A071681 -id:A071681 - cp's OEIS frontend

A061357 Number of 0

0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 4, 1, 3, 4, 3, 3, 5, 4, 3, 5, 3, 3, 6, 2, 5, 6, 2, 5, 6, 4, 5, 7, 4, 4, 8, 4, 4, 9, 4, 4, 7, 3, 6, 8, 5, 5, 8, 6, 7, 10, 6, 5, 12, 3, 5, 10, 3, 7, 9, 5, 5, 8, 7, 7, 11, 5, 5, 12, 4, 8, 11, 4, 8, 10, 5, 5, 13, 9, 6, 11, 7, 6, 14, 6, 8, 13, 5, 8, 11, 6, 9

Offset: 1

Amarnath Murthy, Apr 28 2001

Number of prime pairs (p,q) with p < n < q and q-n = n-p.
The same as the number of ways n can be expressed as the mean of two distinct primes.
Conjecture: for n>=4 a(n)>0. - Benoit Cloitre, Apr 29 2003
Conjectures from Rick L. Shepherd, Jun 24 2003: (Start)
1) For each integer N>=1 there exists a positive integer m(N) such that for n>=m(N) a(n)>a(N). (After the first m(N)-1 terms, a(N) does not reappear). In particular, for N=1 (or 2 or 3), m(N)=4 and a(N)=0, giving Benoit Cloitre's conjecture. (cont.)
(cont.) Conjectures based upon observing a(1),...,a(10000):
m(4)=m(5)=m(6)=m(7)=m(19)=20 for a(4)=a(5)=a(6)=a(7)=a(19)=1,
m(8)=...(7 others)...=m(34)=35 for a(8)=...(7 others)...=a(34)=2,
m(12)=...(10 others)...=m(64)=65 for a(12)=...(10 others)...=a(64)=3,
m(18)=...(10 others)...=m(79)=80 for a(18)=...(10 others)...=a(79)=4,
m(24)=...(14 others)...=m(94)=95 for a(24)=...(14 others)...=a(94)=5,
m(30)=...(17 others)...=m(199)=200 for a(30)=...(17 others)...=a(199)=6, etc.
2) Each nonnegative integer appears at least once in the current sequence.
3) Stronger than 2): A001477 (nonnegative integers) is a subsequence of the current sequence. (Supporting evidence: I've observed that 0,1,2,...,175 is a subsequence of a(1),...,a(10000)).
(End)
a(n) is also the number of k such that 2*k+1=p and 2*(n-k-1)+1=q are both odd primes with p < q with p*q = n^2 - m^2. [Pierre CAMI, Sep 01 2008]
Also: Number of ways n^2 can be written as b^2+pq where 0a(n) = sum (A010051(2*n - p): p prime < n). [Reinhard Zumkeller, Oct 19 2011]
a(n) is also the number of partitions of 2*n into two distinct primes. See the first formula by T. D. Noe, and the Alois P. Heinz, Nov 14 2012, crossreference. - Wolfdieter Lang, May 13 2016
All 0Jamie Morken, Jun 02 2017
a(n) is the number of appearances of n in A143836. - Ya-Ping Lu, Mar 05 2023

			a(10)= 2: there are two such pairs (3,17) and (7,13), as 10 = (3+17)/2 = (7+13)/2.

Pierre CAMI, Table of n, a(n) for n = 1..60000

Cf. A071681 (subsequence for prime n only).
Cf. A092953.
Bisection of A117929 (even part). - Alois P. Heinz, Nov 14 2012

a061357 n = sum $
   zipWith (\u v -> a010051 u * a010051 v) [n+1..] $ reverse [1..n-1]
-- Reinhard Zumkeller, Nov 10 2012, Oct 19 2011

Table[Count[Range[n - 1], k_ /; And[PrimeQ[n - k], PrimeQ[n + k]]], {n, 98}] (* Michael De Vlieger, May 14 2016 *)

a(n)=my(s);forprime(p=2,n-1,s+=isprime(2*n-p));s \\ Charles R Greathouse IV, Mar 08 2013

from sympy import primerange, isprime
def A061357(n): return sum(1 for p in primerange(n) if isprime((n<<1)-p)) # Chai Wah Wu, Sep 03 2024

a(n) = A045917(n) - A010051(n). - T. D. Noe, May 08 2007
a(n) = sum(A010051(n-k)*A010051(n+k): 1 <= k < n). - Reinhard Zumkeller, Nov 10 2012
a(n) = sum_{i=2..n-1} A010051(i)*A010051(2n-i). [Wesley Ivan Hurt, Aug 18 2013]

More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001

A126204 Least prime representable as the arithmetic mean of two other primes in n different ways.

Original entry on oeis.org

2, 5, 11, 17, 37, 53, 89, 71, 101, 179, 137, 193, 233, 257, 251, 401, 311, 373, 347, 457, 661, 479, 547, 599, 617, 641, 683, 907, 881, 773, 827, 809, 941, 1033, 977, 1289, 1201, 1187, 1229, 1223, 1277, 1361, 1597, 1493, 1447, 1499, 1451, 1549, 1487, 1709, 1933

Offset: 0

Robert G. Wilson v, Mar 22 2007

nonn

a(n)=A000040(A137700(n)); A071681(A137700(n))=n and A071681(m)<>n for m < A137700(n). - Reinhard Zumkeller, Feb 07 2008

Robert G. Wilson v, Table of n, a(n) for n = 0..11300

Cf. A071681.

f[n_] := Block[{c = 0, k = PrimePi@n - 1}, While[k > 0, If[ PrimeQ[2n - Prime@k], c++ ]; k-- ]; c]; t = Table[0, {1000}]; Do[ p = Prime@n; a = f@p; If[a < 1001 && t[[a]] == 0, t[[a]] = p; Print[{a, p}]], {n, 8900}]

Start of b-file modified to match sequence by N. J. A. Sloane, Aug 31 2009

A071680 Primes that are the arithmetic mean of their prime predecessor and another prime.

Original entry on oeis.org

5, 37, 53, 67, 157, 173, 211, 257, 263, 277, 373, 479, 563, 593, 607, 613, 631, 653, 733, 809, 947, 977, 1009, 1103, 1123, 1187, 1223, 1297, 1367, 1471, 1511, 1607, 1663, 1721, 1747, 1753, 1783, 1867, 1901, 1907, 1931, 1993, 2137, 2287, 2377, 2411, 2417

Offset: 1

Reinhard Zumkeller, May 31 2002; revised Jul 16 2003

nonn

prime(n) where 2*prime(n) - prime(n-1) is prime. - Robert Israel, Dec 01 2015

			A000040(12) = 37, A000040(12-1) = 31, 37 = (31 + 43)/2, therefore 37 is a term.

Robert Israel, Table of n, a(n) for n = 1..11000

Cf. A071681, A006562 is a subsequence.

Primes:= select(isprime, [2,seq(i,i=1..10^4,2)]):
Primes[select(i -> isprime(2*Primes[i]-Primes[i-1]), [$2..nops(Primes)])]; # Robert Israel, Dec 01 2015

p = q = 2; lst = {}; Do[q = Prime@n; If[PrimeQ[2q - p], AppendTo[lst, q]]; p = q, {n, 2, 400}]; lst (* Robert G. Wilson v, Mar 22 2007 *)

lista(nn) = {forprime(p=5, nn, if (isprime(2*p-precprime(p-1)), print1(p, ", ")););} \\ Michel Marcus, Dec 01 2015

Thanks to Sven Simon for noticing errors in the original version.

A078497 The member r of a triple of primes (p,q,r) in arithmetic progression which sum to 3*prime(n) = A001748(n) = p + q + r.

Original entry on oeis.org

7, 11, 17, 19, 23, 31, 29, 41, 43, 43, 53, 67, 53, 59, 71, 79, 73, 83, 79, 97, 107, 107, 127, 113, 109, 113, 139, 137, 151, 149, 167, 151, 167, 163, 163, 199, 197, 179, 191, 199, 233, 223, 227, 241, 223, 283, 257, 277, 239, 251, 271, 263, 263, 269, 281, 313

Offset: 3

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), Nov 27 2002

nonn

In case more than one triple of primes p, q=p+d and r=p+2*d exists, we take r=a(n) from the triple with the smallest d. This shows the difference from A092940, which would take the maximum r over all triples. - R. J. Mathar, May 19 2007

			a(1) = 7 because 3+5+7 = 15;
a(2) = 11 because 3+7+11 = 21;
a(3) = 17 because 5+11+17= 33.

Cf. A078496, A078498, A001748, A092940, A071681, A078611.

A078497 := proc(n) local p3, i,d,r,p; p3 := ithprime(n) ; i := n+1 ; while true do r := ithprime(i) ; d := r-p3 ; p := p3-d ; if isprime(p) then RETURN(r) ; fi ; i := i+1 ; od ; RETURN(-1) ; end: for n from 3 to 60 do printf("%d, ",A078497(n)) ; od ; # R. J. Mathar, May 19 2007

f[n_] := Block[{p = Prime[n], k}, k = p + 1; While[ !PrimeQ[k] || !PrimeQ[2p - k], k++ ]; k]; Table[ f[n], {n, 3, 60}]

Edited and extended by Robert G. Wilson v, Nov 29 2002

Further edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A116619 a(n) = number of ways of representing 2*prime(n) as the unordered sum of two primes.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 2, 4, 4, 3, 5, 5, 5, 5, 6, 6, 4, 6, 8, 6, 5, 6, 7, 7, 9, 7, 8, 7, 7, 9, 9, 11, 7, 11, 9, 9, 7, 11, 9, 10, 8, 10, 12, 11, 7, 11, 12, 12, 9, 13, 11, 11, 15, 14, 15, 14, 10, 11, 14, 13, 13, 15, 17, 12, 14, 14, 15, 19, 14, 19, 15, 15, 18, 15, 17, 15, 17, 16, 17, 17, 18, 17

Offset: 1

Jonathan Vos Post, Mar 14 2006

2*prime(n) = A100484(n), the n-th even semiprime.

a(n) = A071681(n) + 1. - Reinhard Zumkeller, Mar 27 2015

			2*prime(23) = 166 can be represented in 6 ways as the unordered sum of two primes: 166 = 3+163 = 17+149 = 29+137 = 53+113 = 59+107 = 83+83, so a(23) = 6.
2*prime(54) = 502 can be represented in 15 ways as the unordered sum of two primes: 502 = 3+499 = 11+491 = 23+479 = 41+461 = 53+449 = 59+443 = 71+431 = 83+419 = 101+401 = 113+389 = 149+353 = 191+311 = 233+269 = 239+263 = 251+251, so a(54) = 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A045917, A100484.

Cf. A071681, A010051.

a116619 = (+ 1) . a071681  -- Reinhard Zumkeller, Mar 27 2015

{for(n=1,83,c=0;k=2*prime(n);forprime(p=2,prime(n),if(isprime(k-p),c++));print1(c,","))} \\ Klaus Brockhaus, Dec 23 2006

a(n) = A045917(A100484(n)).

Edited, corrected and extended by Klaus Brockhaus, Dec 23 2006

A071704 Number of ways to represent the n-th prime as arithmetic mean of three other odd primes.

Original entry on oeis.org

0, 0, 0, 2, 5, 7, 10, 14, 16, 24, 29, 31, 42, 40, 43, 52, 62, 70, 75, 87, 82, 96, 102, 112, 127, 137, 136, 142, 154, 154, 186, 199, 204, 215, 233, 248, 250, 262, 272, 284, 309, 324, 344, 334, 348, 358, 406, 414, 430, 446, 441, 489, 486, 511, 508

Offset: 1

Reinhard Zumkeller, Jun 03 2002

nonn

			a(5)=5 as A000040(5)=11 and there are no more representations not containing 11 than 11 = (3+7+23)/3 = (3+13+17)/3 = (5+5+23)/3 = (7+7+19)/3 = (7+13+13)/3.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A071681, A071703, A000040, A065091, A258233.

a071704 n = z (us ++ vs) 0 (3 * q)  where
   z _ 3 m = fromEnum (m == 0)
   z ps'@(p:ps) i m = if m < p then 0 else z ps' (i+1) (m - p) + z ps i m
   (us, _:vs) = span (< q) a065091_list; q = a000040 n
-- Reinhard Zumkeller, May 24 2015

N:= 300: # to get the first A000720(N) terms
P:= select(isprime, [seq(i,i=3..3*N,2)]):
nP:= nops(P):
V:= Vector(N):
for i from 1 to nP do
  for j from i to nP do
    for k from j to nP while P[i]+P[j]+P[k] <= 3*N do
      r:= (P[i]+P[j]+P[k])/3;
      if r::integer and isprime(r) and r <> P[j] and r <= N then V[r]:= V[r]+1 fi
od od od:
seq(V[ithprime(i)],i=1..numtheory:-pi(N)); # Robert Israel, Aug 09 2018

M = 300; (* to get the first A000720(M) *)
P = Select[Range[3, 3*M, 2], PrimeQ]; nP = Length[P]; V = Table[0, {M}];
For[i = 1, i <= nP, i++,
For[j = i, j <= nP, j++,
For[k = j, k <= nP && P[[i]] + P[[j]] + P[[k]] <= 3*M , k++, r = (P[[i]] + P[[j]] + P[[k]])/3; If[IntegerQ[r] && PrimeQ[r] && r != P[[j]] && r <= M, V[[r]] = V[[r]]+1]
]]];
Table[V[[Prime[i]]], {i, 1, PrimePi[M]}] (* Jean-François Alcover, Mar 09 2019, after Robert Israel *)

a(n, p=prime(n))=my(s=0); forprime(q=p+2, 3*p-4, my(t=3*p-q); forprime(r=max(t-q, 3), (3*p-q)\2, if(t!=p+r && isprime(t-r), s++))); s \\ Charles R Greathouse IV, Jun 04 2015

Definition corrected by Zak Seidov, May 24 2015

A071703 Number of ways to represent the n-th prime as arithmetic mean of three odd primes.

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 14, 16, 20, 28, 32, 36, 47, 45, 48, 58, 68, 74, 81, 95, 88, 101, 108, 119, 134, 146, 143, 150, 161, 161, 195, 208, 215, 222, 244, 257, 259, 269, 283, 293, 319, 332, 354, 346, 359, 365, 417, 426, 442, 455, 454, 500, 497, 526

Offset: 1

Reinhard Zumkeller, Jun 03 2002

nonn

			a(4)=4 as A000040(4)=7 and there are no more representations than 7 = (3+5+13)/3 = (3+7+11)/3 = (5+5+11)/3 = (7+7+7)/3.

Cf. A071681, A071704.

Cf. A010051, A000040, A065091.

a071703 = z a065091_list 0 . (* 3) . a000040 where
   z _ 3 m = fromEnum (m == 0)
   z ps'@(p:ps) i m = if m < p then 0 else z ps' (i+1) (m - p) + z ps i m
-- Reinhard Zumkeller, May 24 2015

Definition, initial term and example corrected. Thanks to Zak Seidov, who found the mistake. - Reinhard Zumkeller, May 24 2015

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.

A137700 Smallest k such that the k-th prime has exactly n distinct representations as arithmetic mean of two primes.

Original entry on oeis.org

1, 3, 5, 7, 12, 16, 24, 20, 26, 41, 33, 44, 51, 55, 54, 79, 64, 74, 69, 88, 121, 92, 101, 109, 113, 116, 124, 155, 152, 137, 144, 140, 160, 174, 165, 209, 197, 195, 201, 200, 206, 218, 251, 238, 229, 239, 230, 244, 236, 267, 295, 281, 299, 301, 307, 312, 313, 325

Offset: 0

Reinhard Zumkeller, Feb 07 2008

nonn

A071681(a(n))=n and A071681(m)<>n for mA000040(a(n))=A126204(n).

Corrected description suggested by Zak Seidov, May 16 2008

A253138 Number of ways to represent the n-th prime as the arithmetic mean of two semiprimes.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 4, 3, 4, 3, 3, 5, 7, 6, 5, 5, 8, 8, 7, 9, 7, 10, 10, 12, 11, 15, 12, 14, 14, 13, 11, 13, 15, 15, 14, 15, 20, 14, 15, 19, 20, 16, 17, 17, 17, 21, 24, 23, 24, 28, 23, 25, 24, 27, 25, 32, 29, 25, 21, 26, 31, 31, 29, 36, 32

Offset: 1

Michel Lagneau, Mar 23 2015

nonn

Conjecture: a(n)>0 for n>5.
Note that a(n) = A241535(n) = A241536(n) = 0 for n=1,2 and 5. - Michel Marcus, Mar 26 2015
Among the a(n) decompositions of prime(n) into two semiprimes (prime(n)+ k)/2 and (prime(n)-k)/2, there is one where k is minimum with k = A241536(n) and there is one where k is maximum with k = prime(n) - A241535(n).

			a(12)=3 as prime(12) = 37 = (9+65)/2 = (25+49)/2 =(35+39)/2 where 9, 25, 35, 39, 49 and 65 are semiprime.

Michel Lagneau, Table of n, a(n) for n = 1..1000

Cf. A001358, A071681, A241535, A241536.

Cf. A064911, A000040.

a253138 n = sum $ map a064911 $
   takeWhile (> 0) $ map (2 * p -) $ dropWhile (< p) a001358_list
   where p = a000040 n
-- Reinhard Zumkeller, Mar 27 2015

with(numtheory):for n from 1 to 100 do:c:=0:p:=ithprime(n):for m from 1 to p-1 do:p1:=p-m:p2:=p+m:if bigomega(p1)=2 and bigomega(p2)=2 then c:=c+1:else fi:od:printf(`%d, `,c):od:

Reap[For[n=1, n <= 100, n++, c=0; p = Prime[n]; For[m=1, m <= p-1, m++, p1 = p-m; p2 = p+m; If[PrimeOmega[p1] == 2 && PrimeOmega[p2] == 2 , c = c+1]]; Print[c]; Sow[c]]][[2, 1]] (* Jean-François Alcover, Mar 23 2015, translated from Maple *)

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A061357 Number of 0

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A126204 Least prime representable as the arithmetic mean of two other primes in n different ways.

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A071680 Primes that are the arithmetic mean of their prime predecessor and another prime.

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A078497 The member r of a triple of primes (p,q,r) in arithmetic progression which sum to 3*prime(n) = A001748(n) = p + q + r.

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A116619 a(n) = number of ways of representing 2*prime(n) as the unordered sum of two primes.

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A071704 Number of ways to represent the n-th prime as arithmetic mean of three other odd primes.

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A071703 Number of ways to represent the n-th prime as arithmetic mean of three odd primes.

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