Michel Eduardo Beleza Yamagishi - cp's OEIS frontend

A380545 Cumulative sum of the smallest prime in the minimal Goldbach partition for 2*n, n>=2.

2, 5, 8, 11, 16, 19, 22, 27, 30, 33, 38, 41, 46, 53, 56, 59, 64, 71, 74, 79, 82, 85, 90, 93, 98, 105, 108, 113, 120, 123, 126, 131, 138, 141, 146, 149, 152, 157, 164, 167, 172, 175, 180, 187, 190, 195, 202, 221, 224, 229, 232, 235, 240, 243, 246, 251, 254, 259

Offset: 2

Michel Eduardo Beleza Yamagishi, Jun 23 2025

nonn

			For n = 2, 4 = 2 + 2, the smallest prime p_1 = 2, so a(2) = A020481(2) = 2 = 2.
For n = 3, 6 = 3 + 3, the smallest prime p_2 = 3, so a(3) = a(2) + A020481(3) = 2 + 3 = 5.
For n = 4, 8 = 3 + 5, the smallest prime p_3 = 3, so a(4) = a(3) + A020481(4) = 5 + 3 = 8.

Partial sums of A020481.

GoldbachMinPrimeCumSum[N_] := If[N < 4, {}, Accumulate[Table[Select[Prime[Range[PrimePi[n]]], PrimeQ[n - #] &, 1][[1]], {n, 4, N, 2}]]]

a(2) = A020481(2) and a(n) = a(n-1) + A020481(n) for n>2.

A380546 Cumulative sum of the greatest prime in the minimal Goldbach partition for 2*n, n>=2.

Original entry on oeis.org

2, 5, 10, 17, 24, 35, 48, 61, 78, 97, 116, 139, 162, 185, 214, 245, 276, 307, 344, 381, 422, 465, 508, 555, 602, 649, 702, 755, 808, 867, 928, 989, 1050, 1117, 1184, 1255, 1328, 1401, 1474, 1553, 1632, 1715, 1798, 1881, 1970, 2059, 2148, 2227, 2324, 2421, 2522

Offset: 2

Michel Eduardo Beleza Yamagishi, Jun 23 2025

nonn

Partial sums of A020482.

GoldbachMaxPrimeCumSum[N_] := If[N < 4, {}, Accumulate[Table[n - Select[Prime[Range[PrimePi[n]]], PrimeQ[n - #] &, 1][[1]], {n, 4, N, 2}]]]

a(2) = A020482(2) and a(n) = a(n-1) + A020482(n) for n>2.

A382766 Odd primes p such that p + 4, p + 6 and p + 8 are composite.

Original entry on oeis.org

113, 137, 139, 179, 181, 197, 199, 211, 239, 241, 281, 283, 293, 317, 337, 409, 419, 421, 467, 509, 521, 523, 547, 577, 617, 619, 631, 659, 661, 691, 709, 773, 787, 797, 809, 811, 827, 829, 839, 863, 887, 919, 953, 997, 1019, 1021, 1039, 1049, 1051, 1069

Offset: 1

Michel Eduardo Beleza Yamagishi, Apr 04 2025

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A382765, A083371, A124582, A049591, A067774.

P:= select(isprime,{seq(i,i=3..10008,2)}):
R:= P minus (P -~ 4) minus (P -~ 6) minus (P -~ 8):
sort(convert(R,list)); # Robert Israel, Apr 28 2025

Select[Table[
  Module[{p = 2, q},
   While[True, q = 2 n - p; If[PrimeQ[p] && PrimeQ[q], Break[]];
    p = NextPrime[p]]; If[p == 11, q, Nothing]], {n, 2, 1000}], # =!=
   Nothing &]

isok(p) = (p%2) && isprime(p) && !isprime(p+4) && !isprime(p+6) && !isprime(p+8); \\ Michel Marcus, Apr 07 2025

A377842 a(n) = q - 2p, where q is the greatest prime such that p=2n - q is also prime.

Original entry on oeis.org

-2, -3, -1, 1, -3, 5, 7, 3, 11, 13, 9, 17, 13, 9, 23, 25, 21, 17, 31, 27, 35, 37, 33, 41, 37, 33, 47, 43, 39, 53, 55, 51, 47, 61, 57, 65, 67, 63, 59, 73, 69, 77, 73, 69, 83, 79, 75, 41, 91, 87, 95, 97, 93, 101, 103, 99, 107, 103, 99, 83, 91, 87, 71, 121, 117, 125, 121, 117, 131, 133

Offset: 2

Michel Eduardo Beleza Yamagishi, Dec 10 2024

sign

M. E. B. Yamagishi, Goldbach's Conjecture: A Simple and Fast Algorithm, Preprint, 2024.

Cf. A020481, A020482, A378896.

Table[Module[{p = 2, q},
  While[True, q = 2 n - p; If[PrimeQ[p] && PrimeQ[q], Break[]];
   p = NextPrime[p]]; q - 2 p], {n, 2, 100}]

a(n) = my(q=precprime(2*n)); while (!isprime(2*n - q), q = precprime(q-1)); q - 2*(2*n-q); \\ Michel Marcus, Dec 12 2024

A378201 a(n) = 2n mod q, where q is the greatest prime such that 2n - q is also prime.

Original entry on oeis.org

0, 0, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 7, 3, 3, 5, 7, 3, 5, 3, 3, 5, 3, 5, 7, 3, 5, 7, 3, 3, 5, 7, 3, 5, 3, 3, 5, 7, 3, 5, 3, 5, 7, 3, 5, 7, 19, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 7, 13, 11, 13, 19, 3, 5, 3, 5, 7, 3, 3, 5, 7, 11, 11, 3, 3, 5, 7, 3, 5, 7, 3, 5, 3, 5, 7, 3, 5, 7, 3, 3, 5, 7, 11, 11, 3, 3, 5

Offset: 2

Michel Eduardo Beleza Yamagishi, Nov 19 2024

nonn

Cf. A002373.

Table[Module[{p = 2, q},
  While[True, q = 2 n - p; If[PrimeQ[p] && PrimeQ[q], Break[]];
   p = NextPrime[p]]; Mod[2 n, q]], {n, 2, 100}]

A378020 a(n) = pi(A020482(n)) - pi(A020481(n)).

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 4, 3, 5, 6, 5, 7, 6, 5, 8, 9, 8, 7, 10, 9, 11, 12, 11, 13, 12, 11, 14, 13, 12, 15, 16, 15, 14, 17, 16, 18, 19, 18, 17, 20, 19, 21, 20, 19, 22, 21, 20, 14, 23, 22, 24, 25, 24, 26, 27, 26, 28, 27, 26, 23, 25, 24, 21, 29, 28, 30, 29, 28, 31, 32, 31, 30, 28, 29, 33, 34

Offset: 1

Michel Eduardo Beleza Yamagishi, Nov 14 2024

nonn

Cf. A000720, A020481, A020482, A377758, A377972.

Table[Module[{p = 2, q},
  While[True, q = 2 n - p; If[PrimeQ[p] && PrimeQ[q], Break[]];
   p = NextPrime[p]]; PrimePi[q] - PrimePi[p]], {n, 2, 100}]

a(n) = A377972(n) - A377758(n).

A377972 a(n) is the greatest i such that 2n-prime(i) is also a prime, where prime(i) is the i-th prime.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 9, 10, 11, 11, 11, 12, 12, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 19, 19, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 22, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 30, 29, 30, 30, 29, 31, 31, 32

Offset: 2

Michel Eduardo Beleza Yamagishi, Nov 13 2024

nonn

			For n=2, 2*2 - 2 = 2 and pi(2) = 1.
For n=3, 2*3 - 3 = 3, pi(3)=2.

Cf. A000040, A000720, A020482, A377758.

Table[Max[PrimePi[Flatten[Select[IntegerPartitions[2 n, {2}], AllTrue[#, PrimeQ] &]]]], {n, 2, 101}]

a(n) = forprime(q=2, n, if(isprime(2*n-q), return(primepi(2*n-q)))); \\ Michel Marcus, Nov 16 2024

a(n) = A000720(A020482(n)).

prime(a(n)) + prime(A377758(n)) = 2*n.

A377758 a(n) is the least i such that 2n-prime(i) is also a prime, where prime(i) is the i-th prime.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 4, 2, 2, 3, 4, 2, 3, 2, 2, 3, 2, 3, 4, 2, 3, 4, 2, 2, 3, 4, 2, 3, 2, 2, 3, 4, 2, 3, 2, 3, 4, 2, 3, 4, 8, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 4, 6, 5, 6, 8, 2, 3, 2, 3, 4, 2, 2, 3, 4, 5, 5, 2, 2, 3, 4, 2, 3, 4, 2, 3, 2, 3

Offset: 2

Michel Eduardo Beleza Yamagishi, Nov 06 2024

nonn

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A000040, A000720, A020481, A377972.

f:= proc(n) local i,p;
  p:= 1:
  for i from 1 do
    p:= nextprime(p);
    if isprime(2*n-p) then return i fi
  od
end proc:
map(f, [$2..100]); # Robert Israel, Nov 19 2024

Table[Module[{i = 1}, While[i <= PrimePi[n] && ! PrimeQ[n - Prime[i]], i++]; If[i <= PrimePi[n], i, None]], {n, 4, 1000, 2}]

a(n) = my(i=1); while (!isprime(2*n-prime(i)), i++); i; \\ Michel Marcus, Nov 06 2024

from sympy import primerange, isprime
def A377758(n): return next(i for i, p in enumerate(primerange(2*n),1) if isprime((n<<1)-p)) # Chai Wah Wu, Nov 19 2024

a(n) = pi(A020481(n)).

A377540 Numbers k such that at least one of the numbers 6k-1 or 6k+1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 32, 33, 35, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 87

Offset: 1

Michel Eduardo Beleza Yamagishi, Oct 31 2024

Union of A024898 and A024899.
Cf. A002476, A007528.
Complement of A060461 (with respect to the positive integers) or A171696 (with respect to the nonnegative integers).

Select[Range[100], PrimeQ[6 # - 1] || PrimeQ[6 # + 1] &]

isok(k) = isprime(6*k-1) || isprime(6*k+1); \\ Michel Marcus, Oct 31 2024

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User: Michel Eduardo Beleza Yamagishi

A380545 Cumulative sum of the smallest prime in the minimal Goldbach partition for 2*n, n>=2.

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A380546 Cumulative sum of the greatest prime in the minimal Goldbach partition for 2*n, n>=2.

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A382766 Odd primes p such that p + 4, p + 6 and p + 8 are composite.

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A377842 a(n) = q - 2*p, where q is the greatest prime such that p=2*n - q is also prime.

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A378201 a(n) = 2*n mod q, where q is the greatest prime such that 2*n - q is also prime.

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A378020 a(n) = pi(A020482(n)) - pi(A020481(n)).

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A377972 a(n) is the greatest i such that 2n-prime(i) is also a prime, where prime(i) is the i-th prime.

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A377758 a(n) is the least i such that 2n-prime(i) is also a prime, where prime(i) is the i-th prime.

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Maple

Mathematica

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Python

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A377540 Numbers k such that at least one of the numbers 6k-1 or 6k+1 is prime.

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Author

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A377842 a(n) = q - 2p, where q is the greatest prime such that p=2n - q is also prime.

A378201 a(n) = 2n mod q, where q is the greatest prime such that 2n - q is also prime.