A178954 -id:A178954 - cp's OEIS frontend

A178670 Number of ways to express prime(n) as (prime(n+k) + prime(n-k))/2.

0, 0, 1, 0, 2, 1, 1, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 1, 3, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 2, 2, 2, 1, 0, 1, 3, 0, 1, 3, 2, 1, 4, 2, 1, 0, 4, 0, 0, 0, 2, 0, 0, 2, 1, 1, 0, 0, 1, 0, 0, 0, 3, 4, 0, 0, 1, 1, 1, 2, 3, 3, 2, 4, 2, 2, 1, 3, 4, 4, 1, 1, 1, 3, 2, 2, 2, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 3, 0, 2

Offset: 1

Juri-Stepan Gerasimov, Jan 04 2011

nonn

			a(5) = 2 because the 5th prime (11) is half the sum of the 7th and 3rd prime (17+5) or half the sum of the 8th and 2nd prime (19+3).
a(8) = 0 because the 8th prime (19) cannot be expressed as (1/2)*(prime(8+k) + prime(8-k)) for any k.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000040, A129363, A178609, A178954.

nn=1000; p=Prime[Range[2*nn]]; Table[s=Take[p,n-1] + Reverse[Take[p, {n+1,2n-1}]]; Count[s,2*p[[n]]], {n,nn}]

a(n)={s=2*prime(n);a=0;for(i=1,n-1,if(prime(n+i)+prime(n-i)==s,a=a+1));a}

A178953 Indices n such that 2*prime(n) cannot be written as a sum of two distinct prime(n-k) and prime(n+k).

Original entry on oeis.org

1, 2, 4, 8, 9, 14, 15, 21, 22, 29, 30, 35, 38, 46, 48, 49, 50, 52, 53, 57, 58, 60, 61, 62, 65, 66, 90, 91, 95, 96, 97, 99, 114, 120, 121, 122, 123, 124, 125, 128, 145, 146, 149, 153, 154, 163, 176, 179, 180, 186, 187, 189, 191, 192, 197

Offset: 1

Juri-Stepan Gerasimov, Jan 02 2011

nonn

Snapshots: a(1000) = 6922, a(2000) = 16376, a(3000) = 25951, a(4000) = 37266, a(5000) = 51926, a(6000) = 69928. - R. J. Mathar, Jan 08 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..500

Cf. A178609, A071681, A178954.

a178953 n = a178953_list !! (n-1)
a178953_list = filter ((== 0) . a178609) [1..]
-- Reinhard Zumkeller, Jan 30 2014

A178609 := proc(n) for k from n-1 to 0 by -1 do if ithprime(n-k)+ithprime(n+k)=2*ithprime(n) then return k; end if; end do: end proc:
for n from 1 to 200 do if A178609(n) = 0 then printf("%d,",n) ; end if; end do: # R. J. Mathar, Jan 05 2011

A178609[n_] := For[k = n-1, k >= 0, k--, If[Prime[n-k] + Prime[n+k] == 2*Prime[n], Return[k]]]; Reap[For[n = 1, n <= 200, n++, If[A178609[n] == 0, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 13 2018, after R. J. Mathar *)

A178609(a(n))=0.

A127925 Primes p such that 2p < prime(k-i) + prime(k+i) for i=1..k-1, where p=prime(k).

Original entry on oeis.org

3, 7, 19, 23, 43, 47, 73, 109, 113, 199, 283, 293, 313, 317, 463, 467, 503, 509, 523, 619, 661, 683, 691, 887, 1063, 1069, 1109, 1129, 1303, 1307, 1321, 1327, 1613, 1621, 1627, 1637, 1669, 1789, 2143, 2161, 2383, 2393, 2399, 2477, 2731, 2753, 2803, 2861, 2971

Offset: 1

T. D. Noe, Feb 06 2007

nonn

One of several sets of "good primes" in section A14 of Guy.

McNew calls these numbers "midpoint convex primes". - Peter Munn, Jul 04 2025

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed. Springer, 2004.

T. D. Noe, Table of n, a(n) for n=1..10001
Nathan McNew, Popular values of the largest prime divisor function (corrected version), page 16, November 2015.

Cf. A028388.
A246033 is a subset.
Subset of A124661, A178954.

t={}; n=1; While[Length[t]<100, n++; p=Prime[n]; i=1; While[i

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A178670 Number of ways to express prime(n) as (prime(n+k) + prime(n-k))/2.

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A178953 Indices n such that 2*prime(n) cannot be written as a sum of two distinct prime(n-k) and prime(n+k).

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A127925 Primes p such that 2p < prime(k-i) + prime(k+i) for i=1..k-1, where p=prime(k).

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