A061357 -id:A061357 - cp's OEIS frontend

A143788 Prime numbers p such that p + q = 117810 with q prime and p*q=58905^2-x^2.

13, 23, 31, 37, 47, 53, 59, 79, 83, 89, 101, 107, 109, 131, 137, 139, 151, 167, 191, 193, 233, 239, 269, 271, 281, 293, 307, 311, 313, 367, 373, 379, 383, 397, 421, 439, 449, 457, 479, 491, 503, 541, 569, 571, 587, 601, 607, 617, 619, 643, 647, 677, 683, 691

Offset: 1

Pierre CAMI, Sep 01 2008

The sequence has 2613 terms.

P. CAMI, Table of n, a(n) for n = 1..2613 (complete sequence)

A061357

okQ[n_]:=Module[{q=117810-n},PrimeQ[q]&&IntegerQ[Sqrt[3469799025- n*q]]]; Select[Prime[Range[200]],okQ] (* Harvey P. Dale, Jul 18 2011 *)

n = 58905, a(i) = odd primes k = 0: i = 0: j = 11300 10 k = k + 1 If k > (n - 3) / 2 Then End p = 2 * k + 1: q = 2 * (n - k - 1) + 1 15 i = i + 1 If p < a(i) Then i = i - 1: GoTo 10 If p = a(i) Then GoTo 20 GoTo 15 20 j = j - 1 If q > a(j) Then j = j + 1: GoTo 10 If q = a(j) Then Print p: GoTo 10 GoTo 20 GoTo 10

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

Pierre CAMI, Sep 02 2008

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

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

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

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

Name simplified by Ya-Ping Lu, Mar 05 2023

A234649 Difference between the first members of the widest and the narrowest prime pair having an arithmetic mean of n.

Original entry on oeis.org

2, 2, 4, 2, 6, 4, 6, 6, 10, 8, 12, 0, 14, 14, 10, 14, 14, 16, 18, 16, 16, 12, 22, 16, 20, 24, 24, 26, 26, 28, 26, 32, 30, 26, 36, 16, 36, 36, 28, 36, 36, 18, 44, 38, 40, 44, 42, 40, 50, 48, 40, 42, 52, 30, 42, 46, 42, 56, 56, 58, 48, 60, 64, 56, 66, 60, 48, 60, 70, 68, 68, 54, 68, 74, 60, 56

Offset: 8

Ralf Stephan, Dec 29 2013

nonn

The widest prime pair with a mean of n is (A002373(n),A020482(n)) and the narrowest is (A078587(n),A078496(n)).
Existence of a(n) for all n depends on A061357(n) > 0.
Even numbers missing in the subsequence with n<10^5 are 34,62,82,88,112,116,118,122,130,140,152...
a(n) = 0 for n=4,5,6,7,19 because A061357(n) = 1.

			The prime pairs with an arithmetic mean of 18 are (17,19), (13,23), (7,29), and (5,31), so a(18) = 17-5 = 31-19 = 12. The only pair with mean of 19 is (7,31) so a(19) = 0.

Ralf Stephan, Table of n, a(n) for n = 8..100000

Cf. A045917.

a(n)=mi=0;ma=0;forprime(p=3,n-1,if(isprime(2*n-p),if(!mi,mi=2*n-p);ma=2*n-p));if(!ma,-1,mi-ma)

a(n) = A078587(n) - A002373(n) = A078496(n) - A020482(n).

A352305 a(n) is the (conjectured) largest even number that can be expressed as the sum of two distinct primes in exactly n ways.

Original entry on oeis.org

6, 38, 68, 128, 158, 188, 398, 362, 458, 542, 632, 692, 602, 992, 808, 908, 1112, 1238, 1412, 1418, 1718, 1544, 1574, 1622, 1682, 2048, 2252, 2018, 2672, 2042, 2558, 2936, 2504, 2978, 2966, 3092, 3218, 3242, 3272, 3506, 3632, 3754, 4022, 4058, 4052, 4412, 4448, 4478

Offset: 0

Ilya Gutkovskiy, Mar 11 2022

nonn

			a(5) = 188 because 188 = 7 + 181 = 31 + 157 = 37 + 151 = 61 + 127 = 79 + 109 and it is conjectured that 188 is the last term of A080854.

Cf. A000954, A038609, A061357, A077914, A077969, A078299, A080854, A080862, A087747, A117929, A212613.

More terms from Hugo Pfoertner, Dec 18 2024

A121611 Number of ways to express n as an average of three distinct primes.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 2, 3, 2, 4, 3, 6, 3, 6, 3, 8, 3, 10, 3, 12, 5, 13, 5, 14, 5, 17, 4, 21, 4, 22, 4, 21, 6, 23, 5, 27, 7, 32, 5, 36, 5, 35, 7, 34, 5, 36, 7, 43, 5, 46, 8, 47, 8, 44, 5, 52, 6, 55, 6, 61, 8, 58, 8, 57, 9, 67, 8, 71, 7, 79, 7, 72, 9, 69, 6, 76, 7, 83, 9, 94, 9, 91, 9, 89, 8, 94, 8

Offset: 1

Zak Seidov, Sep 08 2006

nonn

For odd n's a(n) are much larger than a(n-/+1). Cf. A061357 = number of ways n can be expressed as the mean of two distinct primes, A061357 = number of ways the even integer 2n can be written as the sum of two primes for all even integers >6.

			a(4)=1 because 4 = (2+3+7)/3 (1 way),
a(6)=2 because 6 = (2+3+13)/3 = (2+5+11)/3 (2 ways)
a(9)=3 because 9 = (3+5+19)/3 = (3+7+17)/3 = (3+11+13)/3 (3 ways)
a(11)=4 because 11 = (3+7+23)/3 = (3+11+19)/3 = (3+13+17)/3 = (5+11+19)/3 (4 ways), etc.

Cf. A061357, A061357.

A140955 Even integers that do not have at least two partitions into 2 distinct primes.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 38

Offset: 1

Gil Broussard, Jul 25 2008

If A056636(3) = 128 (as is conjectured), then 38 is the last term in the sequence. - Charles R Greathouse IV, Sep 07 2022

			8 is a term because 3+5 is the only sum of primes = 8.
16 is not in the sequence because 16 = 3+13 and 5+11.
The only ways to express 10 as a sum of two unordered primes are 3+7 and 5+5. In one of the sums the primes are distinct. Thus, 10 is in this sequence. - _Tanya Khovanova_, Sep 07 2022

Cf. A056636, A061357, A077914, A117929.

Select[Range[0,100,2],Length[Select[Union/@IntegerPartitions[#,{2}],AllTrue[#,PrimeQ]&&Length[#]==2&]]<2&] (* James C. McMahon, Jul 15 2025 *)

is(n)=if(n%2, return(0)); my(t); forprime(p=3, n\2-1, if(isprime(n-p) && t++>1, return(0))); 1 \\ Charles R Greathouse IV, Sep 07 2022

Offset changed to 1 by Alois P. Heinz, Sep 07 2022

A214844 Number of partitions of 2^n into three distinct primes.

Original entry on oeis.org

0, 0, 0, 1, 3, 2, 10, 8, 32, 18, 75, 51, 292, 140, 518, 534, 2167, 1292, 6055, 4318, 23899, 16589, 53108, 46683, 312340, 159483, 567857, 639256, 2810965, 1826974

Offset: 1

Zak Seidov, Mar 08 2013

nonn

			a(4) = 1 because 2^4 = 16 = 2 + 3 + 11 (1 partition),
a(5) = 3 because 2^5 = 32 = 2 + 7 + 23 = 2 + 11 + 19 = 2 + 13 + 17 (3 partitions).

Index entries for sequences related to Goldbach conjecture

Cf. A125688.

Do[Print[{k, n=2^k; s=0; Do[p=Prime[i]; Do[q=Prime[j]; r=n-p-q; If[r>q && PrimeQ[r], s++], {j, i+1, PrimePi[(n-p)/2]}], {i,1}]; s}], {k,30}]
Table[m = 2^n - 2; cnt = 0; p = 3; While[p < m/2, If[PrimeQ[m - p], cnt++]; p = NextPrime[p]]; cnt, {n, 20}] (* T. D. Noe, Mar 08 2013 *)

a(n)=my(N=2^n-2,s);forprime(p=3,N/2-1,s+=ispseudoprime(N-p));s \\ Charles R Greathouse IV, Mar 08 2013

a(n) = A061357(2^(n-1) - 1) for n > 2. - Charles R Greathouse IV, Mar 08 2013

A269329 Number of partitions of a positive integer n into two distinct primes such that for even n, it is of the form n = p + q and for odd n, it is of the form n = 2p' + q'.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 4, 2, 2, 2, 2, 2, 3, 3, 3, 2, 4, 2, 4, 3, 2, 2, 4, 3, 3, 4, 4, 1, 3, 3, 5, 4, 4, 3, 6, 3, 4, 5, 4, 4, 6, 3, 5, 5, 3, 3, 6, 3, 3, 6, 3, 2, 7, 5, 7, 6, 5, 2, 6, 5, 4, 6, 4, 4, 8, 5, 7, 7, 5, 4, 8, 4, 4, 8, 7, 4, 7, 4, 7, 9, 4, 4, 10, 4, 5, 7, 6, 3, 9

Offset: 1

Frank M Jackson and M. B. Rees, Feb 23 2016

nonn

This sequence combines Levy's conjecture for odd positive numbers with the Goldbach conjecture for even positive numbers and strenghtens both by restricting the prime pairs to be distinct. I.e., every positive integer n > 6 is the sum of two distinct primes p and q such that for n even, it is of the form n = p + q and for n odd, it is of the form n = 2p' + q'.

			a(23)=3. Hence there are 3 partitions (as defined above) of the odd integer 23, namely 19+2+2, 17+3+3 and 13+5+5. a(24)=3. Hence there are 3 partitions of the even integer 24, namely 19+5, 17+7 and 13+11.

Cf. A002375, A061357, A194830, A223893, A225522.

parts[n_, a_, b_] := Select[IntegerPartitions[n, {a+b}, Prime@Range[PrimePi[n]]], Length[Union@#]==2&&MemberQ[Values@Counts@#, a] &]; lst1=Table[Length@parts[2n-1, 1, 2], {n, 1, 200}]; lst2=Table[Length@parts[2n, 1, 1], {n, 1, 200}]; Riffle[lst1, lst2]

A306878 Number of 0 < k < n such that n-k and n+k are both nonprimes.

Original entry on oeis.org

0, 0, 0, 0, 2, 1, 2, 3, 3, 3, 5, 5, 6, 6, 7, 6, 9, 10, 8, 10, 11, 10, 12, 13, 13, 13, 15, 14, 16, 18, 15, 18, 20, 16, 20, 21, 20, 21, 24, 21, 23, 26, 24, 24, 29, 25, 27, 30, 26, 30, 32, 29, 31, 33, 31, 33, 36, 33, 34, 41, 34, 36, 42, 35, 40, 42, 40, 40, 43, 42, 44, 48, 44, 44

Offset: 1

Gionata Neri, Mar 14 2019

nonn

Cf. A061357, A291564.

a(n)=my(s); for(c=1, n-1, s+=(bigomega(n-c)-1)*(bigomega(n+c)-1)!=0); s

a(n) = n - 1 - A061357(n) - A291564(n).

A352596 Conjecturally the number of positive even integers that can be expressed as the sum of two distinct primes in exactly n ways.

Original entry on oeis.org

3, 5, 9, 12, 12, 16, 19, 13, 24, 19, 21, 25, 15, 29, 28, 16, 31, 22, 34, 32, 20, 29, 26, 24, 28, 36, 34, 35, 37, 22, 29, 37, 36, 34, 39, 32, 39, 35, 28, 31, 28

Offset: 0

Ilya Gutkovskiy, Mar 22 2022

Cf. A000974, A038609, A061357, A077914, A077969, A078299, A080854, A080862, A117929, A229492, A352305.

cp's OEIS Frontend

A143788 Prime numbers p such that p + q = 117810 with q prime and p*q=58905^2-x^2.

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

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A234649 Difference between the first members of the widest and the narrowest prime pair having an arithmetic mean of n.

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A352305 a(n) is the (conjectured) largest even number that can be expressed as the sum of two distinct primes in exactly n ways.

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A121611 Number of ways to express n as an average of three distinct primes.

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A140955 Even integers that do not have at least two partitions into 2 distinct primes.

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PARI

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A214844 Number of partitions of 2^n into three distinct primes.

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A269329 Number of partitions of a positive integer n into two distinct primes such that for even n, it is of the form n = p + q and for odd n, it is of the form n = 2p' + q'.

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A306878 Number of 0 < k < n such that n-k and n+k are both nonprimes.

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