A024675 -id:A024675 - cp's OEIS frontend

A063934 Numbers which are either prime or the average of consecutive odd primes.

2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 26, 29, 30, 31, 34, 37, 39, 41, 42, 43, 45, 47, 50, 53, 56, 59, 60, 61, 64, 67, 69, 71, 72, 73, 76, 79, 81, 83, 86, 89, 93, 97, 99, 101, 102, 103, 105, 107, 108, 109, 111, 113, 120, 127, 129, 131, 134, 137, 138

Offset: 1

Henry Bottomley, Aug 21 2001

nonn

Numbers n such that nextprime(n-1) + prevprime(n+1) = 2n. - Wesley Ivan Hurt, May 13 2017

			7 is prime, 9 is the average of 7 and 11, 11 is prime, 12 is the average of 11 and 13; so 7, 9, 11 and 13 are in the sequence.

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

Cf. A063932, A063933.

Function[p, Union@ Join[p, Rest@ Map[Mean, Partition[p, 2, 1]]]]@ Prime@ Range@ 34 (* Michael De Vlieger, May 13 2017 *)

{ for (n=1, 1000, if (n==1, a=2; p=3, if (n%2, a=(q + p=nextprime(q + 1))/2, a=q=p)); write("b063934.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 03 2009

For n >= 1: a(2n)=A000040(n+1); a(2n+1)=(A000040(n+1)+A000040(n+2))/2 =A024675(n).

A093343 Primes of form (prime(n)^2 + prime(n+1)^2)/2.

Original entry on oeis.org

17, 37, 229, 2029, 14449, 22501, 25609, 28909, 32401, 42061, 57601, 72901, 116989, 176401, 181501, 265261, 304729, 324901, 378229, 462409, 497041, 695581, 804709, 1089961, 1299721, 1416109, 1664101, 1742401, 1932181, 1971241, 2712709, 2873029, 3062509, 3186229

Offset: 1

Giovanni Teofilatto, Apr 26 2004

nonn

Except for the first term, all terms == 1 mod 6. - Zak Seidov, Dec 02 2009

Except 17, all terms == 1 mod 12. Primes of the form A028334(n+1)^2 + A024675(n)^2. - Thomas Ordowski, Jun 28 2013

Zak Seidov, Table of n, a(n) for n=1..1000

Cf. A103739.

Select[Mean/@Partition[Prime[Range[500]]^2,2,1],PrimeQ] (* Harvey P. Dale, Jun 16 2021 *)

Conjecture: a(n) ~ A224888(n). - Thomas Ordowski, Jul 25 2013

Corrected and extended by Rick L. Shepherd, Nov 24 2004

A126554 Arithmetic mean of two consecutive balanced primes (of order one).

Original entry on oeis.org

29, 105, 165, 192, 234, 260, 318, 468, 578, 600, 630, 693, 840, 962, 1040, 1113, 1155, 1205, 1295, 1439, 1629, 1750, 1830, 2097, 2352, 2547, 2790, 2933, 3135, 3310, 3475, 3685, 3873, 4211, 4433, 4527, 4627, 4674, 4842, 5050, 5110, 5208, 5345, 5390, 5478

Offset: 1

Artur Jasinski, Dec 27 2006

nonn

Might be called interprimes of order two, since the arithmetic means of two consecutive odd primes (A024675) sometimes are called interprimes.
Balanced primes of order two (A082077) and doubly balanced primes (A051795) have different definitions.
For primes in this sequence (prime interprimes of order two) see A126555.

Muniru A Asiru and Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Muniru A Asiru)

Cf. A006562, A024675, A082077, A051795, A126555.

P:=Filtered([1..6000],IsPrime);;P1:=List(Filtered(List([0..Length(P)-3],k->List([1..3],j->P[j+k])),i->Sum(i)/3=i[2]),m->m[2]);;
a:=List([1..Length(P1)-1],n->(P1[n+1]+P1[n])/2); # Muniru A Asiru, Mar 31 2018

b = {}; a = {}; Do[If[PrimeQ[((Prime[n + 2] + Prime[n + 1])/2 + (Prime[n + 1] + Prime[n])/2)/2], AppendTo[a, ((Prime[n + 2] + Prime[n + 1])/2 + (Prime[n + 1] + Prime[n])/2)/2]], {n, 1, 1000}]; Do[AppendTo[b, (a[[k + 1]] + a[[k]])/2], {k, 1, Length[a] - 1}]; b

{m=6000;a=0;p=2;q=3;r=5;while(r<=m,if((p+r)/2==q,if(a>0,print1((a+q)/2,","));a=q);p=q;q=r;r=nextprime(r+1))} \\ Klaus Brockhaus, Jan 05 2007

a(n) = (A006562(n+1)+A006562(n))/2.

Edited by Klaus Brockhaus, Jan 05 2007

A058296 Average of consecutive primes.

Original entry on oeis.org

2, 4, 9, 15, 21, 30, 39, 45, 56, 64, 72, 81, 93, 102, 108, 120, 134, 144, 154, 165, 176, 186, 195, 205, 225, 231, 240, 254, 266, 274, 282, 300, 312, 324, 342, 351, 363, 376, 386, 399, 414, 426, 436, 446, 459, 465, 483, 495, 506, 522, 544, 560, 570, 582, 596

Offset: 1

Donald Mills (dmills(AT)math.siu.edu), Feb 16 2003

2 together with average of odd primes taken two at a time without overlaps, i.e., 2 together with average of (3,5), (7,11), (13,17), etc. - Harvey P. Dale, Apr 09 2018

Harry J. Smith, Table of n, a(n) for n = 1..20000
Donald Mills, Some Interesting Sequences.

A bisection of A024675.

Cf. A000040, A079424.

with(linalg): v := linalg[vector](100): v[1] := 2: for j from 2 to 100 do v[j] := (ithprime(2*j-2)+ithprime(2*j-1))/2: od: print(v);

Join[{2},Mean/@Partition[Prime[Range[2,121]],2]] (* Harvey P. Dale, Apr 09 2018 *)

{ write("b058296.txt", 1, " ", 2); p2=2; for (n=2, 20000, p1=nextprime(p2+1); p2=nextprime(p1+1); a=(p1+p2)/2; write("b058296.txt", n, " ", a); ); } \\ Harry J. Smith, May 30 2009

a(1)=2, a(n) = (prime(2n-2) + prime(2n-1))/2 for n>1, where prime(i) is the i-th prime.

A078443 Numbers which are both interprime and semiprime.

Original entry on oeis.org

4, 6, 9, 15, 21, 26, 34, 39, 69, 86, 93, 111, 129, 134, 205, 217, 254, 274, 309, 334, 381, 386, 393, 446, 453, 473, 489, 501, 515, 566, 667, 687, 723, 771, 803, 879, 933, 939, 974, 1003, 1011, 1126, 1167, 1207, 1226, 1234, 1243, 1286, 1294, 1299, 1313, 1465

Offset: 1

Lior Manor, Dec 31 2002

nonn

Ratio of the number of odd terms to the number of even terms increases with increasing n. - Zak Seidov, May 22 2015

			a[8]=39 because 39=(37+41)/2 and 39=3*13

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

Intersection of A001358 and A024675. - Zak Seidov, May 22 2015

Select[Mean/@Partition[Prime[Range[250]],2,1],PrimeOmega[#]==2&] (* Harvey P. Dale, Oct 08 2013 *)

lista(nn) = {prevp = 2; forprime (p=3, nn, n = p + prevp; if (n % 2 == 0, if (bigomega(n/2) == 2, print1(n/2, ", "););); prevp = p;);} \\ Michel Marcus, Jun 09 2013

A126556 Arithmetic mean of two consecutive prime interprimes of second order: interprimes of third order.

Original entry on oeis.org

734, 2825, 5957, 10305, 13932, 15830, 18825, 25084, 30205, 32121, 34901, 40640, 47984, 70842, 102897, 120165, 125973, 130250, 138924, 145480, 148894, 154236, 161676, 167730, 174737, 180632, 183077, 191253, 210375, 224327, 232817, 246285

Offset: 1

Artur Jasinski, Dec 27 2006

nonn

For primes in this sequence (prime interprimes of third order) see A126557.

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

Cf. A006562 (balanced primes), A024675 (interprimes), A126554 (interprimes of second order), A126555 (prime interprimes of second order).

{m=250000;a=0;g=0;p=2;q=3;r=5;while(r<=m,if((p+r)/2==q,if(a>0,b=(a+q)/2;if(isprime(b),if(g>0,print1(h=(g+b)/2,","));g=b));a=q);p=q;q=r;r=nextprime(r+1))} \\ Klaus Brockhaus, Jan 11 2007

a(n) = (A126555(n)+A126555(n+1))/2.

Edited by Klaus Brockhaus, Jan 11 2007

A074927 a(n) such that p(n)*p(n+1)+a(n) is a minimal square.

Original entry on oeis.org

3, 1, 1, 4, 1, 4, 1, 4, 9, 1, 9, 4, 1, 4, 9, 9, 1, 9, 4, 1, 9, 4, 9, 16, 4, 1, 4, 1, 4, 49, 4, 9, 1, 25, 1, 9, 9, 4, 9, 9, 1, 25, 1, 4, 1, 36, 36, 4, 1, 4, 9, 1, 25, 9, 9, 9, 1, 9, 4, 1, 25, 49, 4, 1, 4, 49, 9, 25, 1, 4, 9, 16, 9, 9, 4, 9, 16, 4, 16, 25, 1, 25, 1, 9, 4, 9, 16, 4, 1, 4, 36, 16, 4

Offset: 1

Zak Seidov, Oct 02 2002

nonn

When a(n)=1, p(n) and p(n+1) are twin primes.

a(n+1) = A072681(A024675(n)). - Reinhard Zumkeller, Mar 04 2009

			a(154) = 100 because p(154)*p(155) + 100 = 804609 = 897^2.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Flatten[{3,Table[((Prime[n+1]-Prime[n])/2)^2,{n,2,100}]}] (* Vaclav Kotesovec, Mar 23 2014 *)
Join[{3},((#[[2]]-#[[1]])/2)^2&/@Partition[Prime[Range[2,100]],2,1]] (* Harvey P. Dale, Dec 04 2016 *)

For n>1: a(n) = ((p(n+1)-p(n))/2)^2. - Reinhard Zumkeller, Oct 22 2002

A075541 Numbers k such that prime(k) + prime(k+1) + prime(k+2) is a multiple of 3.

Original entry on oeis.org

2, 15, 36, 39, 46, 54, 55, 73, 96, 99, 102, 107, 110, 118, 129, 160, 164, 167, 179, 184, 187, 194, 199, 202, 218, 231, 238, 239, 242, 271, 272, 273, 274, 290, 291, 292, 311, 326, 339, 356, 357, 358, 362, 387, 419, 426, 437, 438, 449, 452, 464, 465, 489, 508

Offset: 1

Zak Seidov, Sep 21 2002

nonn

Not every three successive primes have an integer average. The integer averages are in A075540.

Not all of these 3-averages are prime: the prime 3-averages are in A006562 (balanced primes). There are surprisingly many prime 3-averages: among first 117 3-averages, there are 59 primes. Indices i(n) of first prime in sequence of three primes with integer average are in sequence A064113. Interprimes (s-averages with s=2) are all composite, see A024675.

			a(2) = 15 because (prime(15)+prime(16)+prime(17)) = (1/3)*(47 + 53 + 59) = 53 (integer average of three successive primes).

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

Cf. A006562, A024675, A075540, A064113.

R:= NULL: count:= 0:
q:= 2: r:= 3:
for i from 1 while count < 100 do
  p:= q; q:= r; r:= nextprime(r);
  if p+q+r mod 3 = 0 then
     R:= R,i; count:= count+1
  fi
od:
R; # Robert Israel, Nov 10 2024

A075541= {}; Do[If[IntegerQ[s3 = (Prime[i] + Prime[i + 1] + Prime[i + 2])/3], A075541 = Append[A075541, i]], {i, 1000}]; (* 119 terms*)

A162800 a(n) = n-th grid point that is covered by the zig-zag function for prime numbers such that the grid point is a vertex in the graph of the function.

Original entry on oeis.org

0, 2, 4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99, 102, 105, 108, 111, 120, 129, 134, 138, 144, 150, 154, 160, 165, 170, 176, 180, 186, 192, 195, 198, 205, 217, 225, 228, 231, 236, 240, 246, 254, 260, 266, 270, 274, 279

Offset: 0

Omar E. Pol, Jul 16 2009

Also {0, 2} together the numbers A024675.

See A162345 for the first differences.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000040, A006005, A024675, A058296, A079424, A162345, A162801, A162802.

Join[{0, 2}, Most[#] + Differences[#]/2] & [Prime[Range[2, 100]]] (* Paolo Xausa, Jun 17 2024 *)

Edited by Omar E. Pol, Jul 18 2009

A072681 a(n) = (n - A007917(n)) * (A007918(n) - n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 3, 4, 3, 0, 1, 0, 3, 4, 3, 0, 1, 0, 3, 4, 3, 0, 5, 8, 9, 8, 5, 0, 1, 0, 5, 8, 9, 8, 5, 0, 3, 4, 3, 0, 1, 0, 3, 4, 3, 0, 5, 8, 9, 8, 5, 0, 5, 8, 9, 8, 5, 0, 1, 0, 5, 8, 9, 8, 5, 0, 3, 4, 3, 0, 1, 0, 5, 8, 9, 8, 5, 0, 3, 4, 3, 0, 5, 8, 9, 8, 5, 0, 7, 12, 15, 16, 15, 12, 7, 0, 3, 4, 3, 0, 1, 0

Offset: 2

Reinhard Zumkeller, Jul 01 2002

a(n)=0 iff n is prime.
Local maxima occur at interprimes: a(A024675(n)) = A074927(n+1). - Reinhard Zumkeller, Mar 04 2009
Expanding upon the maxima comment, repetitive subset triplets (like 3,4,3) of form (k,k+1,k) occur when the middle value is a square. - Bill McEachen, Apr 14 2025

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A007917, A007918, A007920, A024675, A064722, A072680, A074927.

a[n_] := (n - NextPrime[n+1, -1])*(NextPrime[n] - n); Table[a[n], {n, 2, 103}] (* Jean-François Alcover, Jun 14 2013 *)

a(n) = A064722(n) * A007920(n).

a(n) = A064722(n) * (A072680(n) - A064722(n)).

cp's OEIS Frontend

A063934 Numbers which are either prime or the average of consecutive odd primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A093343 Primes of form (prime(n)^2 + prime(n+1)^2)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A126554 Arithmetic mean of two consecutive balanced primes (of order one).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Mathematica

PARI

Formula

Extensions

A058296 Average of consecutive primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A078443 Numbers which are both interprime and semiprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A126556 Arithmetic mean of two consecutive prime interprimes of second order: interprimes of third order.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A074927 a(n) such that p(n)*p(n+1)+a(n) is a minimal square.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica