A024675 -id:A024675 - cp's OEIS frontend

A126557 Primes in A126556.

174737, 224327, 433813, 541447, 787243, 969667, 980081, 1080787, 1286581, 1372979, 1534513, 1567037, 1570649, 1577189, 1659673, 1726993, 1931291, 2242883, 2282041, 2415557, 2460827, 3162503, 3711047, 4090787, 4450373

Offset: 1

Artur Jasinski, Dec 27 2006

nonn

Prime interprimes of third order.

Primes that are the arithmetic mean of two consecutive prime interprimes of second order; primes of the form (A126555(k)+A126555(k+1))/2.

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), A126556 (interprimes of third order).

{m=5000000;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,if(isprime(h=(g+b)/2),print1(h,",")));g=b));a=q);p=q;q=r; r=nextprime(r+1))} \\ Klaus Brockhaus, Jan 11 2007

Edited and extended by Klaus Brockhaus, Jan 11 2007

A129752 Triangular numbers t which are average of two consecutive primes p and p+4.

Original entry on oeis.org

15, 21, 45, 105, 231, 351, 465, 741, 861, 1431, 1485, 3081, 3321, 4005, 7875, 10731, 11175, 11781, 13695, 14535, 17205, 17391, 18915, 21321, 22155, 23871, 30135, 33411, 36585, 39621, 42195, 51681, 58311, 80601, 90525, 92235, 97461, 108345

Offset: 1

Zak Seidov, May 14 2007

nonn

All terms are multiples of 3.

			15-/+2=(13,17), 21-/+2=(19,23), 45-/+2=(43,47) are pairs of consecutive primes.

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

Cf. A000217, A024675, A130178.

select(t -> isprime(t-2) and isprime(t+2), [seq(i*(i+1)/2,i=1..1000)]); # Robert Israel, Oct 18 2020

t-/+2 are pair of consecutive primes.

A263676 Numbers that are both interprime and oblong.

Original entry on oeis.org

6, 12, 30, 42, 56, 72, 240, 342, 420, 462, 506, 552, 600, 650, 870, 1056, 1190, 1482, 1722, 1806, 2550, 2652, 2970, 3540, 4422, 6320, 7140, 8010, 10302, 12656, 13572, 14042, 17292, 18360, 19182, 19460, 20022, 22952, 23562, 24180, 27060, 29070, 29756, 31152, 33306, 35156, 35532, 39006

Offset: 1

Antonio Roldán, Oct 23 2015

			342 is in this sequence because 342 = 18*19 is oblong, and 342 = (337 + 347)/2, with 337 and 347 consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Interprime
Eric Weisstein's World of Mathematics, Pronic Number

Intersection of A024675 and A002378. - Omar E. Pol, Oct 24 2015
Lesser of consecutive primes is in the sequence A242383.
Cf. A075190, A075277, A075296, A078443, A130178, A263674, A263675.

lim = 40000; Intersection[Plus @@@ Partition[Table[Prime@ n, {n, 2, PrimePi@ lim}], 2, 1]/2, Table[n (n + 1), {n, 0, lim}]] (* Michael De Vlieger, Nov 18 2015, after Clark Kimberling at A024675 and Robert G. Wilson v at A002378 *)
obQ[n_]:=With[{divs=Partition[Divisors[n],2,1]},Length[Select[divs,#[[2]]-#[[1]]== 1 && Times@@#==n&]]>0]; Select[Mean/@Partition[Prime[ Range[ 2,40000]],2,1],obQ] (* Harvey P. Dale, Nov 01 2022 *)

{for(i=1,500,n=i*(i+1);if(n==(precprime(n-1)+nextprime(n+1))/2, print1(n,", ")))}

A264721 Composite numbers that are greater than the average of their closest flanking primes.

Original entry on oeis.org

10, 16, 22, 27, 28, 35, 36, 40, 46, 51, 52, 57, 58, 65, 66, 70, 77, 78, 82, 87, 88, 94, 95, 96, 100, 106, 112, 121, 122, 123, 124, 125, 126, 130, 135, 136, 145, 146, 147, 148, 155, 156, 161, 162, 166, 171, 172, 177, 178, 187, 188, 189, 190, 196, 206, 207, 208

Offset: 1

Chris Boyd, Nov 21 2015

Composite numbers that are nearer to the immediately next prime than to the immediately previous prime.
Members of this sequence are the numbers C, necessarily composite, such that I_n < C < P_n+1, where P_n is the n-th odd prime and I_n the interprime (A024675) between P_n and P_n+1.
Prime-free subsequence of A264719.

			a(7) = 36 because 36 > (31 + 37)/2 = 34.

Chris Boyd, Table of n, a(n) for n = 1..10000

Cf. A264719, A264720, A264722.

Select[Range@ 208, And[CompositeQ@ #, # > (Abs@ NextPrime[#, -1] + NextPrime@ #)/2] &] (* Michael De Vlieger, Nov 22 2015 *)

test(n)= {if(n-precprime(n-1)>nextprime(n+1)-n&&n>2&&!isprime(n),return(1),return(0))}
for(i=1,200,if(test(i),print1(i,", ")))

A264722 Composite numbers that are less than the average of their closest flanking primes.

Original entry on oeis.org

8, 14, 20, 24, 25, 32, 33, 38, 44, 48, 49, 54, 55, 62, 63, 68, 74, 75, 80, 84, 85, 90, 91, 92, 98, 104, 110, 114, 115, 116, 117, 118, 119, 128, 132, 133, 140, 141, 142, 143, 152, 153, 158, 159, 164, 168, 169, 174, 175, 182, 183, 184, 185, 194, 200, 201, 202, 203

Offset: 1

Chris Boyd, Nov 21 2015

Composite numbers that are nearer to the immediately previous prime than to the immediately next prime.
Members of this sequence are the numbers C, necessarily composite, such that P_n < C < I_n, where P_n is the n-th odd prime and I_n the interprime (A024675) between P_n and P_n+1.
Prime-free subsequence of A264720.

			a(7) = 33 because 33 < (31 + 37)/2 = 34.

Chris Boyd, Table of n, a(n) for n = 1..10000

Cf. A264719, A264720, A264721.

Select[Range@ 204, And[CompositeQ@ #, # < (NextPrime[#, -1] + NextPrime@ #)/2] &] (* Michael De Vlieger, Nov 22 2015 *)
Range[#[[1]]+1,Total[#]/2 -1]&/@Select[Partition[Prime[Range[50]],2,1], #[[2]]- #[[1]]>2&]//Flatten  (* Harvey P. Dale, Jul 28 2020 *)

test(n)= {if(n-precprime(n-1)2&&!isprime(n),return(1),return(0))}
for(i=1,200,if(test(i),print1(i,", ")))

A063932 Average of largest prime less than or equal to n and smallest prime greater than or equal to n.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 9, 9, 11, 12, 13, 15, 15, 15, 17, 18, 19, 21, 21, 21, 23, 26, 26, 26, 26, 26, 29, 30, 31, 34, 34, 34, 34, 34, 37, 39, 39, 39, 41, 42, 43, 45, 45, 45, 47, 50, 50, 50, 50, 50, 53, 56, 56, 56, 56, 56, 59, 60, 61, 64, 64, 64, 64, 64, 67, 69, 69, 69, 71, 72, 73

Offset: 2

Henry Bottomley, Aug 21 2001

nonn

			a(7) = (7 + 7)/2 = 7;
a(8) = (7 + 11)/2 = 9.

Harry J. Smith, Table of n, a(n) for n = 2..1000

Interleaving of A000040 and A001223-1 copies of A024675. Cf. A063934.

Table[Mean[{NextPrime[n-1],NextPrime[n+1,-1]}],{n,2,80}] (* Harvey P. Dale, Nov 22 2011 *)

{ for (n=2, 1000, write("b063932.txt", n, " ", (precprime(n) + nextprime(n))/2) ) } \\ Harry J. Smith, Sep 02 2009

a(n) = (A007917(n) + A007918(n))/2 = n - A063933(n).

A063933 Difference between n and the average of largest prime less than or equal to n and smallest prime greater than or equal to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -2, -1, 0, 1, 2, 0, 0, 0, -2, -1, 0, 1, 2, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -2, -1, 0, 1, 2, 0, -2, -1, 0, 1, 2, 0, 0, 0, -2, -1, 0, 1, 2, 0, -1, 0, 1, 0, 0, 0, -2, -1, 0, 1, 2, 0, -1, 0, 1, 0, -2, -1, 0, 1, 2, 0, -3, -2, -1, 0, 1, 2, 3, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0

Offset: 2

Henry Bottomley, Aug 21 2001

sign

			a(10) = 10 - (11 - 7)/2 = 1; a(11) = 11 - (11 + 11)/2 = 0.

Harry J. Smith, Table of n, a(n) for n = 2..1000

{ for (n=2, 1000, write("b063933.txt", n, " ", n - (precprime(n) + nextprime(n))/2) ) } \\ Harry J. Smith, Sep 03 2009

a(n) = n - (A007917(n) + A007918(n))/2 = n - A063932(n).

a(n) = 0 for numbers in A063934 (i.e., in A000040 or A024675).

A072570 Even interprimes i = (p+q)/2 (where p, q are consecutive primes) such that (q-p)/2 is not divisible by 3.

Original entry on oeis.org

4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 120, 138, 144, 150, 180, 186, 192, 198, 228, 240, 246, 270, 282, 288, 300, 312, 324, 342, 348, 414, 420, 426, 432, 462, 522, 552, 570, 582, 600, 618, 636, 642, 660, 696, 714, 780, 792, 810, 816, 822, 828, 834, 846, 858

Offset: 1

Marco Matosic, Jun 24 2002

nonn

A superset of A014574. [R. J. Mathar, Mar 03 2009]

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

Cf. A024675, A072571. A072568 is union of A072571 and this sequence.

a = Table[Prime[n], {n, 2, 200}]; b = {}; Do[d = (a[[n + 1]] - a[[n]])/2; If[ EvenQ[ a[[n]] + d] && (Mod[d, 6] == 5 || Mod[d, 6] == 1), b = Append[b, a[[n]] + d]], {n, 1, 198}]; b
Mean/@Select[Partition[Prime[Range[200]],2,1],EvenQ[Mean[#]] && !Divisible[ (#[[2]]-#[[1]])/2,3]&] (* Harvey P. Dale, Sep 27 2017 *)

q=3;forprime(p=5,1e3,(s=q+q=p)%4==0 && (s-2*p)%3 && print1(s/2",")) \\ M. F. Hasler, Nov 29 2013

is_A072570(n)=my(p=precprime(n));nextprime(n)+p==2*n && (n-p)%3 && !bittest(n,0) \\ M. F. Hasler, Nov 30 2013

If d = (P_{n+1} - P_n)/2 is even & d/2 == +/- 1 (mod 6), then P_n + d = (P_{n+1} + P_n)/2 is in the sequence. [Corrected by M. F. Hasler, Nov 29 2013]

Edited by N. J. A. Sloane and Robert G. Wilson v, Jun 27 2002

A072571 Even interprimes i = (p+q)/2 (where p, q are consecutive primes) such that (q-p)/2 is divisible by 3.

Original entry on oeis.org

26, 34, 50, 56, 64, 76, 86, 134, 154, 160, 170, 176, 236, 254, 260, 266, 274, 334, 356, 370, 376, 386, 436, 446, 506, 532, 544, 560, 566, 574, 590, 596, 604, 610, 650, 656, 680, 730, 736, 754, 944, 950, 974, 980, 994, 1016, 1036, 1066, 1078, 1100, 1106

Offset: 1

Marco Matosic, Jun 24 2002

nonn

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

Cf. A024675, A072570. A072568 is union of A072570 and this sequence.

a = Table[Prime[n], {n, 2, 200}]; b = {}; Do[d = (a[[n + 1]] - a[[n]])/2; If[ EvenQ[ a[[n]] + d] && Mod[d, 6] == 3, b = Append[b, a[[n]] + d]], {n, 1, 198}]; b
Select[Mean/@Select[Partition[Prime[Range[200]],2,1],Divisible[(#[[2]]- #[[1]])/ 2,3]&],EvenQ] (* Harvey P. Dale, May 09 2021 *)

(P_n+1 - P_n)/2 is even but not divisible by 4.

Edited by N. J. A. Sloane and Robert G. Wilson v, Jun 27 2002

A126558 Arithmetic mean of two consecutive prime interprimes of third order: interprimes of fourth order.

Original entry on oeis.org

199532, 329070, 487630, 664345, 878455, 974874, 1030434, 1183684, 1329780, 1453746, 1550775, 1568843, 1573919, 1618431, 1693333, 1829142, 2087087, 2262462, 2348799, 2438192, 2811665, 3436775, 3900917, 4270580, 4830665

Offset: 1

Artur Jasinski, Dec 27 2006

nonn

For primes in this sequence (prime interprimes of fourth order) see A127364.

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), A126556 (interprimes of third order), A126557 (prime interprimes of third order).

{m=5000000;a=0;g=0;e=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,h=(g+b)/2;if(isprime(h),if(e>0,f=(e+h)/2;print1(f,","));e=h));g=b));a=q);p=q;q=r;r=nextprime(r+1))} \\ Klaus Brockhaus, Jan 11 2007

a(n) = (A126557(n)+A126557(n+1))/2.

Edited and extended by Klaus Brockhaus, Jan 11 2007

cp's OEIS Frontend

A126557 Primes in A126556.

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A129752 Triangular numbers t which are average of two consecutive primes p and p+4.

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A263676 Numbers that are both interprime and oblong.

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Mathematica

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A264721 Composite numbers that are greater than the average of their closest flanking primes.

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A264722 Composite numbers that are less than the average of their closest flanking primes.

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A063932 Average of largest prime less than or equal to n and smallest prime greater than or equal to n.

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A063933 Difference between n and the average of largest prime less than or equal to n and smallest prime greater than or equal to n.

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A072570 Even interprimes i = (p+q)/2 (where p, q are consecutive primes) such that (q-p)/2 is not divisible by 3.

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