A024675 -id:A024675 - cp's OEIS frontend

A075190 Numbers k such that k^2 is an interprime = average of two successive primes.

2, 3, 8, 9, 12, 15, 18, 21, 25, 33, 41, 51, 60, 64, 72, 78, 92, 112, 117, 129, 138, 140, 159, 165, 168, 172, 192, 195, 198, 213, 216, 218, 228, 237, 273, 295, 298, 303, 304, 309, 322, 327, 330, 338, 342, 356, 360, 366, 387, 393, 408, 416, 429, 432, 441, 447, 456

Offset: 1

Zak Seidov, Sep 09 2002

nonn

Interprimes are in A024675, even interprimes are in A072568, odd interprimes are in A072569 n^3 as interprimes are in A075191, n^4 as interprimes are in A075192, n^5 as interprimes are in A075228, n^6 as interprimes are in A075229, n^7 as interprimes are in A075230, n^8 as interprimes are in A075231, n^9 as interprimes are in A075232, n^10 as interprimes are in A075233, a(n) such that a(n)^n = smallest interprime (of the form a^n) are in A075234.

			3 is a term because 3^2 = 9 is the average of two successive primes 7 and 11.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1317 from Zak Seidov)

Cf. A024675, A072568, A072569, A075191, A075192.

Cf. A075228, A075229, A075230, A075231, A075232, A075233, A075234.

s := 2: for n from 2 to 1000 do if prevprime(n^s)+nextprime(n^s)=2*n^s then print(n) else; fi; od;

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Select[ Range[464], 2#^2 == PrevPrim[ #^2] + NextPrim[ #^2] &] (* Robert G. Wilson v, Sep 14 2002 *)
n2ipQ[n_]:=Module[{n2=n^2},(NextPrime[n2]+NextPrime[n2,-1])/2==n2]; Select[Range[500],n2ipQ] (* Harvey P. Dale, May 04 2011 *)
Select[Sqrt[Mean[#]]&/@Partition[Prime[Range[30000]],2,1],IntegerQ] (* Harvey P. Dale, May 26 2013 *)

a(n) = sqrt(A069495(n)).

Edited by Robert G. Wilson v, Sep 14 2002

A072569 Odd interprimes.

Original entry on oeis.org

9, 15, 21, 39, 45, 69, 81, 93, 99, 105, 111, 129, 165, 195, 205, 217, 225, 231, 279, 309, 315, 351, 363, 381, 393, 399, 405, 441, 453, 459, 465, 473, 483, 489, 495, 501, 515, 615, 625, 645, 667, 675, 687, 705, 723, 741, 747, 759, 765, 771, 803, 825, 855, 861

Offset: 1

Marco Matosic, Jun 24 2002

nonn

The interprimes (A024675) are those integers that lie at the midpoint between consecutive odd primes.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Interprime

Odd members of A024675. Sequence is union of A072572 and A072573.

Cf. A024675, A072568.

a = Table[ Prime[n], {n, 2, 200}]; b = {}; Do[d = (a[[n + 1]] - a[[n]])/2; If[ OddQ[ a[[n]] + d], b = Append[b, a[[n]] + d]], {n, 1, 198}]; b
Select[Plus @@@ Partition[Table[Prime[n], {n, 2, 100}], 2, 1]/2, OddQ]
Select[Mean/@Partition[Prime[Range[2,200]],2,1],OddQ] (* Harvey P. Dale, Jan 22 2019 *)

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

A054643 Primes prime(n) such that prime(n) + prime(n+1) + prime(n+2) == 0 (mod 3).

Original entry on oeis.org

3, 47, 151, 167, 199, 251, 257, 367, 503, 523, 557, 587, 601, 647, 727, 941, 971, 991, 1063, 1097, 1117, 1181, 1217, 1231, 1361, 1453, 1493, 1499, 1531, 1741, 1747, 1753, 1759, 1889, 1901, 1907, 2063, 2161, 2281, 2393, 2399, 2411, 2441, 2671, 2897, 2957

Offset: 1

Labos Elemer, May 15 2000

nonn

The 2 differences of these 3 primes should be congruent of 6, except the first prime 3, for which 3 + 5 + 7 = 15 holds. Sequences like A047948, A052198 etc. are subsequences here.

			For prime(242) = 1531, the sum is 4623, the mean is 1541 and the successive differences are 6a=12 or 6b=6 resp.

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

Cf. A034961, A034707, A024675, A052288, A047948, A052198.
A122535 is a subsequence.
Cf. A075541 (for their indices).

Select[Partition[Prime@ Range@ 430, 3, 1], Divisible[Total@ #, 3] &][[All, 1]] (* Michael De Vlieger, Jun 29 2017 *)

A072568 Even interprimes.

Original entry on oeis.org

4, 6, 12, 18, 26, 30, 34, 42, 50, 56, 60, 64, 72, 76, 86, 102, 108, 120, 134, 138, 144, 150, 154, 160, 170, 176, 180, 186, 192, 198, 228, 236, 240, 246, 254, 260, 266, 270, 274, 282, 288, 300, 312, 324, 334, 342, 348, 356, 370, 376, 386, 414, 420, 426, 432

Offset: 1

Marco Matosic, Jun 21 2002

The interprimes (A024675) are those integers that lie at the midpoint between consecutive odd primes.

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

Even members of A024675.

Cf. A024675, A072569, A072570, A072571, A072572, A072573.

Select[Plus @@@ Partition[Table[Prime[n], {n, 2, 100}], 2, 1]/2, EvenQ]

Offset corrected by Amiram Eldar, Mar 23 2020

A069495 Squares which are the arithmetic mean of two consecutive primes.

Original entry on oeis.org

4, 9, 64, 81, 144, 225, 324, 441, 625, 1089, 1681, 2601, 3600, 4096, 5184, 6084, 8464, 12544, 13689, 16641, 19044, 19600, 25281, 27225, 28224, 29584, 36864, 38025, 39204, 45369, 46656, 47524, 51984, 56169, 74529, 87025, 88804, 91809, 92416, 95481, 103684

Offset: 1

Amarnath Murthy, Mar 30 2002

nonn

			144 = (139 + 149)/2 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A062703, A075190.

Intersection of A000290 and A024675.

a:= proc(n) option remember; local k, kk, p, q;
      for k from 1 +`if`(n=1, 1, iroot(a(n-1), 2))
      do kk:= k^2; p, q:= prevprime(kk), nextprime(kk);
         if (p+q)/2=kk then return kk fi
      od
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Dec 21 2013

p = -1; Do[q = Prime[n]; If[ IntegerQ[ Sqrt[(p + q)/2]], Print[(p + q)/2]]; p = q, {n, 1, 10000} ]
Select[Mean/@Partition[Prime[Range[11000]],2,1],IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Jan 23 2019 *)

a(n) = (A075190(n))^2. - Zak Seidov

Edited and extended by Robert G. Wilson v, Apr 01 2002

A072572 Odd interprimes divisible by 3.

Original entry on oeis.org

9, 15, 21, 39, 45, 69, 81, 93, 99, 105, 111, 129, 165, 195, 225, 231, 279, 309, 315, 351, 363, 381, 393, 399, 405, 441, 453, 459, 465, 483, 489, 495, 501, 615, 645, 675, 687, 705, 723, 741, 747, 759, 765, 771, 825, 855, 861, 879, 885, 897, 909, 915, 933

Offset: 1

Marco Matosic, Jun 25 2002

nonn

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

Cf. A072569, A072573, A024675.

a = Table[ Prime[n], {n, 2, 200}]; b = {}; Do[d = (a[[n + 1]] - a[[n]])/2; If[ OddQ[a[[n]] + d] && Mod[d, 3] != 0, b = Append[b, a[[n]] + d]], {n, 1, 198}]; b
Select[Mean/@Partition[Prime[Range[4,200]],2,1],OddQ[#]&&Divisible[ #,3]&] (* Harvey P. Dale, Oct 29 2013 *)

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

Corrected by T. D. Noe, Nov 01 2006

A072573 Odd interprimes not divisible by 3.

Original entry on oeis.org

205, 217, 473, 515, 625, 667, 803, 1003, 1207, 1243, 1313, 1465, 1505, 1517, 1537, 1681, 1715, 1795, 1817, 1895, 2075, 2105, 2191, 2303, 2405, 2453, 2585, 2627, 2783, 2933, 3055, 3073, 3175, 3197, 3265, 3337, 3353, 3505, 3565, 3665, 3937, 3995, 4085

Offset: 1

Marco Matosic, Jun 25 2002

nonn

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

Cf. A072569, A072572, A024675.

a = Table[ Prime[n], {n, 2, 600}]; b = {}; Do[d = (a[[n + 1]] - a[[n]])/2; If[ OddQ[ a[[n]] + d] && Mod[d, 3] == 0, b = Append[b, a[[n]] + d]], {n, 1, 598}]; b

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

A075191 Numbers k such that k^3 is an interprime = average of two successive primes.

Original entry on oeis.org

4, 12, 16, 26, 28, 36, 48, 58, 66, 68, 74, 78, 102, 106, 112, 117, 124, 126, 129, 130, 148, 152, 170, 174, 184, 189, 190, 192, 224, 273, 280, 297, 321, 324, 369, 372, 373, 399, 408, 410, 421, 426, 429, 435, 447, 449, 450, 470, 475, 496, 504, 507, 531, 537

Offset: 1

Zak Seidov, Sep 09 2002

nonn

Interprimes are in A024675, even interprimes are in A072568, odd interprimes are in A072569 n^2 as interprimes are in A075190, n^4 as interprimes are in A075192, n^5 as interprimes are in A075228, n^6 as interprimes are in A075229, n^7 as interprimes are in A075230, n^8 as interprimes are in A075231, n^9 as interprimes are in A075232, n^10 as interprimes are in A075233, a(n) such that a(n)^n = smallest interprime (of the form a^n) are in A075234.

			4 is a term because 4^3 = 64 is the average of two successive primes 61 and 57.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A024675, A072568, A072569, A075190, A075192.

Cf. A075228, A075229, A075230, A075231, A075232, A075233, A075234.

s := 3: for n from 2 to 1000 do if prevprime(n^s)+nextprime(n^s)=2*n^s then print(n) else; fi; od;

Select[ Range[548], 2#^3 == PrevPrim[ #^3] + NextPrim[ #^3] &]
n3ipQ[n_]:=Mean[{NextPrime[n^3],NextPrime[n^3,-1]}]==n^3; Select[ Range[ 600],n3ipQ] (* Harvey P. Dale, Oct 05 2017 *)
Select[Surd[Mean[#],3]&/@Partition[Prime[Range[8*10^6]],2,1],IntegerQ] (* Harvey P. Dale, Apr 07 2023 *)

is(n)=n=n^3;nextprime(n)+precprime(n)==2*n \\ Charles R Greathouse IV, Aug 25 2014

Edited by Robert G. Wilson v Sep 14 2002

A075192 Numbers k such that k^4 is an interprime = average of two successive primes.

Original entry on oeis.org

3, 5, 8, 21, 55, 66, 87, 99, 104, 105, 110, 120, 129, 135, 141, 144, 152, 168, 172, 186, 187, 192, 211, 222, 243, 279, 283, 295, 297, 321, 342, 385, 395, 398, 408, 425, 426, 460, 520, 541, 559, 597, 626, 627, 638, 642, 657, 666, 673, 680, 713, 755, 759, 765

Offset: 1

Zak Seidov, Sep 09 2002

nonn

Interprimes are in A024675, even interprimes are in A072568, odd interprimes are in A072569 n^2 as interprimes are in A075190, n^3 as interprimes are in A075191, n^5 as interprimes are in A075228, n^6 as interprimes are in A075229, n^7 as interprimes are in A075230, n^8 as interprimes are in A075231, n^9 as interprimes are in A075232, n^10 as interprimes are in A075233, a(n) such that a(n)^n = smallest interprime (of the form a^n) are in A075234.

			3 belongs to this sequence because 3^4 = 81 is the average of two successive primes 79 and 83.

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

Cf. A024675, A072568, A072569, A075190, A075191.

Cf. A075228, A075229, A075230, A075231, A075232, A075233, A075234.

s := 4: for n from 2 to 1000 do if prevprime(n^s)+nextprime(n^s)=2*n^s then print(n) else; fi; od;

intprQ[n_]:=Module[{c=n^4},c==Mean[{NextPrime[c],NextPrime[c,-1]}]]; Select[Range[800],intprQ] (* Harvey P. Dale, Dec 01 2013 *)

Edited by Robert G. Wilson v Sep 14 2002

A075228 Numbers k such that k^5 is an interprime = average of two successive primes.

Original entry on oeis.org

20, 42, 77, 81, 186, 198, 200, 220, 248, 266, 270, 294, 300, 387, 411, 477, 498, 537, 630, 678, 682, 696, 728, 741, 774, 819, 872, 985, 987, 1001, 1014, 1037, 1060, 1083, 1084, 1087, 1098, 1140, 1155, 1162, 1232, 1245, 1278, 1316, 1370, 1392, 1397, 1402

Offset: 1

Zak Seidov, Sep 09 2002

nonn

Interprimes are in A024675, even interprimes are in A072568, odd interprimes are in A072569 n^2 as interprimes are in A075190, n^3 as interprimes are in A075191, n^4 as interprimes are in A075192, n^6 as interprimes are in A075229, n^7 as interprimes are in A075230, n^8 as interprimes are in A075231, n^9 as interprimes are in A075232, n^10 as interprimes are in A075233, a(n) such that a(n)^n = smallest interprime (of the form a^n) are in A075234.

			20 is a term because 20^5 = 3200000 is the average of two successive primes 3199997 and 3200003.

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

Cf. A024675, A072568, A072569, A075190, A075191, A075192.

Cf. A075229, A075230, A075231, A075232, A075233, A075234.

s := 5: for n from 2 to 1000 do if prevprime(n^s)+nextprime(n^s)=2*n^s then print(n) else; fi; od;

More terms from Jason Earls, Sep 09 2002

Edited by Robert G. Wilson v Sep 14 2002

cp's OEIS Frontend

A075190 Numbers k such that k^2 is an interprime = average of two successive primes.

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A072569 Odd interprimes.

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A054643 Primes prime(n) such that prime(n) + prime(n+1) + prime(n+2) == 0 (mod 3).

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A072568 Even interprimes.

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A069495 Squares which are the arithmetic mean of two consecutive primes.

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A072572 Odd interprimes divisible by 3.

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A072573 Odd interprimes not divisible by 3.

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A075191 Numbers k such that k^3 is an interprime = average of two successive primes.

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