A060863 -id:A060863 - cp's OEIS frontend

A050221 a(n) = number of sets of consecutive primes whose arithmetic mean is A060863(n).

1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 3, 2, 2, 1, 1, 2, 2, 2, 2, 5, 2, 3, 2, 4, 2, 1, 3, 2, 1, 1, 2, 2, 1, 5, 1, 4, 2, 2, 1, 3, 1, 2, 1, 1, 4, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 1, 1, 5, 3, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 4, 1, 2, 2, 1, 3, 3, 1, 3, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 3, 1, 1, 2, 4, 4, 2, 4, 1, 3, 2

Offset: 1

Naohiro Nomoto, May 08 2003

Essentially A122821 with the 0's removed.

			For n=4; A060863(4) = 5. the two sets are 5/1 = 5, (3+5+7)/3 = 5. so a(4)=2.

Cf. A060863, A122821.

f[n_]:=Block[{i=1,j,c=0,m},While[Prime[i]<=n, j=1; While[m=Sum[Prime[k],{k,i,i+j-1}]/j; If[m==n,c++ ]; m0&] (* Ray Chandler, Oct 03 2006 *)

a(n) = A122821(A060863(n)).

Extended by Ray Chandler, Oct 03 2006

A082596 a(n) is the number of values of k such that A077389(k) = A060863(n).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Naohiro Nomoto, May 14 2003

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A060863, A077389, A082653.

A082653 For smallest numbers k such that A082596(k) = n, sequence gives A060863(k).

Original entry on oeis.org

2, 30, 910, 15203

Offset: 1

Naohiro Nomoto, May 17 2003

			For n=2; smallest number k = 20, A082596(20) = 2. so a(2) = A060863(20) = 30.

Cf. A060863, A077389, A082592.

A082661 Numbers n such that A082596(n) = A050221(n), sequence gives A060863(n).

Original entry on oeis.org

2, 4, 52, 70, 95, 100, 124, 153, 169, 177, 201, 230, 252, 261, 272, 273, 275, 314, 316, 322, 377, 378, 384, 434, 450, 451, 507, 517, 527, 657, 702, 711, 720, 833, 884, 896, 904, 910, 930, 1005, 1025, 1047, 1081, 1137, 1189, 1202, 1228, 1351, 1358, 1415

Offset: 1

Naohiro Nomoto, May 18 2003

nonn

A346399 a(n) is the number of symmetrically distributed consecutive primes centered at n (including n itself if n is prime).

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 1, 0, 4, 0, 1, 6, 1, 0, 6, 0, 1, 4, 1, 0, 2, 0, 1, 0, 0, 2, 0, 0, 1, 10, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 6, 1, 0, 2, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0, 1, 4, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 2, 1, 0, 0, 2, 0, 0, 1, 0, 4, 0, 1, 0, 0, 2, 0

Offset: 1

Ya-Ping Lu, Sep 18 2021

nonn

a(n) is the number of consecutive primes in Goldbach pairs of 2n centered at n.
a(n) is odd if n is prime; otherwise, a(n) is even.
n is prime if a(n) = 1 and n is composite if a(n) = 0.
a(n) = 14 is not seen until n = 8021811 (with none higher through 4*10^7). - Bill McEachen, Jul 26 2024

			a(1) = 0 because no prime is <= 1.
a(2) = 1 because no prime is < 2 and {2} is the only symmetrically distributed prime centered at 2.
a(30) = 10 because there are 10 symmetrically distributed consecutive primes, {13, 17, 19, 23, 29, 31, 37, 41, 43, 47}, centered at 30.

Jason Yuen, Table of n, a(n) for n = 1..10000

Cf. A045917, A050237, A060863, A122821.

from sympy import isprime
for n in range(1, 100):
    d = 1 if n%2 == 0 else 2
    ct = 1 if isprime(n) else 0
    while n - d > 2:
        k = isprime(n+d) + isprime(n-d)
        if k == 2: ct += 2
        elif k == 1: break
        d += 2
    print(ct)

A122821 Number of ways n can be represented as the arithmetic mean of consecutive primes.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 0, 2, 0, 1, 3, 1, 0, 3, 0, 2, 2, 1, 0, 1, 2, 2, 0, 0, 2, 0, 0, 2, 5, 2, 0, 0, 3, 0, 0, 2, 4, 2, 0, 1, 3, 2, 0, 1, 1, 2, 0, 2, 1, 5, 1, 4, 0, 2, 2, 0, 0, 1, 3, 1, 0, 0, 2, 0, 0, 1, 1, 4, 1, 2, 1, 1, 0, 0, 1, 1, 3, 2, 0, 2, 2, 1, 0, 0, 1, 5, 0, 3, 0, 1, 1, 1, 0, 2, 2, 2, 0, 1, 1, 3, 3, 1, 0, 4

Offset: 1

Ray Chandler, Sep 28 2006

nonn

Cf. A050221, A050237, A060863, A060864, A082370.

f[n_]:=Block[{i=1,j,c=0,m},While[Prime[i]<=n, j=1; While[m=Sum[Prime[k],{k,i,i+j-1}]/j; If[m==n,c++ ]; m

A050237 a(n) = the smallest number m such that there are exactly n sets of consecutive primes, each of which has an arithmetic mean of m.

Original entry on oeis.org

1, 2, 5, 12, 38, 30, 173, 165, 12259, 8803, 36735, 67263, 5296771, 32975, 1147233

Offset: 0

Naohiro Nomoto, May 08 2003

First appearance of n in A122821.

			a(4) = 38 because there are exactly four sets of consecutive primes which have means of 38: {31,37,41,43}, {29,...,47}, {23,...,53} and {2,...,83},

Cf. A060863, A082431, A122821.

{a(n)= m=2; starting_index=1; k=starting_index; sum_of_primes=0; prime_count=0; sets=0; until( (prime(starting_index)>m) && (sets==n), if( (prime(starting_index)> m) || (sets>n), m++; sets=0; starting_index=1; k=starting_index); sum_of_primes=sum_of_primes+prime(k); prime_count++; mean=sum_of_primes/ prime_count; if(meanRick L. Shepherd, Jun 14 2004

Edited by Don Reble, Jun 17 2003

A060864 Positive integers that are not an average of consecutive primes.

Original entry on oeis.org

1, 8, 10, 14, 16, 20, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 48, 54, 57, 58, 62, 63, 65, 66, 74, 75, 80, 84, 85, 88, 90, 94, 98, 104, 118, 119, 121, 128, 136, 140, 141, 142, 146, 147, 148, 152, 156, 158, 159, 161, 162, 164, 168, 171, 172, 174, 178, 182, 184, 188

Offset: 1

David W. Wilson, May 04 2001

nonn

Complement of A060863.

A122821(a(n)) = 0.

A082370 a(n) = number of sets of consecutive primes whose arithmetic mean is A000040(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 4, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 3, 1, 2, 4, 3, 3, 5, 1, 1, 6, 2, 3, 1, 2, 1, 2, 3, 2, 3, 2, 1, 1, 2, 2, 2, 4, 2, 1, 2, 4, 3, 3, 3, 2, 2, 1, 2, 1, 4, 3, 5, 2, 1, 2, 1, 3, 1, 3, 1, 3, 3, 2, 3, 2, 3, 1, 1, 2, 1, 5, 2, 1, 2, 3, 1, 2, 1, 3, 3, 2, 1, 1, 5, 2, 2

Offset: 1

Naohiro Nomoto, May 11 2003

			For n=3; A000040(3) = 5. the two sets are 5/1 = 5, (3+5+7)/3 = 5. so a(3)=2.

Robert Israel, Table of n, a(n) for n = 1..4000

Cf. A050221, A060863, A082431, A122821.

N:= 300:
P:= [0,seq(ithprime(i),i=1..N)]:
S:= ListTools:-PartialSums(P):
mmax:= numtheory:-pi(floor(S[N]/N)):
V:= Vector(1..mmax,1):
for i from 1 to N+1 do
  for j from i+2 to N+1 do
    r:= (S[j]-S[i])/(j-i);
    if r::integer and isprime(r) then
      k:= numtheory:-pi(r);
      if k <= mmax then
        V[k]:= V[k]+1
      fi
    fi
od od:
convert(V,list); # Robert Israel, Mar 18 2018

a(n) = A122821(A000040(n)).

Extended by Ray Chandler, Oct 03 2006

A082592 A077389 sorted and duplicates removed.

Original entry on oeis.org

2, 4, 5, 9, 12, 17, 23, 29, 30, 31, 37, 38, 39, 49, 51, 52, 56, 70, 71, 79, 89, 95, 100, 105, 110, 111, 113, 124, 125, 133, 137, 139, 149, 151, 153, 155, 157, 169, 177, 179, 185, 187, 199, 200, 201, 214, 227, 230, 242, 251, 252, 261, 272, 273, 274, 275, 280, 281

Offset: 1

Naohiro Nomoto, May 13 2003

Cf. A060863.

cp's OEIS Frontend

A050221 a(n) = number of sets of consecutive primes whose arithmetic mean is A060863(n).

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A082596 a(n) is the number of values of k such that A077389(k) = A060863(n).

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A082653 For smallest numbers k such that A082596(k) = n, sequence gives A060863(k).

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A082661 Numbers n such that A082596(n) = A050221(n), sequence gives A060863(n).

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A346399 a(n) is the number of symmetrically distributed consecutive primes centered at n (including n itself if n is prime).

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A122821 Number of ways n can be represented as the arithmetic mean of consecutive primes.

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Mathematica

A050237 a(n) = the smallest number m such that there are exactly n sets of consecutive primes, each of which has an arithmetic mean of m.

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PARI

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A060864 Positive integers that are not an average of consecutive primes.

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A082370 a(n) = number of sets of consecutive primes whose arithmetic mean is A000040(n).

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Maple

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A082592 A077389 sorted and duplicates removed.

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