89, 99, 111, 119, 127, 137, 143, 151, 161, 171, 185, 197, 203, 209, 215, 221, 239, 257, 267, 275, 287, 299, 307, 319, 329, 339, 351, 359, 371, 383, 389, 395, 409, 433, 449, 455, 461, 471, 479, 491, 507, 519, 531

Offset: 0

Reinhard Zumkeller, Oct 04 2002

Least m such that A076272(m) > A076272(m-1) for n>0; a(0)=1.

A076272(a(k)+j) = A008578(k) for k>0 and 0<=j < A075527(k-1).

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A001043.

nxt[{a_,b_}]:={a+1,Prime[a+1]+Prime[a]-1}; Join[{1},Transpose[ NestList[ nxt,{1,2},60]][[2]]] (* or *) Join[{1,2},Total/@Partition[Prime[ Range[ 60]],2,1]-1] (* Harvey P. Dale, Jun 12 2012 *)

a(n)=if(n<1,1,if(n==1,2,prime(n)+prime(n-1)-1)) \\ Lambert Klasen, Jan 14 2005; corrected by Michel Marcus, Nov 05 2023

a(n) = A001043(n-1)-1, n>1. - R. J. Mathar, Jun 04 2020

Simpler description from Vladeta Jovovic, Mar 29 2003

More terms from Lambert Klasen (lambert.klasen(AT)gmx.de), Jan 14 2005

A261468 a(n) = prime(n+2) mod prime(n).

Original entry on oeis.org

1, 1, 1, 6, 6, 6, 6, 10, 8, 8, 10, 6, 6, 10, 12, 8, 8, 10, 6, 8, 10, 10, 14, 12, 6, 6, 6, 6, 18, 18, 10, 8, 12, 12, 8, 12, 10, 10, 12, 8, 12, 12, 6, 6, 14, 24, 16, 6, 6, 10, 8, 12, 16, 12, 12, 8, 8, 10, 6, 12, 24, 18, 6, 6, 18, 20, 16, 12, 6, 10, 14, 14, 12

Offset: 1

Altug Alkan, Aug 20 2015

			a(3) = 11 mod 5 = 1.

Cf. A031131.

[NthPrime(n+2) mod NthPrime(n): n in [1..80]]; // Vincenzo Librandi, Aug 20 2015

a[n_]:=PowerMod[Prime[n + 2], 1, Prime[n]]; Table[a[n], {n, 80}] (* Vincenzo Librandi, Aug 20 2015 *)
Mod[#[[3]],#[[1]]]&/@Partition[Prime[Range[80]],3,1] (* Harvey P. Dale, Mar 14 2020 *)

first(m)=vector(m,i,prime(i+2)% prime(i)); \\ Anders Hellström, Aug 20 2015

a(n) = A031131(n) = A075527(n) for n>3. - Alois P. Heinz, Aug 20 2015

cp's OEIS Frontend

A075526 a(n) = A008578(n+2) - A008578(n+1).

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A031131 Difference between n-th prime and (n+2)-nd prime.

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A076272 Largest prime factor of A076271(n): A006530(A076271(n)).

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A076273 a(0) = 1, a(1) = 2; for n>1, a(n) = prime(n)+prime(n-1)-1.

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A261468 a(n) = prime(n+2) mod prime(n).

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