A063091 -id:A063091 - cp's OEIS frontend

A098974 Primes p such that q-p = 24, where q is the next prime after p.

1669, 2179, 4177, 4523, 4759, 5237, 6173, 6397, 6737, 7079, 7369, 7793, 8123, 8329, 9067, 11003, 11633, 11839, 12073, 12119, 13009, 13267, 16033, 16193, 16453, 16763, 16787, 17053, 17683, 17989, 18593, 18637, 19183, 19507, 20483, 22409, 22877, 23227

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Oct 23 2004

Lower prime of a difference of 24 between consecutive primes.

23 successive numbers after prime number p are composite. - Artur Jasinski, Jan 15 2007

Remi Eismann, Table of n, a(n) for n = 1..10000
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
Index entries for primes, gaps between

Cf. A000040, A001223, A054541, A075526, A001359, A054799, A063091, A096292, A015913, A023200, A029710, A031934, A031936, A031938.

Cf. A000230, A023200, A031924, A031926, A031928, A031930, A031932, A061779.

a = {}; Do[If[Prime[x + 1] - Prime[x] == 24, AppendTo[a, Prime[x]]], {x, 1, 10000}]; a (* Artur Jasinski, Jan 15 2007 *)

Entry revised by N. J. A. Sloane, Feb 13 2007

A204099 Number of integers between successive twin prime pairs.

Original entry on oeis.org

0, 3, 3, 9, 9, 15, 9, 27, 3, 27, 9, 27, 9, 3, 27, 9, 27, 9, 27, 33, 69, 9, 27, 57, 45, 27, 15, 21, 15, 147, 9, 3, 27, 21, 135, 9, 15, 9, 27, 57, 75, 45, 9, 9, 15, 105, 21, 27, 3, 117, 9, 45, 27, 21, 63, 81, 3, 51, 15, 45, 27, 51, 3, 21, 15, 9, 93, 27, 39

Offset: 1

Michel Lagneau, Jan 10 2012

nonn

a(n) is divisible by 3.

			a(1) = 0 because (3,5) is adjacent to (5,7); a(2) = 3 because the numbers 8, 9 and 10 are between (5,7) and (11,13), ...

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

Cf. A046933, A063091, A167132, A204100.

T:=array(1..100,1..2):k:=0:for n from 1 to 1000 do:p1:=ithprime(n):p2:=ithprime(n+1):if p2-p1 = 2 then k:=k+1:T[k,1]:=p1:T[k,2]:=p2:else fi:od: for p from 2 to k do:x:= T[p+1,1]- T[p,2]: printf(`%d, `,x-1):od:

Module[{tr=Transpose[Select[Partition[Prime[Range[450]],2,1],#[[2]]- #[[1]] == 2&]],fir,las},fir=Rest[tr[[1]]];las=Most[tr[[2]]];Flatten[Abs[ Differences/@ Thread[{fir,las}]]]-1/.{-1->0}] (* Harvey P. Dale, Jun 11 2014 *)

a(n) = A167132(n) - 1.

a(n) = A063091(n+1) - A063091(n) - 3.

A204100 Number of integers between successive twin primes, divided by 3.

Original entry on oeis.org

0, 1, 1, 3, 3, 5, 3, 9, 1, 9, 3, 9, 3, 1, 9, 3, 9, 3, 9, 11, 23, 3, 9, 19, 15, 9, 5, 7, 5, 49, 3, 1, 9, 7, 45, 3, 5, 3, 9, 19, 25, 15, 3, 3, 5, 35, 7, 9, 1, 39, 3, 15, 9, 7, 21, 27, 1, 17, 5, 15, 9, 17, 1, 7, 5, 3, 31, 9, 13, 9, 13, 55, 13, 21, 9, 7, 5, 19

Offset: 1

Michel Lagneau, Jan 10 2012

nonn

			a(2) = 1 because there exists three numbers 8, 9 and 10 between (5,7) and (11,13) => a(2) = 3/3 = 1.

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

Cf. A028334, A063091, A046933, A204099.

T:=array(1..100,1..2):k:=0:for n from 1 to 1000 do:p1:=ithprime(n):p2:=ithprime(n+1):if p2-p1 = 2 then k:=k+1:T[k,1]:=p1:T[k,2]:=p2:else fi:od: for p from 2 to k do:x:= T[p+1,1]- T[p,2]: printf(`%d, `,(x-1)/3):od:

Join[{0},Rest[(#[[2]]-#[[1]]-1)/3&/@Partition[Rest[Flatten[Select[ Partition[ Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&]]],2]]] (* Harvey P. Dale, Jan 10 2016 *)

a(n) = (A063091(n+1)- A063091(n)-3)/3 = A204099(n)/3

A066728 a(n) is the number of integers of the form (n+k+n*k)/(n-k) for k = 1,2,...,n-1.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 2, 4, 2, 7, 1, 7, 3, 5, 3, 8, 1, 11, 3, 7, 3, 9, 2, 9, 5, 7, 3, 15, 1, 13, 3, 6, 7, 11, 3, 11, 3, 9, 3, 19, 1, 15, 5, 7, 5, 11, 2, 17, 5, 11, 3, 15, 3, 19, 7, 9, 3, 15, 1, 15, 5, 7, 11, 15, 3, 15, 3, 15, 3, 29, 1, 14, 5, 7, 11, 15, 3, 23, 4, 11, 4, 15, 3, 15, 7, 9, 3, 29, 3, 23

Offset: 1

Benoit Cloitre, Jan 15 2002

a(n) = 1 iff n is 2 or the lesser of twin primes (for n >= 3, n follows the sequence A001359).

Also the number of factors of n*(n+2) which are less than n. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 02 2003

			(4 + 1 + 4*1)/(4 - 1), (4 + 2 + 4*2)/(4 - 2), and (4 + 3 + 4*3)/(4 - 1) are integers, hence a(4) = 3.

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

Cf. A000005, A001359, A063091.

with(numtheory):A066728 := n->ceil(tau(n*(n+2))/2)-1;

a[n_] := Ceiling[DivisorSigma[0, n*(n+2)] / 2]  - 1; Array[a, 100] (* Amiram Eldar, Feb 01 2025 *)

a(n) = ceil(numdiv(n*(n+2))/2) - 1; \\ Amiram Eldar, Feb 01 2025

a(n) = ceiling( d(n*(n+2)) / 2 ) - 1, where d(n) = number of divisors of n (A000005). - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 02 2003

A176180 Primes p such that gcd(p(n)-1, p(n+1)-1) != gcd(p(n)+1, p(n+1)+1).

Original entry on oeis.org

7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 89, 97, 103, 109, 113, 127, 131, 139, 151, 157, 163, 167, 173, 181, 193, 199, 211, 223, 229, 233, 241, 251, 257, 263, 271, 277, 293, 307, 313, 331, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 421

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 11 2010

nonn

Cf. A063091

lst={};Do[p0=Prime[n];p1=Prime[n+1];If[GCD[p0-1,p1-1]!=GCD[p0+1,p1+1],AppendTo[lst,p0]],{n,6!}];lst

A176181 Primes p(n) such that gcd(p(n)-1, p(n+1)-1) > gcd(p(n)+1, p(n+1)+1).

Original entry on oeis.org

13, 31, 37, 61, 73, 89, 97, 109, 113, 151, 157, 181, 193, 199, 211, 229, 241, 271, 277, 313, 331, 349, 367, 373, 389, 397, 401, 421, 433, 449, 457, 523, 541, 571, 601, 607, 613, 619, 631, 661, 673, 691, 701, 727, 733, 751, 757, 761, 769, 811, 853, 877, 929

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 11 2010

nonn

Sequence does not contain any lesser of twin primes A001359. (Proof. If p(n+1) = p(n)+2, then gcd(p(n)-1, p(n+1)-1) = 2 = gcd(p(n)+1, p(n+1)+1), so p(n) is not a term.) - Jonathan Sondow, Feb 03 2012

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

Cf. A063091, A176180.

lst={}; Do[p0=Prime[n]; p1=Prime[n+1]; If[GCD[p0-1,p1-1] > GCD[p0+1,p1+1], AppendTo[lst,p0]], {n,200}]; lst
Transpose[Select[Partition[Prime[Range[200]],2,1],GCD@@(#-1)>GCD@@(#+1)&]] [[1]] (* Harvey P. Dale, Sep 30 2014 *)

Definition clarified by Jonathan Sondow, Feb 03 2012

A176182 Primes p(n) such that gcd(p(n)-1, p(n+1)-1) < gcd(p(n)+1, p(n+1)+1).

Original entry on oeis.org

7, 19, 23, 43, 47, 53, 67, 79, 83, 103, 127, 131, 139, 163, 167, 173, 223, 233, 251, 257, 263, 293, 307, 353, 359, 379, 383, 409, 439, 443, 463, 467, 479, 487, 491, 499, 503, 509, 557, 563, 587, 593, 643, 647, 653, 677, 683, 709, 719, 739, 743, 797, 823, 829

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 11 2010

nonn

Cf. A063091, A176180, A176181.

lst={};Do[p0=Prime[n];p1=Prime[n+1];If[GCD[p0-1,p1-1]Harvey P. Dale, Jul 06 2011 *)

cp's OEIS Frontend

A098974 Primes p such that q-p = 24, where q is the next prime after p.

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A204099 Number of integers between successive twin prime pairs.

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A204100 Number of integers between successive twin primes, divided by 3.

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A066728 a(n) is the number of integers of the form (n+k+n*k)/(n-k) for k = 1,2,...,n-1.

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A176180 Primes p such that gcd(p(n)-1, p(n+1)-1) != gcd(p(n)+1, p(n+1)+1).

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A176181 Primes p(n) such that gcd(p(n)-1, p(n+1)-1) > gcd(p(n)+1, p(n+1)+1).

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A176182 Primes p(n) such that gcd(p(n)-1, p(n+1)-1) < gcd(p(n)+1, p(n+1)+1).

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