A102552 -id:A102552 - cp's OEIS frontend

A122535 Smallest prime of a triple of successive primes, where the middle one is the arithmetic mean of the other two.

3, 47, 151, 167, 199, 251, 257, 367, 557, 587, 601, 647, 727, 941, 971, 1097, 1117, 1181, 1217, 1361, 1499, 1741, 1747, 1901, 2281, 2411, 2671, 2897, 2957, 3301, 3307, 3631, 3727, 4007, 4397, 4451, 4591, 4651, 4679, 4987, 5101, 5107, 5297, 5381, 5387

Offset: 1

Miklos Kristof, Sep 18 2006

nonn

Subsets are A047948, A052188, A052189, A052190, A052195, A052197, A052198, etc. - R. J. Mathar, Apr 11 2008
Could be generated by searching for cases A001223(i) = A001223(i+1), writing down A000040(i). - R. J. Mathar, Dec 20 2008
a(n) = A006562(n) - A117217(n). - Zak Seidov, Feb 12 2013
These are primes for which the subsequent prime gaps are equal, so (p(k+2)-p(k+1))/(p(k+1)-p(k)) = 1. It is conjectured that prime gaps ratios equal to one are less frequent than those equal to 1/2, 2, 3/2, 2/3, 1/3 and 3. - Andres Cicuttin, Nov 07 2016

			The prime 7 is not in the list, because in the triple (7, 11, 13) of successive primes, 11 is not equal (7 + 13)/2 = 10.
The second term, 47, is the first prime in the triple (47, 53, 59) of primes, where 53 is the mean of 47 and 59.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primes in arithmetic progressions

Cf. A006562, A062839, A102552, A117217, A181424.

a122535 = a000040 . a064113  -- Reinhard Zumkeller, Jan 20 2012

Clear[d2, d1, k]; d2[n_] = Prime[n + 2] - 2*Prime[n + 1] + Prime[n]; d1[n_] = Prime[n + 1] - Prime[n]; k[n_] = -d2[n]/(1 + d1[n])^(3/2); Flatten[Table[If[k[n] == 0, Prime[n], {}], {n, 1, 1000}]] (* Roger L. Bagula, Nov 13 2008 *)
Transpose[Select[Partition[Prime[Range[750]], 3, 1], #[[2]] == (#[[1]] + #[[3]])/2 &]][[1]]  (* Harvey P. Dale, Jan 09 2011 *)

A122535()={n=3;ctr=0;while(ctr<50, avgg=( prime(n-2)+prime(n) )/2;
if( prime(n-1) ==avgg, ctr+=1;print( ctr,"  ",prime(n-2) )  );n+=1); } \\ Bill McEachen, Jan 19 2015

{A000040(i): A000040(i+1)= (A000040(i)+A000040(i+2))/2 }. - R. J. Mathar, Dec 20 2008

a(n) = A000040(A064113(n)). - Reinhard Zumkeller, Jan 20 2012

More terms from Roger L. Bagula, Nov 13 2008

Definition rephrased by R. J. Mathar, Dec 20 2008

A102553 Numbers k such that for all prime-factors p: p = (k AND p), bitwise.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 51, 53, 59, 61, 63, 67, 71, 73, 79, 83, 85, 89, 95, 97, 101, 103, 107, 109, 111, 113, 119, 123, 125, 127, 131, 135, 137, 139, 143, 149, 151, 157, 163, 167, 173, 175, 179, 181, 187, 191, 193, 197, 199

Offset: 1

Reinhard Zumkeller, Jan 14 2005

Numbers k such that A102550(k) = A001221(k).
Apart from first term, subsequence of A102552;
A000040 is a subsequence.
Numbers k such that the bitwise OR of k with all prime divisors of k is equal to k. - Chai Wah Wu, Dec 18 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A001221, A102550, A102552, A102554, A007088, A004676.

okQ[n_] := AllTrue[FactorInteger[n][[All, 1]], # == BitAnd[n, #]&];
Select[Range[200], okQ] (* Jean-François Alcover, Nov 16 2021 *)

from itertools import count, islice
from operator import ior
from functools import reduce
from sympy import primefactors
def A102553_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n == 1 or n|reduce(ior,primefactors(n))==n,count(max(startvalue,1)))
A102553_list = list(islice(A102553_gen(),20)) # Chai Wah Wu, Dec 18 2022

cp's OEIS Frontend

A122535 Smallest prime of a triple of successive primes, where the middle one is the arithmetic mean of the other two.

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