A005115 -id:A005115 - cp's OEIS frontend

A126989 Gaps associated with the first and smallest arithmetic progressions of n consecutive primes in A006560.

Original entry on oeis.org

0, 1, 2, 6, 30, 30, 210

Offset: 1

Artur Jasinski, Jan 01 2007

The gap for the first and smallest AP of 7 consecutive primes is at least 210 (so the 7th term is not definitive).

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

H. Dubner et al., Ten consecutive primes in arithmetic progression, Mathematics of Computation, Volume 71, Number 239, Pages 1323-1328.

Cf. A006560, A005115, A093364.

a(7) corrected by Stephen Tucker, Jan 25 2009

A376109 a(n) is the length of the longest arithmetic progression ending at n consisting of numbers with the same number of prime factors as n, counted with multiplicity.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 2, 3, 2, 1, 3, 2, 3, 2, 3, 3, 4, 2, 2, 3, 2, 3, 5, 2, 3, 1, 3, 3, 3, 2, 2, 3, 2, 2, 4, 3, 4, 3, 2, 4, 3, 2, 3, 2, 3, 3, 5, 2, 4, 3, 3, 5, 4, 2, 3, 3, 3, 1, 3, 3, 3, 3, 4, 3, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 2, 4, 4, 3, 3, 4, 5, 3, 3, 2, 4, 4, 4, 3, 3, 2, 4, 3, 3

Offset: 1

Robert Israel, Sep 10 2024

nonn

a(n) is the greatest k such that there exists d >= 1 with A001222(n-i*d) = A001222(n) for 0 <= i < k.
If m divides n, then a(n) >= a(m).
a(n) = 1 if and only if n is a power of 2.

			a(7) = 3 because 7 is prime and there is an arithmetic progression of 3 primes, namely 3, 5, 7, ending with 7 but no such arithmetic progression of 4 primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001222, A005115, A096003, A373887, A373888.

M:= Array(1..10):
for n from 2 to 100 do
  v:= numtheory:-bigomega(n);
  if M[v] = 0 then M[v]:= n else M[v]:= M[v],n fi;
od:
for i from 1 to 10 do M[i]:= [M[i]] od:
f:= proc(s) local n,i,m,d,v,j;
   m:= 1;
   v:= numtheory:-bigomega(s);
   member(s,M[v],n);
   for i from n-1 to 1 by -1 do
     d:= s - M[v][i];
     if s - m*d < M[v][1] then return m fi;
     for j from 2 while ListTools:-BinarySearch(M[v],s-j*d) <> 0 do od:
     m:= max(m,j);
   od;
  m;
end proc:
f(1):= 1:
map(f, [$1..100]);

A093365 Least number which is the end of an arithmetic progression of n numbers that are the sums of two nonzero squares.

Original entry on oeis.org

2, 5, 8, 26, 34, 65, 146, 170, 194, 218, 242, 1445, 2225, 2309, 2393, 2477, 2561, 2645, 2729, 2813, 2897, 71633, 479581, 664445, 685697, 1141625, 1184129, 4281133, 4344889, 4408645, 31694041, 32519173, 33344305

Offset: 1

Hugo Pfoertner, Apr 27 2004

nonn

The next term is > 225000000.

			Example: a(6)=65: 5=2^2+1^2, 17=4^2+1^2, 29=5^2+2^2, 41=5^2+4^2, 53=7^2+2^2, 65=7^2+4^2.

Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions

Arithmetic progressions in A000404. For gaps see A093366.

Cf. A005115, arithmetic progressions of primes.

More terms from Hugo Pfoertner, Apr 29 2004

Corrected erroneous terms starting at a(28) Hugo Pfoertner, Oct 15 2010

A093366 Gaps associated with the arithmetic progressions in A093365.

Original entry on oeis.org

0, 3, 3, 8, 8, 12, 24, 24, 24, 24, 24, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 2772, 21252, 21252, 21252, 42504, 42504, 63756, 63756, 63756, 825132, 825132, 825132

Offset: 1

Hugo Pfoertner, Apr 27 2004

Cf. A000404, A005115, A093364, A093365.

Erroneous terms a(31) and a(32) corrected by Hugo Pfoertner, Oct 15 2010

A261149 a(n) = 515486946529943 + (n-1)*30526020494970.

Original entry on oeis.org

515486946529943, 546012967024913, 576538987519883, 607065008014853, 637591028509823, 668117049004793, 698643069499763, 729169089994733, 759695110489703, 790221130984673, 820747151479643, 851273171974613, 881799192469583, 912325212964553, 942851233459523

Offset: 1

Marco Ripà, Aug 10 2015

The terms n = 1..24 are prime. This is the longest known sequence of 24 primes in arithmetic progression with minimal end known as of August 10, 2015.

			a(24) = 515486946529943 + 23*30526020494970 = 1217585417914253 is prime.

Colin Barker, Table of n, a(n) for n = 1..1000
Jens Kruse Andersen, All known AP24 to AP26.
Wikipedia, Largest known primes in AP.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005115, A002110, A204189.

[515486946529943+(n-1)*30526020494970: n in [1..20]];

Table[515486946529943 + (n - 1) 30526020494970, {n, 1, 20}]

Vec(-x*(484960926034973*x-515486946529943)/(x-1)^2 + O(x^40)) \\ Colin Barker, Aug 25 2015

[515486946529943+(n-1)*30526020494970 for n in (1..20)] #

a(n) = 515486946529943 + (n-1)*136831*A002110(9).

G.f.: -x*(484960926034973*x-515486946529943) / (x-1)^2. - Colin Barker, Aug 25 2015

A261150 a(n) = 403185216600637 + (n-1)*2124513401010.

Original entry on oeis.org

403185216600637, 405309730001647, 407434243402657, 409558756803667, 411683270204677, 413807783605687, 415932297006697, 418056810407707, 420181323808717, 422305837209727, 424430350610737, 426554864011747, 428679377412757, 430803890813767, 432928404214777

Offset: 1

Marco Ripà, Aug 10 2015

The terms n = 1..23 are prime. This is the longest known sequence of 23 primes in arithmetic progression with minimal end known as of August 10, 2015.

			a(23) = 403185216600637 + 22*2124513401010 = 449924511422857 is prime.

Colin Barker, Table of n, a(n) for n = 1..1000
Jens Kruse Andersen, All known AP24 to AP26.
Wikipedia, Largest known primes in AP.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005115, A002110, A204189.

[403185216600637+(n-1)*2124513401010: n in [1..20]];

Table[403185216600637 + (n - 1) 2124513401010, {n, 1, 23}]

Vec(-x*(401060703199627*x-403185216600637)/(x-1)^2 + O(x^40)) \\ Colin Barker, Aug 25 2015

[403185216600637+(n-1)*2124513401010 for n in (1..20)]

a(n) = 403185216600637 + (n-1)*9523*A002110(9).

G.f.: -x*(401060703199627*x-403185216600637) / (x-1)^2. - Colin Barker, Aug 25 2015

A261151 a(n) = 11410337850553 + (n-1)*4609098694200.

Original entry on oeis.org

11410337850553, 11871247719973, 12332157589393, 12793067458813, 13253977328233, 13714887197653, 14175797067073, 14636706936493, 15097616805913, 15558526675333, 16019436544753, 16480346414173, 16941256283593, 17402166153013, 17863076022433, 18323985891853

Offset: 1

Marco Ripà, Aug 10 2015

The terms n = 1..22 are prime. This is the longest known sequence of 22 primes in arithmetic progression with minimal end known as of August 10, 2015.

			a(22) = 11410337850553 + 21*4609098694200 = 108201410428753 is prime.

Colin Barker, Table of n, a(n) for n = 1..1000
Jens Kruse Andersen, All known AP24 to AP26.
Wikipedia, Largest known primes in AP.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005115, A002110, A204189.

[11410337850553+(n-1)*4609098694200: n in [1..20]];

Table[11410337850553 + (n - 1) 4609098694200, {n, 1, 20}]

Vec(-x*(10949427981133*x-11410337850553) / (x-1)^2 + O(x^40)) \\ Colin Barker, Aug 25 2015

[11410337850553+(n-1)*4609098694200 for n in (1..20)]

a(n) = 11410337850553 + (n-1)*475180*A002110(8).

G.f.: -x*(10949427981133*x-11410337850553) / (x-1)^2. - Colin Barker, Aug 25 2015

A290967 Smallest known example of a 3 X 3 X 3 generalized arithmetic progression (GAP) of 27 primes, listed in increasing order.

Original entry on oeis.org

929, 3833, 4079, 6737, 6983, 7229, 8369, 9887, 10133, 11273, 11519, 13037, 14177, 14423, 14669, 15809, 17327, 17573, 18713, 18959, 20477, 21617, 21863, 22109, 24767, 25013, 27917

Offset: 1

Hugo Pfoertner, Aug 15 2017

27 primes arranged in a 3 X 3 X 3 cube such that the differences between the numbers in the 3 coordinate directions are constants. The 3 constants are 2904, 3150, and 7440.

The arrangement was found in an undergraduate project at Westfield State College by Jeffrey P. Vanasse and Michael E. Guenette, working under the direction of Mathematics Department faculty members Marcus Jaiclin and Julian F. Fleron.

			:           -> +2904 ->
:          +-----+-----+-----+
:      |   |  929| 3833| 6737|\
:    +3150 | 4079| 6983| 9887| \
:      V   | 7229|10133|13037|  \
:          +-----+-----+-----+   \
:       \   \   +-----+-----+-----+
:      +7440 \  | 8369|11273|14177|\
:         \   \ |11519|14423|17327| \   \
:              \|14669|17573|20477|  \ +7440
:               +-----+-----+-----+   \   \
:                \   +-----+-----+-----+
:                 \  |15809|18713|21617|   |
:                  \ |18959|21863|24767| +3150
:                   \|22109|25013|27917|   V
:                    +-----+-----+-----+
:                     -> +2904 ->

Hugo Pfoertner, Illustration of arrangement, copied from announcement of original authors.
Westfield State College, Mathematics Students Make Prime Discovery, ScienceDaily, 18 November 2008.

Cf. A005115.

sort([seq(seq(seq(929+i*3150+k*2904+j*7440, k=0..2), i=0..2), j=0..2)])[];
# Alois P. Heinz, Aug 16 2017

A291163 a(n) = smallest number k > a(n-1) maximizing the number of primes in all sums a(j)+k, j=1..(n-1), with a(1)=2.

Original entry on oeis.org

2, 3, 4, 9, 10, 27, 34, 69, 70, 429, 430, 1059, 1484, 3537, 8284, 65169, 98464, 2061999, 2210564, 10919799, 11521580, 495385137, 567955604, 1112946057, 4926960394, 365847990027

Offset: 1

Rainer Rosenthal and Hugo Pfoertner, Oct 07 2017

			a(6)=27 because it is the smallest number producing 3 primes in the sums with all previous terms: a(1)+27 = 2+27 = 29, a(3)+27 = 4+27 = 31, a(5)+27 = 10+27 = 37;
a(7)=34: a(2)+34 = 3+34 = 37, a(4)+34 = 9+34 = 43, a(6)+34 = 27+34 = 61;
a(8)=69 because it is the smallest number producing 4 primes in the sums with all previous terms: a(1)+69 = 2+69 = 71, a(3)+69 = 4+69 = 73, a(5)+69 = 10+69 = 79, a(7)+69 = 34+69 = 103.

Cf. A005115, A103828.

PT(x)={print1(x,", ");write("b291163.txt",n++," ",x)};
n=0;
ae=vector(11);
ao=vector(11);
ae[1]=2;PT(ae[1]);
ao[1]=3;PT(ao[1]);
for (m=1,10,\
   start=(ao[m]+1)/2;\
   for (kh=start,100*start,k=kh+kh;\
      for(jj=1,m,j=m-jj+1;if(!isprime(k+ao[j]),next(2)));\
      ae[m+1]=k;PT(k);break(1));\
   start=ae[m+1]/2;\
   for (kh=start,100*start,k=kh+kh+1;\
      for(jj=1,m+1,j=m-jj+2;if(!isprime(k+ae[j]),next(2)));\
      ao[m+1]=k;PT(k);break(1)))
\\ Hugo Pfoertner, Oct 10 2017

A291525 a(n) is the largest number in an n-term AP of Chen primes.

Original entry on oeis.org

2, 3, 7, 23, 29, 257, 1439, 2351, 26561, 146639, 1891949, 2062889, 341708489, 2062232987

Offset: 1

Charles R Greathouse IV, Aug 25 2017

Zhou proves that a(n) exists for each n, generalizing Green & Tao (2008) from primes to Chen primes and generalizing Green & Tao (2006) from 3-AP to n-AP. Sequence is increasing by definition.

			3, 5, 7 = a(3)
5, 11, 17, 23 = a(4)
5, 11, 17, 23, 29 = a(5)
107, 137, 167, 197, 227, 257 = a(6)
179, 389, 599, 809, 1019, 1229, 1439 = a(7)
881, 1091, 1301, 1511, 1721, 1931, 2141, 2351 = a(8)
4721, 7451, 10181, 12911, 15641, 18371, 21101, 23831, 26561 = a(9)
122069, 124799, 127529, 130259, 132989, 135719, 138449, 141179, 143909, 146639 = a(10)
182549, 353489, 524429, 695369, 866309, 1037249, 1208189, 1379129, 1550069, 1721009, 1891949 = a(11)
182549, 353489, 524429, 695369, 866309, 1037249, 1208189, 1379129, 1550069, 1721009, 1891949, 2062889 = a(12)
205492409, 216843749, 228195089, 239546429, 250897769, 262249109, 273600449, 284951789, 296303129, 307654469, 319005809, 330357149, 341708489 = a(13)
19712507, 176829467, 333946427, 491063387, 648180347, 805297307, 962414267, 1119531227, 1276648187, 1433765147, 1590882107, 1747999067, 1905116027, 2062232987 = a(14)

B. Green and T. Tao, Restriction theory of the Selberg sieve, with applications, J. Théor. Nombres Bordeaux 18:1 (2006), pp. 147-182. arXiv:math/0405581 [math.NT]
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics 167 (2008), pp. 481-547. arXiv:math/0404188 [math.NT], 2004-2007.
Binbin Zhou, The Chen primes contain arbitrarily long arithmetic progressions, Acta Arithmetica 138 (2009), pp. 301-315.

Cf. A109611, A005115.

primorial(n)=vecprod(primes(primepi(n)));
listChen(lim)=my(v=List([2]), semi=List(), L=lim+2, p=3); forprime(q=3, L\3, forprime(r=3, min(L\q, q), listput(semi, q*r))); semi=Set(semi); forprime(q=7, lim, if(setsearch(semi, q+2), listput(v, q))); forprime(q=5, L, if(q-p==2, listput(v, p)); p=q); Set(v)
chen=listChen(1e6); \\ Increase as needed to find more terms
a(n,startAt=n)=n--; my(div=lcm(primorial(n+1),n)); for(i=startAt,#chen, for(j=1,i-n, my(d=chen[i]-chen[j],g); if(d%div,next); g=d/n; forstep(k=chen[j]+g, chen[i]-g, g, if(!setsearch(chen,k), next(2))); return(chen[i])))

a(14) from Charles R Greathouse IV, Sep 06 2017

cp's OEIS Frontend

A126989 Gaps associated with the first and smallest arithmetic progressions of n consecutive primes in A006560.

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A376109 a(n) is the length of the longest arithmetic progression ending at n consisting of numbers with the same number of prime factors as n, counted with multiplicity.

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A093365 Least number which is the end of an arithmetic progression of n numbers that are the sums of two nonzero squares.

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A093366 Gaps associated with the arithmetic progressions in A093365.

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A261149 a(n) = 515486946529943 + (n-1)*30526020494970.

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A261150 a(n) = 403185216600637 + (n-1)*2124513401010.

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A261151 a(n) = 11410337850553 + (n-1)*4609098694200.

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Programs

Magma

Mathematica

PARI

Sage

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A290967 Smallest known example of a 3 X 3 X 3 generalized arithmetic progression (GAP) of 27 primes, listed in increasing order.

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