A072568 -id:A072568 - cp's OEIS frontend

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

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

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

cp's OEIS Frontend

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

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