A024675 -id:A024675 - cp's OEIS frontend

A075229 Numbers k such that k^6 is an interprime = average of two successive primes.

2, 4, 6, 18, 24, 27, 30, 53, 96, 122, 175, 195, 213, 231, 265, 300, 408, 420, 426, 450, 492, 532, 570, 614, 618, 657, 682, 705, 774, 777, 822, 858, 915, 946, 948, 1001, 1008, 1061, 1073, 1107, 1145, 1186, 1233, 1269, 1323, 1352, 1374, 1406, 1413, 1440, 1480

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^5 as interprimes are in A075228, 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.

			2 is a term because 2^6 = 64 is the average of two successive primes 63 and 67.

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

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

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

s := 6: 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[1500], 2#^6 == NextPrime[#^6,-1] + NextPrime[#^6] &]

Edited by Robert G. Wilson v Sep 14 2002

A075230 Numbers k such that k^7 is an interprime = average of two successive primes.

Original entry on oeis.org

20, 33, 41, 71, 82, 99, 151, 165, 254, 267, 283, 316, 345, 462, 486, 496, 516, 630, 657, 668, 676, 681, 687, 724, 760, 945, 1004, 1050, 1085, 1167, 1305, 1314, 1316, 1326, 1335, 1389, 1414, 1420, 1454, 1456, 1512, 1638, 1644, 1726, 1803, 1874, 1905, 1963

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^5 as interprimes are in A075228, n^6 as interprimes are in A075229, 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 20^7 = 1280000000 is the average of two successive primes 1279999997 and 1280000003.

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

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

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

s := 7: 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[2000],Mean[{NextPrime[#^7],NextPrime[#^7,-1]}]==#^7&] (* Harvey P. Dale, Aug 09 2013 *)

Edited by Robert G. Wilson v Sep 14 2002

A075231 Numbers k such that k^8 is an interprime = average of two successive primes.

Original entry on oeis.org

12, 111, 116, 175, 183, 205, 246, 305, 313, 406, 438, 593, 594, 620, 696, 714, 788, 824, 844, 969, 1014, 1023, 1061, 1080, 1153, 1176, 1204, 1288, 1367, 1456, 1470, 1515, 1533, 1538, 1572, 1626, 1659, 1689, 1692, 1695, 1734, 1759, 1788, 1860, 1928

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^5 as interprimes are in A075228, n^6 as interprimes are in A075229, n^7 as interprimes are in A075230, 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.

			12 is a term because 12^8 = 429981696 is the average of two successive primes 429981691 and 429981701.

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

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

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

s := 8: 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[2000], 2#^8 == NextPrime[#^8, -1] + NextPrime[#^8] &]

Edited by Robert G. Wilson v Sep 14 2002

A075232 Numbers k such that k^9 is an interprime = average of two successive primes.

Original entry on oeis.org

9, 74, 110, 141, 340, 370, 411, 423, 546, 687, 720, 723, 725, 744, 813, 834, 966, 1033, 1054, 1137, 1178, 1233, 1264, 1284, 1287, 1320, 1335, 1460, 1636, 1642, 1768, 1934, 2046, 2053, 2064, 2103, 2214, 2397, 2447, 2465, 2486, 2496, 2510, 2716, 2741, 2775

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^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^10 as interprimes are in A075233, a(n) such that a(n)^n = smallest interprime (of the form a^n) are in A075234.

			9 is a term because 9^9 = 387420489 is the average of two successive primes 387420479 and 387420499.

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

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

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

s := 9: 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[2869], 2#^9 == NextPrime[#^9, -1] + NextPrime[#^9] &]

Edited by Robert G. Wilson v Sep 14 2002

A075233 Numbers k such that k^10 is an interprime = average of two successive primes.

Original entry on oeis.org

9, 42, 87, 105, 108, 141, 144, 166, 215, 250, 381, 387, 482, 490, 500, 645, 748, 792, 831, 860, 876, 968, 990, 1377, 1448, 1468, 1526, 1769, 1780, 1922, 1968, 2084, 2118, 2228, 2245, 2252, 2373, 2381, 2478, 2565, 2672, 2781, 2883, 2915, 2972, 2988, 3008

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^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, a(n) such that a(n)^n = smallest interprime (of the form a^n) are in A075234.

			9 is a term because 9^10 = 3486784401 is the average of two successive primes 3486784393 and 3486784409.

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

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

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

s := 10: 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[3087], 2#^10 == NextPrime[#^10, -1] + NextPrime[#^10] &]
Select[Range[3100],With[{c=#^10},c==Mean[{NextPrime[c],NextPrime[c,-1]}]]&] (* Harvey P. Dale, May 21 2025 *)

Edited by Robert G. Wilson v Sep 14 2002

A075234 Least k such that k^n is the smallest interprime which is an n-th power.

Original entry on oeis.org

4, 2, 4, 3, 20, 2, 20, 12, 9, 9, 24, 2, 23, 26, 20, 66, 10, 3, 16, 3, 92, 13, 18, 48, 230, 129, 78, 181, 315, 33, 231, 19, 14, 152, 78, 39, 39, 4, 144, 9, 143, 55, 106, 25, 10, 91, 17, 7, 107, 91, 35, 44, 426, 81, 380, 97, 265, 237, 611, 1034, 122, 1072, 298, 1213, 18, 51

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^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(1)=4 because 4^1 = 4 is the smallest interprime of the form k^1.
a(2)=2 because 2^2 = 4 is the smallest interprime of the form k^2.
a(3)=4 because 4^3 = 64 is the smallest interprime of the form k^3.
a(5)=20 because 20^5 = 3200000 is the smallest interprime of the form k^5.
a(29)=315 because 315^29 is the smallest interprime of the form k^29.

Amiram Eldar, Table of n, a(n) for n = 1..250 (terms 1..100 from Zak Seidov)

Cf. A072568, A072569.

The first 10 terms in this sequence are the first terms in A024675, A075190, A075191, A075192, A075228, A075229, A075230, A075231, A075232, A075233.

s := 10: 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]; a = {}; Do[k = 2; While[2k^n != PrevPrim[k^n] + NextPrim[k^n], k++ ]; a = Append[a, k], {n, 1, 67}]; a

Edited and extended by Robert G. Wilson v, Sep 14 2002

Typos in EXAMPLE fixed by Zak Seidov, Feb 09 2012

A075540 Integers that are the average of three successive primes.

Original entry on oeis.org

5, 53, 157, 173, 211, 257, 263, 373, 511, 537, 563, 593, 607, 653, 733, 947, 977, 999, 1073, 1103, 1123, 1187, 1223, 1239, 1367, 1461, 1501, 1511, 1541, 1747, 1753, 1763, 1773, 1899, 1907, 1917, 2071, 2181, 2287, 2401, 2409, 2417, 2449, 2677, 2903, 2963

Offset: 1

Zak Seidov, Sep 21 2002

nonn

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

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 the first 10000 terms of the sequence there are 2417 primes. Indices i(n) of first prime in sequence of three primes with integer average are in A075541, for prime 3-averages i(n) are in A064113. Interprimes (s-averages with s=2) are all composite, see A024675. (Edited by Zak Seidov, Sep 01 2015 )

			a(1) = 5 = (1/3)(3+5+7), first integer average of three successive primes; next is: a(2) = 53 = (1/3)(47 + 53 + 59); up to n=8 all terms are prime; while a(9) = 511 = (1/3)( 503 + 509 + 521) is the first nonprime 3-average: 511=7*73.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A006562, A024675, A075541, A064113.

Cf. A102655.

a075540 n = a075540_list !! (n-1)
a075540_list = map fst $ filter ((== 0) . snd) $
   zipWith3 (\x y z -> divMod (x + y + z) 3)
            a000040_list (tail a000040_list) (drop 2 a000040_list)
-- Reinhard Zumkeller, Jan 20 2012

N:= 10^4: # to get all terms using primes <= N
Primes:= select(isprime,[2,seq(2*i+1, i=1..(N-1)/2)]):
select(type,(Primes[1..-3] + Primes[2..-2] + Primes[3..-1])/3,integer); # Robert Israel, Sep 01 2015

Select[MovingAverage[Prime[Range[500]],3],IntegerQ] (* Harvey P. Dale, Aug 10 2012 *)

a(n) = (1/3) (p(i)+p(i+1)+p(i+2)), for some i(n).

Comment and example edited, inefficient Mma removed by Zak Seidov, Sep 01 2015

A130178 Triangular numbers which are the average of two consecutive primes.

Original entry on oeis.org

6, 15, 21, 45, 105, 120, 231, 300, 351, 465, 741, 780, 861, 1176, 1431, 1485, 3081, 3240, 3321, 3828, 4005, 4278, 5460, 6786, 6903, 7140, 7381, 7503, 7875, 8001, 10731, 11175, 11325, 11781, 12246, 12561, 13530, 13695, 14535, 14706, 17205, 17391

Offset: 1

Zak Seidov, May 14 2007

nonn

Intersection of A000217 (triangular numbers) and A024675 (average of two consecutive primes).

			6-/+1, 15-/+2,..., 300-/+7 are pairs of consecutive primes.

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

Cf. A000217, A024675.

A205153 Least k such that n divides s(k)-s(j) for some j

Original entry on oeis.org

2, 2, 3, 4, 3, 4, 6, 4, 5, 8, 5, 6, 9, 6, 7, 10, 7, 9, 10, 8, 9, 8, 14, 9, 10, 9, 11, 10, 14, 10, 20, 14, 11, 16, 11, 12, 20, 12, 13, 22, 13, 16, 17, 14, 16, 14, 15, 16, 17, 15, 16, 15, 30, 16, 17, 16, 18, 17, 23, 17, 20, 31, 18, 20, 18, 19, 20, 19, 21, 20, 22, 20, 24

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

See A204892 for a discussion and guide to related sequences.

Cf. A024675, A204892, A205375.

s[n_] := s[n] = (Prime[n + 1] + Prime[n + 2])/2; z1 = 1100; z2 = 80;
Table[s[n], {n, 1, 30}]     (* A024675 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]     (* A204980 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
Table[d[n], {n, 1, z2}]     (* A205152 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}]       (* A205153 *)
Table[j[n], {n, 1, z2}]       (* A205154 *)
Table[s[k[n]], {n, 1, z2}]    (* A205372 *)
Table[s[j[n]], {n, 1, z2}]    (* A205373 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205374 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}]  (* A205375 *)

A162345 Length of n-th edge in the graph of the zig-zag function for prime numbers.

Original entry on oeis.org

2, 2, 2, 3, 3, 3, 3, 3, 5, 4, 4, 5, 3, 3, 5, 6, 4, 4, 5, 3, 4, 5, 5, 7, 6, 3, 3, 3, 3, 9, 9, 5, 4, 6, 6, 4, 6, 5, 5, 6, 4, 6, 6, 3, 3, 7, 12, 8, 3, 3, 5, 4, 6, 8, 6, 6, 4, 4, 5, 3, 6, 12, 9, 3, 3, 9, 10, 8, 6, 3, 5, 7, 7, 6, 5, 5, 7, 6, 6, 9, 6, 6, 6, 4, 5, 5

Offset: 1

Omar E. Pol, Jul 04 2009

Also, first differences of A162800.
Also {2, 2, } together with the numbers A052288.
Note that the graph of the zig-zag function for prime numbers is similar to the graph of the mountain path function for prime numbers but with exactly a vertex between consecutive odd noncomposite numbers (A006005).
This is the same as A115061 if n>1 (and also essentially equal to A052288). Proof: Because this is the first differences of A162800, which is {0,2} together with A024675, this sequence (for n>=3) is given by a(n) = (prime(n+1) - prime(n-1))/2. Similarly, because half the numbers between prime(n-1) and prime(n+1) are closer to prime(n) than any other prime, A115061(n) = (prime(n+1) - prime(n-1))/2 for n>=3 as well. - Nathaniel Johnston, Jun 25 2011

			Array begins:
=====
x, y
=====
2, 2;
2, 3;
3, 3;
3, 3;
5, 4;

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Omar E. Pol, Graph of the mountain path function for prime numbers

Cf. A000040, A006005, A008578, A024675, A052288, A162203, A162800, A162801, A162802.

[2,2] cat[(NthPrime(n+1)-NthPrime(n-1))/2: n in [3..80]]; // Vincenzo Librandi, Dec 19 2016

A162345 := proc(n) if(n<=2)then return 2: fi: return (ithprime(n+1) - ithprime(n-1))/2: end: seq(A162345(n),n=1..100); # Nathaniel Johnston, Jun 25 2011

Join[{2, 2}, Table[(Prime[n+1] - Prime[n-1])/2, {n, 3, 100}]] (* Vincenzo Librandi, Dec 19 2016 *)

a(n) = (prime(n+1) - prime(n-1))/2 for n>=3. - Nathaniel Johnston, Jun 25 2011

Edited by Omar E. Pol, Jul 16 2009

cp's OEIS Frontend

A075229 Numbers k such that k^6 is an interprime = average of two successive primes.

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A075230 Numbers k such that k^7 is an interprime = average of two successive primes.

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A075231 Numbers k such that k^8 is an interprime = average of two successive primes.

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A075232 Numbers k such that k^9 is an interprime = average of two successive primes.

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A075233 Numbers k such that k^10 is an interprime = average of two successive primes.

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A075234 Least k such that k^n is the smallest interprime which is an n-th power.

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A075540 Integers that are the average of three successive primes.

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