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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A076559 Greatest prime divisor of n-th interprime: (prime(n) + prime(n+1))/2.

Original entry on oeis.org

2, 3, 3, 3, 5, 3, 7, 13, 5, 17, 13, 7, 5, 5, 7, 5, 2, 23, 3, 19, 3, 43, 31, 11, 17, 7, 3, 37, 5, 43, 67, 23, 3, 5, 11, 5, 11, 17, 11, 5, 31, 3, 13, 11, 41, 31, 5, 19, 11, 59, 5, 41, 127, 13, 19, 5, 137, 31, 47, 3, 5, 103, 13, 7, 3, 167, 19, 29, 13, 89, 11, 37, 47, 127, 193, 131, 19
Offset: 2

Views

Author

Zak Seidov, Oct 19 2002

Keywords

Links

Crossrefs

Cf. A071216.

Programs

  • Maple
    A076559 := proc(n)
    A006530((ithprime(n)+ithprime(n+1))/2) ;
end proc:
seq(A076559(n),n=2..120) ;  # R. J. Mathar, May 10 2023
  • Mathematica
    gpf[n_] := FactorInteger[n][[-1, 1]]; p = Select[Range[405], PrimeQ]; gpf /@ ((p[[2 ;; -2]] + p[[3 ;; -1]])/2) (* Amiram Eldar, Aug 29 2019 *)

Formula

a(n) = A006530(A024675(n-1)). - R. J. Mathar, May 10 2023

A076693 Numbers n such that the average of prime(n) and prime(n+1) is a perfect cube.

Original entry on oeis.org

18, 269, 564, 2020, 2460, 4821, 10499, 17583, 25030, 27154, 34238, 39583, 82915, 92295, 107454, 121248, 142474, 148961, 159031, 162482, 233036, 250923, 342833, 365844, 427518, 460792, 467647, 481615, 741275
Offset: 1

Views

Author

Joseph L. Pe, Oct 25 2002

Keywords

Examples

			The average of prime(18) and prime(19) is (61 + 67)/2 = 4^3, so 18 is a term of the sequence.

Links

Crossrefs

Cf. A000578, A024675.

Programs

  • Mathematica
    Select[Range[10^6], IntegerQ[((Prime[ # ] + Prime[ # + 1])/2)^(1/3)] &]

A079721 Interemirp sequence: average of emirp pairs, analogous to interprimes.

Original entry on oeis.org

15, 24, 34, 54, 72, 76, 88, 102, 110, 131, 153, 162, 173, 189, 255, 324, 342, 353, 374, 545, 705, 721, 736, 741, 747, 756, 765, 838, 922, 939, 947, 960, 969, 977, 987, 1000, 1015, 1026, 1032, 1047, 1065, 1080, 1094, 1100, 1106, 1130, 1152, 1167, 1187
Offset: 1

Views

Author

Michael Joseph Halm, Feb 17 2003

Keywords

Examples

			a(1) = 15 because the first two emirps are 13 and 17 and (13 + 17)/2 = 15

Links

Crossrefs

Cf. A006567, A024675.

Programs

  • Mathematica
    Map[Apply[Total, {##}]/2 &, Partition[Select[Prime[Range[5, 196]], And[PrimeQ@ IntegerReverse[#], ! PalindromeQ[#] ] &], 2, 1]] (* Michael De Vlieger, Aug 26 2025 *)

Formula

a(n) = (A006567(n) + A006567(n+1))/2 where A006567(n) is the n-th emirp.

Extensions

a(4) corrected by Sean A. Irvine, Aug 25 2025

A092576 a(n) = the least number which is the average of two consecutive primes and has exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

4, 12, 56, 72, 64, 192, 960, 1152, 3840, 7168, 4096, 30720, 36864, 110592, 360448, 663552, 2064384, 786432, 3932160, 5242880, 9437184, 63700992, 138412032, 169869312, 436207616, 3875536896, 1358954496, 1879048192, 10066329600, 8053063680, 14495514624
Offset: 2

Views

Author

Zak Seidov, Feb 29 2004

Keywords

Comments

The sequence is non-monotonic: a(6)

Examples

			a(3)=12 because 12=(11+13)/2 and 12=2*2*3 has 3 prime factors.

Links

Crossrefs

Cf. A024675, A078443.

Programs

  • Mathematica
    Flatten[With[{m=Mean/@Partition[Prime[Range[2,370000]],2,1]},Table[ Select[ m,PrimeOmega[#]==n&,1],{n,2,20}]]] (* To generate 30 rather than 20 terms of the sequence, change 370000 to 458000000 and 20 to 30. *) (* Harvey P. Dale, Jun 22 2013 *)

Extensions

Edited by Don Reble, Mar 17 2007

A097330 In the sequence of prime numbers replace each term p with floor(p/2) and ceiling(p/2).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 14, 15, 15, 16, 18, 19, 20, 21, 21, 22, 23, 24, 26, 27, 29, 30, 30, 31, 33, 34, 35, 36, 36, 37, 39, 40, 41, 42, 44, 45, 48, 49, 50, 51, 51, 52, 53, 54, 54, 55, 56, 57, 63, 64, 65, 66, 68, 69, 69, 70, 74, 75, 75, 76, 78, 79
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 17 2004

Keywords

Comments

a(2*n-1) + a(2*n) = A000040(n);
a(2*n) + a(2*n+1) = A024675(n-1) for n > 1;
a(2*n+1) = A005097(n), a(2*(n+1)) = A006254(n).
This is a subsequence of A180108. - Parthasarathy Nambi, Aug 14 2010
Sum_{n>=1} (-1)^a(n) * log(a(n)) = log(2). - Dimitris Valianatos, Jun 14 2016

Examples

			__ 2; __ 3; __ 5; __ 7; _ 11; _ 13; _ 17; __ 19; ...
1, 1; 1, 2; 2, 3; 3, 4; 5, 6; 6, 7; 8, 9; 9, 10; ...
1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, ....

Programs

  • PARI
    a(n)=if(n<3,return(1),return((prime((n+n%2)/2)+1)/2-n%2)); \\ Dimitris Valianatos, Jun 14 2016

Formula

a(n) = 1 if n < 3, otherwise (prime((n + (n mod 2))/2) + 1)/2 - n mod 2.

A109270 Numbers k such that k^2 > (1/2)*(prevprime(k^2) + nextprime(k^2)).

Original entry on oeis.org

4, 6, 10, 11, 14, 16, 17, 20, 22, 24, 26, 28, 30, 31, 36, 38, 39, 40, 45, 48, 50, 52, 54, 56, 57, 59, 62, 65, 66, 67, 70, 73, 74, 76, 79, 81, 84, 85, 87, 90, 91, 94, 95, 96, 97, 99, 100, 104, 105, 106, 109, 110, 111, 114, 115, 116, 120, 122, 123, 124, 125, 126, 130, 134
Offset: 1

Views

Author

Zak Seidov, Jun 24 2005

Keywords

Comments

One may call these k^2 the "strong squares" by analogy with A051634 (strong primes).

Examples

			4^2=16>(13+17)/2 so 4 is a term;
5^2 < (23+29)/2=26, so 5 is not a term;
6^2=36>(31+37)/2 so 6 is a term, etc.

Crossrefs

Cf. A051634, A033597, A075190, A109269.
Cf. A024675.

Programs

  • Maple
    a:=proc(n) if n^2 > (1/2)*(prevprime(n^2)+nextprime(n^2)) then n else fi end: seq(a(n),n=2..150); # Emeric Deutsch, Jun 26 2005
  • Mathematica
    prQ[n_]:=Module[{n2=n^2},n2>(NextPrime[n2]+NextPrime[n2,-1])/2]; Select[ Range[2,150],prQ] (* Harvey P. Dale, Feb 19 2012 *)

Extensions

More terms from Emeric Deutsch, Jun 26 2005

A114010 a(1) = a(2) = 2, Let k(n) = (prime(n) + prime(n+1))/2. Then a(k(n)) = k(n). a(k(n)-i) = prime(n), a(k(n)+i) = prime(n+1) until the next prime occurs.

Original entry on oeis.org

2, 2, 3, 4, 5, 6, 7, 7, 9, 11, 11, 12, 13, 13, 15, 17, 17, 18, 19, 19, 21, 23, 23, 23, 23, 26, 29, 29, 29, 30, 31, 31, 31, 34, 37, 37, 37, 37, 39, 41, 41, 42, 43, 43, 45, 47, 47, 47, 47, 50, 53, 53, 53, 53, 53, 56, 59, 59, 59, 60, 61, 61, 61, 64, 67, 67, 67, 67, 69, 71, 71, 72
Offset: 1

Views

Author

Amarnath Murthy, Nov 12 2005

Keywords

Comments

a(n) is the nearest prime to n, or n if there is a tie. - Wesley Ivan Hurt, May 15 2021

Examples

			(7 + 11)/2 = 9 hence a(9) = 9, a(8) = 7, a(7) = 7, a(10) = 11, a(11) = 11.

Crossrefs

Cf. A024675.

Programs

  • Maple
    A114010 := proc(n) local i,a024675 ; if n <= 2 then 2 ; else for i from 1 do if n >= ithprime(i) and n <= ithprime(i+1) then a024675 := (ithprime(i)+ithprime(i+1))/2 ; if n = a024675 then RETURN(a024675) ; elif n < a024675 then RETURN(ithprime(i)) ; else RETURN(ithprime(i+1)) ; fi ; fi ; od: fi ; end: seq(A114010(n),n=1..120) ; # R. J. Mathar, Feb 06 2008

Extensions

More terms from R. J. Mathar, Feb 06 2008

A122807 Least positive k such that n is equal to arithmetic mean of k consecutive primes, or 0 if no such k exists.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 2, 4, 1, 0, 0, 2, 0, 0, 1, 2, 1, 0, 0, 2, 0, 0, 1, 4, 2, 0, 1, 2, 1, 0, 2, 4, 1, 0, 11, 2, 8, 22, 1, 0, 4, 2, 0, 0, 1, 2, 1, 0, 0, 2, 0, 0, 1, 4, 2, 28, 1, 2, 1, 0, 0, 2, 6, 10, 1, 0, 2, 6, 1, 0, 0, 2, 4, 0, 1, 0, 12, 6, 2, 0, 21, 8, 1, 0, 2, 40, 1, 2
Offset: 1

Views

Author

Ray Chandler, Sep 25 2006

Keywords

Comments

Inspired by A122480.

Crossrefs

Cf. A060864 (indices of 0 terms), A000040 (indices of 1 terms), A024675 (indices of 2 terms).

A123993 Primes p such that p^2 is an interprime = average of two successive primes.

Original entry on oeis.org

2, 3, 41, 907, 1151, 1553, 1609, 1667, 1801, 1907, 1933, 2351, 2473, 2531, 2953, 3001, 3571, 4007, 4073, 4253, 4663, 5023, 5417, 5881, 6143, 6257, 6329, 6343, 7879, 8461, 8521, 8563, 9041, 9067, 10103, 10781, 11243, 11251, 11257, 12097, 12413, 13217
Offset: 1

Views

Author

Alexander Adamchuk, Oct 30 2006

Keywords

Comments

Primes in A075190 (numbers n such that n^2 is an interprime).

Crossrefs

Cf. A075190, A024675 (interprimes).

Programs

  • Mathematica
    Select[PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Select[ Range[25000], 2#^2 == PrevPrim[ #^2] + NextPrim[ #^2] &],PrimeQ]
atsp[n_]:=Module[{n2=n^2},(NextPrime[n2]+NextPrime[n2,-1])/2==n2]; Select[Prime[Range[2000]],atsp]  (* Harvey P. Dale, Jan 05 2011 *)
  • PARI
    isok(p) = isprime(p) && ((nextprime(p^2) + precprime(p^2)) / 2 - p^2 == 0); \\ Michel Marcus, Dec 11 2020

A143836 Triangle read by rows: T(r,c) = (prime(r+2) + prime(c+1))/2.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 8, 9, 10, 12, 10, 11, 12, 14, 15, 11, 12, 13, 15, 16, 18, 13, 14, 15, 17, 18, 20, 21, 16, 17, 18, 20, 21, 23, 24, 26, 17, 18, 19, 21, 22, 24, 25, 27, 30, 20, 21, 22, 24, 25, 27, 28, 30, 33, 34, 22, 23, 24, 26, 27, 29, 30, 32, 35, 36, 39, 23, 24, 25, 27, 28, 30, 31, 33, 36, 37, 40, 42
Offset: 1

Views

Author

Pierre CAMI, Sep 02 2008

Keywords

Comments

The number of appearances of m >= 1 in this sequence is A061357(m). Conjecture: Every integer >= 4 appears at least once in this sequence. - Ya-Ping Lu, Mar 05 2023
The number of composites between 3 and (r+2)-th prime missing from Row 1 through Row r in the triangle is A334810(r+2). - Ya-Ping Lu, Mar 24 2023

Examples

			Triangle begins:
   4;
   5,  6;
   7,  8,  9;
   8,  9, 10, 12;
  10, 11, 12, 14, 15;
  ...

Crossrefs

Cf. A098090 (1st column, except 1st term), A024675 (right diagonal).

Programs

  • PARI
    T(r,c) = (prime(r+2) + prime(c+1))/2; \\ Michel Marcus, Mar 07 2023

Extensions

Name simplified by Ya-Ping Lu, Mar 05 2023
Previous Showing 71-80 of 101 results. Next

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