A001223 -id:A001223 - cp's OEIS frontend

A347123 Fully multiplicative with a(prime(k)) = prime(1+floor(A001223(k)/2)).

1, 2, 3, 4, 3, 6, 5, 8, 9, 6, 3, 12, 5, 10, 9, 16, 3, 18, 5, 12, 15, 6, 7, 24, 9, 10, 27, 20, 3, 18, 7, 32, 9, 6, 15, 36, 5, 10, 15, 24, 3, 30, 5, 12, 27, 14, 7, 48, 25, 18, 9, 20, 7, 54, 9, 40, 15, 6, 3, 36, 7, 14, 45, 64, 15, 18, 5, 12, 21, 30, 3, 72, 7, 10, 27, 20, 15, 30, 5, 48, 81, 6, 7, 60, 9, 10, 9, 24, 11, 54

Offset: 1

Antti Karttunen, Aug 19 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for primes, gaps between

Cf. A001222, A001223, A003586, A007814, A347101, A347102.

Cf. also A336855.

A347123(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 1] = prime(1+((nextprime(f[i, 1]+1)-f[i,1])\2))); factorback(f); };

For all n >= 1, A001222(a(n)) = A001222(n), A007814(a(n)) = A007814(n).

For all n >= 1, a(A003586(n)) = A003586(n).

A376520 Position of first appearance of 2n in the run-compression (A037201) of the first differences (A001223) of the prime numbers (A000040).

Original entry on oeis.org

2, 3, 8, 22, 32, 42, 28, 259, 91, 141, 172, 242, 341, 400, 556, 692, 198, 1119, 3126, 2072, 1779, 1737, 7596, 2913, 3246, 2101, 3598, 7651, 4383, 4294, 3457, 8284, 14220, 11986, 15101, 3204, 32808, 18217, 16273, 42990, 22303, 37037, 13729, 43117, 32820, 70501

Offset: 1

Gus Wiseman, Sep 26 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, we can remove all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The sequence of prime numbers (A000040) is:
  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, ...
with first differences (A001223):
  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, ...
with run-compression (A037201):
  1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, ...
with first appearance of 2n at (A376520):
  2, 3, 8, 22, 32, 42, 28, 259, 91, 141, 172, 242, 341, 400, 556, 692, 198, 1119, ...

This is the position of first appearance of 2n in A037201.
For positions of twos instead of first appearances we have A376343.
The sorted version is A376521.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
A116861 counts partitions by compressed sum, compositions A373949.
A116608 counts partitions by compressed length, compositions A333755.
A274174 counts contiguous compositions, ranks A374249.
A333254 lists run-lengths of differences between consecutive primes.
A373948 encodes compression using compositions in standard order.
Cf. A007921, A030173, A064113, A373817, A373822, A373947, A376305, A376308, A376312.

mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=First/@Split[Differences[Select[Range[10000],PrimeQ]]];
Table[Position[q,2k][[1,1]],{k,mnrm[Rest[q]/2]}]

A052377 Primes followed by an [8,4,8]=[d,D-d,d] prime difference pattern of A001223.

Original entry on oeis.org

389, 479, 1559, 3209, 8669, 12269, 12401, 13151, 14411, 14759, 21851, 28859, 31469, 33191, 36551, 39659, 40751, 50321, 54311, 64601, 70229, 77339, 79601, 87671, 99551, 102539, 110261, 114749, 114761, 118661, 129449, 132611, 136511

Offset: 1

Labos Elemer, Mar 22 2000

nonn

A subsequence of A031926. [Corrected by Sean A. Irvine, Nov 07 2021]
a(n)=p, the initial prime of two consecutive 8-twins of primes as follows: [p,p+8] and [p+12,p+12+8], d=8, while the distance of the two 8-twins is 12 (minimal; see A052380(4/2)=12).
Analogous sequences are A047948 for d=2, A052378 for d=4, A052376 for d=10 and A052188-A052199 for d=6k, so that in the [d,D-d,d] difference patterns which follows a(n) the D-d is minimal(=0,2,4; here it is 4).

			p=1559 begins the [1559,1567,1571,1579] prime quadruple consisting of two 8-twins [1559,1567] and[1571,1579] which are in minimal distance, min{D}=1571-1559=12=A052380(8/2).

Cf. A031926, A053325, A052380, A052376, A052378, A052188-A052190.

a(n) is the initial term of a [p, p+8, p+12, p+12+8] quadruple of consecutive primes.

A062357 a(n) = np(n+1)-(n+1)p(n) = n*d(n)-p(n), where p(n) is the n-th prime and d(n) is the n-th prime-difference, A001223(n).

Original entry on oeis.org

-1, 1, 1, 9, -1, 11, -3, 13, 31, -9, 35, 11, -15, 13, 43, 43, -25, 47, 9, -31, 53, 9, 55, 103, 3, -49, 5, -51, 7, 307, -3, 61, -71, 201, -79, 65, 65, -11, 67, 67, -97, 239, -105, -17, -107, 353, 353, -31, -129, -29, 73, -135, 289, 73, 73, 73, -155, 77, -41, -161, 327, 575, -55, -183, -53, 607, 71, 343, -209, -69, 73, 217

Offset: 1

Labos Elemer, Jul 13 2001

A sequence based on the solution of the equation: 1+(1+n)*prime(n)/x-n*prime(n+1)/x=0 for x. This is an irrational rotation-like sequence: the sequence is similar to a Beatty sequence. - Roger L. Bagula, Jun 06 2002

			n = 10: a(10) = 10*31-11*29 = 310-319 = -9;
n = 54: a(54) = 54*257-55*251 = 13878-13805 = 73;
n = 55: a(55) = 55*263-56*257 = 14465-14392 = 73; consecutive terms are often equal to each other.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A001223, A062742, A062743.

[n*NthPrime(n + 1) - (n + 1)*NthPrime(n): n in [1..75]]; // Vincenzo Librandi, Jun 29 2018

seq(n*ithprime(n+1)-(n+1)*ithprime(n),n=1..80); # Muniru A Asiru, Jun 29 2018

Table[(Prime[w+1]-Prime[w])*w-Prime[w], {w, 1, 1024}]

a(n)={n*prime(n + 1) - (n + 1)*prime(n)} \\ Harry J. Smith, Aug 06 2009

a(n) = n*A000040(n+1) - (n+1)*A000040(n) = n*A001223(n) - A000040(n).

A064026 Numbers k such that d(k) + d(k+1) + d(k+2) = 8, where d(k) = A001223.

Original entry on oeis.org

2, 3, 5, 26, 43, 142, 234, 286, 313, 458, 484, 743, 1167, 1548, 1823, 1833, 1867, 2066, 2151, 2199, 2362, 2493, 2789, 3410, 3718, 4559, 5251, 5618, 6317, 6696, 6899, 7147, 7590, 7807, 7932, 8083, 8602, 9402, 9517, 9682, 10438, 11006, 11239, 11494, 12618

Offset: 1

Jason Earls, Sep 08 2001

Prime(a(n)) = 3, 5, 11, 101, 191, 821, 1481, 1871, 2081, ...; starting with 11 on, all primes == 11 (mod 30). - Zak Seidov, Jan 25 2013

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. A001223.

t={}; Do[If[Prime[n + 3] == Prime[n] + 8, AppendTo[t, n]], {n, 1000}]; t (* Zak Seidov, Jan 25 2013 *)

d(n) = prime(n+1)-prime(n); e(n) = d(n)+d(n+1)+d(n+2); j=[]; for(n=1,35000, if(e(n)==8,j=concat(j,n))); j

d3(n)= { prime(n + 3) - prime(n) } { n=0; default(primelimit, 12000000); for (m=1, 10^9, if (d3(m)==8, write("b064026.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 06 2009

A084290 Difference between consecutive primes arising before difference (d=2) between twin primes. In A001223, terms before the ones that equal 2.

Original entry on oeis.org

1, 2, 4, 4, 6, 4, 6, 4, 4, 4, 6, 10, 6, 10, 4, 4, 6, 6, 4, 4, 10, 10, 10, 4, 12, 6, 6, 4, 10, 6, 12, 10, 4, 4, 4, 6, 10, 10, 10, 4, 22, 6, 18, 6, 4, 12, 4, 4, 10, 4, 6, 6, 4, 4, 12, 4, 4, 4, 18, 16, 4, 10, 12, 4, 12, 16, 4, 16, 16, 12, 6, 4, 6, 12, 10, 4, 4, 10, 12, 4, 6, 28, 10, 4, 22, 4, 28, 6

Offset: 1

Labos Elemer, May 26 2003

nonn

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

Cf. A001223, A084291.

t = {}; s = 1; Do[s1 = Prime[n] - Prime[n - 1]; If[Equal[s1, 2], AppendTo[t, s]]; s = s1, {n, 2, 1000}]; t

list(lim) = {my(prv = 3, dprv = 1, d); forprime(p = 5, lim, d = p - prv; if(d == 2, print1(dprv, ", ")); prv = p; dprv = d);} \\ Amiram Eldar, Feb 08 2025

A084291 Difference between consecutive primes arising after difference (d=2) between twin primes. In A001223, terms succeeding those that equal 2.

Original entry on oeis.org

2, 4, 4, 4, 6, 4, 6, 6, 4, 4, 10, 6, 10, 4, 12, 4, 10, 6, 10, 4, 4, 10, 6, 4, 18, 6, 6, 12, 4, 12, 10, 4, 10, 4, 4, 10, 6, 10, 6, 4, 10, 6, 4, 6, 4, 6, 4, 6, 4, 4, 4, 6, 24, 10, 10, 12, 4, 10, 16, 22, 4, 10, 4, 10, 16, 6, 10, 4, 4, 22, 6, 6, 6, 16, 4, 4, 6, 10, 6, 16, 28, 10, 16, 12, 4, 12, 6

Offset: 1

Labos Elemer, May 26 2003

nonn

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

Cf. A001223, A084290.

Do[s1=Prime[n]-Prime[n-1]; s=Prime[n+1]-Prime[n]; s2=Prime[n+2]-Prime[n+1]; If[Equal[s, 2], Print[s2]], {n, 1, 10000}]
#[[3]]-#[[2]]&/@Select[Partition[Prime[Range[1000]],3,1],#[[2]]- #[[1]] == 2&] (* Harvey P. Dale, Jul 08 2018 *)
Join[{2},SequenceCases[Differences[Prime[Range[3,500]]],{2,}][[;;,2]]] (* _Harvey P. Dale, Mar 20 2023 *)

list(lim) = {my(prv = 3, dprv = 1, d); forprime(p = 5, lim, d = p - prv; if(dprv == 2, print1(d, ", ")); prv = p; dprv = d);} \\ Amiram Eldar, Feb 08 2025

A128039 Numbers n such that 1 - Sum_{k=1..n-1} A001223(k)*(-1)^k = 0.

Original entry on oeis.org

3, 6, 10, 13, 18, 26, 29, 218, 220, 223, 491, 535, 538, 622, 628, 3121, 3126, 3148, 3150, 3155, 3159, 4348, 4436, 4440, 4444, 4458, 4476, 4485, 4506, 4556, 4608, 4611, 4761, 5066, 5783, 5788, 12528, 1061290, 2785126, 2785691, 2867466, 2867469, 2872437

Offset: 1

Manuel Valdivia, May 07 2007

nonn

Sequence has 294 terms < 10^7. If n is in this sequence then prime(n) = abs(3 + 2*Sum_{k=1..n-1} prime(k)*(-1)^k).

			1 - ( -A001223(1) + A001223(2)) = 1-(-1+2) = 0, hence 3 is a term.
1 - ( -A001223(1) + A001223(2) - A001223(3) + A001223(4) - A001223(5)) = 1-(-1+2-2+4-2) = 0, hence 6 is a term.

Eric Weisstein's World of Mathematics, Prime Difference Function
Eric Weisstein's World of Mathematics, Prime Sums
Eric Weisstein's World of Mathematics, Alternating Series

Cf. A127596, A001223 (differences between consecutive primes), A000101 (increasing gaps between primes, upper end), A002386 (increasing gaps between primes, lower end), A066033.

S=0; a=0; Do[S=S+((Prime[k+1]-Prime[k])*(-1)^k); If[1-S==0, a++; Print[a," ",k+1]], {k, 1, 10^7, 1}]

A147965 a(n) = n + 1 - A001223(n) = n - A046933(n). In words, a(n) is the difference between n+1 and the n-th gap between primes.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 6, 5, 4, 9, 6, 9, 12, 11, 10, 11, 16, 13, 16, 19, 16, 19, 18, 17, 22, 25, 24, 27, 26, 17, 28, 27, 32, 25, 34, 31, 32, 35, 34, 35, 40, 33, 42, 41, 44, 35, 36, 45, 48, 47, 46, 51, 44, 49, 50, 51, 56, 53, 56, 59

Offset: 1

Omar E. Pol, Nov 17 2008

Cf. A000040, A001223, A046933, A141042.

Module[{nn=70,prg},prg=Differences[Prime[Range[nn]]];#[[2]]-#[[1]]&/@ Thread[{prg,Range[nn-1]}]+1] (* Harvey P. Dale, Nov 21 2021 *)

Definition corrected by N. J. A. Sloane, Nov 21 2021, at the suggestion of Harvey P. Dale.

A174433 Triangle read by rows: T(n,k) = prime(n) mod A001223(k), where A001223 are differences between consecutive primes.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 1, 3, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 3, 1, 3, 1, 3, 0, 1, 1, 3, 1, 3, 1, 3, 5, 0, 1, 1, 1, 1, 1, 1, 1, 5, 1, 0, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 0, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3

Offset: 1

Juri-Stepan Gerasimov, Nov 28 2010

The first prime gap is 3-2=1, so the first column is T(n,1)=0. The second and third prime gaps are 5-3=2 and 7-5=2, and since all primes > 2 are odd, T(n,2) = T(n,3) = 1.

			Triangle begins:
  0;
  0,1;
  0,1,1;
  0,1,1,3;
  0,1,1,3,1;

Cf. A000040.

A001223 := proc(n) ithprime(n+1)-ithprime(n) ; end proc:
A174433 := proc(n,k) ithprime(n) mod A001223(k) ; end proc:
seq(seq(A174433(n,k),k=1..n),n=1..14) ;

cp's OEIS Frontend

A347123 Fully multiplicative with a(prime(k)) = prime(1+floor(A001223(k)/2)).

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A376520 Position of first appearance of 2n in the run-compression (A037201) of the first differences (A001223) of the prime numbers (A000040).

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A052377 Primes followed by an [8,4,8]=[d,D-d,d] prime difference pattern of A001223.

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A062357 a(n) = n*p(n+1)-(n+1)*p(n) = n*d(n)-p(n), where p(n) is the n-th prime and d(n) is the n-th prime-difference, A001223(n).

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A064026 Numbers k such that d(k) + d(k+1) + d(k+2) = 8, where d(k) = A001223.

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A084290 Difference between consecutive primes arising before difference (d=2) between twin primes. In A001223, terms before the ones that equal 2.

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A084291 Difference between consecutive primes arising after difference (d=2) between twin primes. In A001223, terms succeeding those that equal 2.

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A128039 Numbers n such that 1 - Sum_{k=1..n-1} A001223(k)*(-1)^k = 0.

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A062357 a(n) = np(n+1)-(n+1)p(n) = n*d(n)-p(n), where p(n) is the n-th prime and d(n) is the n-th prime-difference, A001223(n).