A075231 -id:A075231 - 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

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

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

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

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

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

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

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

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