cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Zak Seidov, Sep 09 2002

Keywords

Comments

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.

Examples

			20 is a term because 20^7 = 1280000000 is the average of two successive primes 1279999997 and 1280000003.

Links

Crossrefs

Cf. A024675, A072568, A072569, A075190, A075191, A075192.
Cf. A075228, A075229, A075231, A075232, A075233, A075234.

Programs

  • Maple
    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;
  • Mathematica
    Select[Range[2000],Mean[{NextPrime[#^7],NextPrime[#^7,-1]}]==#^7&] (* Harvey P. Dale, Aug 09 2013 *)

Extensions

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

Views

Author

Zak Seidov, Sep 09 2002

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A024675, A072568, A072569, A075190, A075191, A075192.
Cf. A075228, A075229, A075230, A075232, A075233, A075234.

Programs

  • Maple
    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;
  • Mathematica
    Select[Range[2000], 2#^8 == NextPrime[#^8, -1] + NextPrime[#^8] &]

Extensions

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

Views

Author

Zak Seidov, Sep 09 2002

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A024675, A072568, A072569, A075190, A075191, A075192.
Cf. A075228, A075229, A075230, A075231, A075233, A075234.

Programs

  • Maple
    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;
  • Mathematica
    Select[Range[2869], 2#^9 == NextPrime[#^9, -1] + NextPrime[#^9] &]

Extensions

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

Views

Author

Zak Seidov, Sep 09 2002

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A024675, A072568, A072569, A075190, A075191, A075192.
Cf. A075228, A075229, A075230, A075231, A075232, A075234.

Programs

  • Maple
    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;
  • Mathematica
    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 *)

Extensions

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

Views

Author

Zak Seidov, Sep 09 2002

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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.

Programs

  • Maple
    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;
  • Mathematica
    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

Extensions

Edited and extended by Robert G. Wilson v, Sep 14 2002
Typos in EXAMPLE fixed by Zak Seidov, Feb 09 2012

A124619 Odd interprimes divisible by 13.

Original entry on oeis.org

39, 195, 351, 741, 897, 1313, 1443, 1599, 2379, 2405, 2535, 2613, 2691, 2847, 3055, 3081, 3627, 3705, 4641, 4771, 5031, 5577, 5655, 5889, 5967, 6045, 6201, 6409, 6825, 6877, 6903, 7007, 7033, 7605, 7943, 8437, 8541, 8931, 8957, 9009, 9035, 9321, 9607
Offset: 1

Views

Author

Artur Jasinski, Dec 21 2006

Keywords

Links

Crossrefs

Cf. A072569, A072572, A072573, A126228, A126229, A126230, A124620, A126231, A124622, A126233.

Programs

  • Mathematica
    Do[If[OddQ[(Prime[n + 1] + Prime[n])/2] && Mod[(Prime[n + 1] + Prime[n])/2, 13] == 0, Print[(Prime[n + 1] + Prime[n])/2]], {n, 1, 3000}] (*Artur Jasinski*)
Select[(Total/@Partition[Prime[Range[1200]],2,1])/2,OddQ[#]&&Mod[#,13] == 0&] (* Harvey P. Dale, May 23 2019 *)

Extensions

Extended by Ray Chandler, Jan 09 2007

A124620 Odd interprimes divisible by 17.

Original entry on oeis.org

459, 765, 969, 1003, 1207, 1377, 1581, 1785, 2295, 2601, 2703, 3213, 3723, 3995, 4131, 4403, 4641, 4947, 5933, 5967, 6409, 6477, 6579, 6613, 6749, 6987, 7701, 8075, 8517, 9435, 9741, 9775, 9877, 9945, 10217, 10455, 10897, 10965, 11305, 11645
Offset: 1

Views

Author

Artur Jasinski, Dec 21 2006

Keywords

Links

Crossrefs

Cf. A072569, A072572, A072573, A126228, A126229, A126230, A124619, A126231, A124622, A126233.

Programs

  • Mathematica
    Do[If[OddQ[(Prime[n + 1] + Prime[n])/2] && Mod[(Prime[n + 1] + Prime[n])/2, 17] == 0, Print[(Prime[n + 1] + Prime[n])/2]], {n, 1, 3000}] (*Artur Jasinski*)
Select[Range[17,12000,34],!PrimeQ[#]&&#-NextPrime[#,-1]==NextPrime[#]-#&] (* Harvey P. Dale, Nov 18 2012 *)

Extensions

Extended by Ray Chandler, Jan 09 2007

A126228 Odd interprimes divisible by 5.

Original entry on oeis.org

15, 45, 105, 165, 195, 205, 225, 315, 405, 465, 495, 515, 615, 625, 645, 675, 705, 765, 825, 855, 885, 915, 1095, 1215, 1305, 1425, 1465, 1485, 1505, 1575, 1665, 1695, 1715, 1785, 1795, 1875, 1895, 1995, 2075, 2085, 2105, 2205, 2295, 2405, 2475, 2535, 2585
Offset: 1

Views

Author

Artur Jasinski, Dec 21 2006

Keywords

Links

Crossrefs

Cf. A072569, A072572, A072573, A126229, A126230, A124619, A124620, A126231, A124622, A126233.

Programs

  • Mathematica
    Do[If[OddQ[(Prime[n + 1] + Prime[n])/2] && Mod[(Prime[n + 1] + Prime[n])/2, 5] == 0, Print[(Prime[n + 1] + Prime[n])/2]], {n, 1, 3000}] (*Artur Jasinski*)
Select[Select[Mean/@Partition[Prime[Range[400]],2,1],OddQ],Divisible[#,5]&]  (* Harvey P. Dale, Feb 06 2011 *)

Extensions

Extended by Ray Chandler, Jan 09 2007

A126229 Odd interprimes divisible by 7.

Original entry on oeis.org

21, 105, 217, 231, 315, 399, 441, 483, 861, 987, 1113, 1197, 1281, 1449, 1491, 1505, 1575, 1715, 1785, 1827, 1869, 1995, 2121, 2191, 2205, 2247, 2303, 2541, 2667, 2709, 2751, 2835, 2933, 3045, 3073, 3129, 3213, 3255, 3353, 3381, 3423, 3465, 3675, 3885
Offset: 1

Views

Author

Artur Jasinski, Dec 21 2006

Keywords

Links

Crossrefs

Cf. A072569, A072572, A072573, A126228, A126230, A124619, A124620, A126231, A124622, A126233.

Programs

  • Mathematica
    Do[If[OddQ[(Prime[n + 1] + Prime[n])/2] && Mod[(Prime[n + 1] + Prime[n])/2, 7] == 0, Print[(Prime[n + 1] + Prime[n])/2]], {n, 1, 3000}]

Extensions

Extended by Ray Chandler, Jan 09 2007

A126230 Odd interprimes divisible by 11.

Original entry on oeis.org

99, 165, 231, 363, 473, 495, 759, 803, 825, 1089, 1243, 1485, 1551, 2277, 2453, 2475, 2541, 2585, 2783, 3399, 3465, 3531, 3795, 4521, 4587, 4697, 4785, 4807, 5093, 5203, 5577, 5841, 6831, 7007, 7315, 7381, 7425, 7689, 7755, 8437, 8635, 8679, 8855, 8877
Offset: 1

Views

Author

Artur Jasinski, Dec 21 2006

Keywords

Links

Crossrefs

Cf. A072569, A072572, A072573, A126228, A126229, A124619, A124620, A126231, A124622, A126233.

Programs

  • Mathematica
    Do[If[OddQ[(Prime[n + 1] + Prime[n])/2] && Mod[(Prime[n + 1] + Prime[n])/2, 11] == 0, Print[(Prime[n + 1] + Prime[n])/2]], {n, 1, 3000}]
Select[Mean/@Partition[Prime[Range[2,2000]],2,1],OddQ[#]&&Divisible[#,11]&] (* Harvey P. Dale, Mar 14 2021 *)
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