A082467 -id:A082467 - cp's OEIS frontend

A377320 a(n) is the smallest positive integer k such that n + k and n - k have the same number of prime factors.

1, 1, 1, 3, 2, 2, 2, 6, 1, 5, 3, 2, 3, 6, 1, 1, 3, 2, 9, 2, 2, 5, 3, 4, 6, 1, 1, 11, 6, 4, 1, 6, 2, 2, 2, 2, 3, 8, 1, 1, 3, 2, 4, 3, 4, 12, 1, 1, 3, 2, 3, 1, 1, 3, 2, 7, 1, 4, 7, 4, 3, 6, 5, 1, 2, 1, 3, 5, 1, 3, 4, 4, 3, 1, 4, 13, 6, 2, 5, 15, 2, 7, 1, 3, 3, 1, 3

Offset: 4

Felix Huber, Nov 17 2024

nonn

If the strong Goldbach conjecture is true, that every even number >= 8 is the sum of two distinct primes, then a positive integer k <= A082467(n) exists for n >= 4.

			a(7) = 3 because 10 and 4 have both two prime factors. 8 and 6 or 9 and 7 respectively have a different number of prime factors.

Felix Huber, Table of n, a(n) for n = 4..10000

Cf. A001222, A082467, A178139, A377319, A377321.

A377320:=proc(n)
   local k;
   for k to n-1 do
      if NumberTheory:-Omega(n+k)=NumberTheory:-Omega(n-k) then
         return k
      fi
   od;
end proc;
seq(A377320(n),n=4..90);

A377320[n_] := Module[{k = 0}, While[PrimeOmega[++k + n] != PrimeOmega[n - k]]; k];
Array[A377320, 100, 4] (* Paolo Xausa, Dec 02 2024 *)

a(n) = my(k=1); while (bigomega(n+k) != bigomega(n-k), k++); k; \\ Michel Marcus, Nov 17 2024

1 <= a(n) <= A082467(n).

A377321 a(n) is the smallest positive integer k such that n + k and n - k have the same number of distinct prime factors.

Original entry on oeis.org

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

Offset: 4

Felix Huber, Nov 17 2024

nonn

If the strong Goldbach conjecture is true, that every even number >= 8 is the sum of two distinct primes, then a positive integer k <= A082467(n) exists for n >= 4.

			a(7) = 2 because 9 and 5 have both one distinct prime factor. 8 and 6 have a different number of distinct prime factors.

Felix Huber, Table of n, a(n) for n = 4..10000

Cf. A001221, A082467, A188348, A377319, A377320.

A377321:=proc(n)
   local k;
   for k to n-1 do
      if NumberTheory:-Omega(n+k,'distinct')=NumberTheory:-Omega(n-k,'distinct') then
         return k
      fi
   od;
end proc;
seq(A377321(n),n=4..90);

A377321[n_] := Module[{k = 0}, While[PrimeNu[++k + n] != PrimeNu[n - k]]; k];
Array[A377321, 100, 4] (* Paolo Xausa, Dec 02 2024 *)

a(n) = my(k=1); while (omega(n+k) != omega(n-k), k++); k; \\ Michel Marcus, Nov 17 2024

1 <= a(n) <= A082467(n).

A104884 Records in A104883.

Original entry on oeis.org

4, 5, 8, 24, 54, 117, 222, 258, 291, 591, 888, 951, 1656, 1674, 2451, 2577, 4212, 4857, 6597, 7398, 10758, 10950, 11601, 19608, 20604, 27411, 35157, 43338, 45174, 46920, 53412, 71661, 90699, 96681, 107385, 123051, 130782, 189741, 225747, 273738, 288096, 362781

Offset: 1

Benoit Cloitre and Robert G. Wilson v, Mar 28 2005

nonn

a(k) has the largest equal 'gap' between the nearest primes so far, i.e.; (the sum of the two nearest primes)/2 equals a(k).

Amiram Eldar, Table of n, a(n) for n = 1..72

Cf. A104883, A082467.

f[n_] := Block[{k}, If[ OddQ[n], k = 2, k = 1]; While[ !PrimeQ[n - k] || !PrimeQ[n + k], k += 2]; k]; t = Table[f[n], {n, 4, 10^4}];u = Table[0, {80}]; Do[a = t[[n]]; If[a < 81 && u[[a]] == 0, u[[a]] = n + 3], {n, 10^4}]; a = 0; lst = {}; Do[ If[u[[n]] > a, a = u[[n]]; AppendTo[lst, a]], {n, 80}]; lst

A239146 Smallest k>0 such that n +/- k and n^2 +/- k are all prime. a(n) = 0 if no such number exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 2, 3, 0, 5, 0, 3, 2, 0, 0, 13, 12, 0, 2, 0, 0, 0, 6, 15, 10, 0, 12, 0, 0, 15, 20, 0, 12, 5, 0, 15, 22, 21, 12, 0, 0, 0, 14, 27, 0, 35, 0, 0, 8, 15, 0, 0, 24, 27, 0, 0, 48, 7, 48, 0, 50, 3, 6, 7, 0, 0, 28, 0, 18, 0, 0, 27, 34

Offset: 1

Derek Orr, Mar 11 2014

nonn

a(n) is always smaller than n due to the requirement on n-k.

			8 +/- 1 (7 and 9) and 8^2 +/- 1 (63 and 65) are not all prime. 8 +/- 2 (6 and 10) and 8^2 +/- 2 (62 and 66) are not all prime. However, 8 +/- 3 (5 and 11) and 8^2 +/- 3 (61 and 67) are all prime. Thus, a(8) = 3.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A082467, A172990, A172989.

A239146 := proc(n)
    local k ;
    for k from 1 do
        if n-k <= 1 then
            return 0;
        end if;
        if isprime(n+k) and isprime(n-k) and isprime(n^2+k)
            and isprime(n^2-k) then
            return k;
        end if;
    end do;
end proc:
seq(A239146(n),n=1..80) ; # R. J. Mathar, Mar 12 2014

a[n_] := Catch@ Block[{k = 1}, While[k < n, And @@ PrimeQ@ {n+k, n-k, n^2+k, n^2-k} && Throw@k; k++]; 0]; Array[a, 75] (* Giovanni Resta, Mar 13 2014 *)

import sympy
from sympy import isprime
def c(n):
  for k in range(1,n):
    if isprime(n+k) and isprime(n-k) and isprime(n**2+k) and isprime(n**2-k):
      return k
n = 1
while n < 100:
  if c(n) == None:
    print(0)
  else:
    print(c(n))
  n += 1

A242165 Smallest k>=0, such that n+/-k are both Fermi-Dirac primes (A050376).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 2, 0, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 3, 0, 1, 0, 9, 4, 3, 6, 5, 0, 9, 2, 3, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 9, 0, 5, 6, 3, 4, 9, 0, 1, 0, 9, 4, 3, 6, 5, 0, 15, 2, 3, 0, 1, 0, 7, 4, 3, 4, 5, 0, 1, 0, 1, 0, 5, 4, 3, 14, 9, 0, 7, 10, 9, 4, 13, 6, 7, 0

Offset: 2

Vladimir Shevelev, May 05 2014

nonn

The existence of a(n)>=0 for all n >= 2 is equivalent to the Goldbach conjecture in Fermi-Dirac arithmetic (cf. comment in A241927) that every even number >= 4 is a sum of two terms of A050376 (it is slightly weaker than Goldbach conjecture for primes).

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (in Russian; MR 2000f: 11097, pp. 3912-3913).

Peter J. C. Moses, Table of n, a(n) for n = 2..10001

Cf. A082467, A241922, A241927, A241947.

a(A050376(n)) = 0.

A298736 a(n) = s(n) - prime(n+1)+3, where s(n) = smallest even number x > prime(n) such that the difference x-p is composite for all primes p <= prime(n).

Original entry on oeis.org

6, 10, 26, 90, 88, 84, 82, 200, 282, 280, 522, 518, 516, 512, 942, 936, 934, 928, 924, 922, 2566, 2562, 2556, 2548, 2544, 2542, 5268, 5266, 5262, 5248, 5244, 5238, 5236, 7280, 7278, 7272, 7266, 7262, 7256, 43356, 43354, 43344, 43342, 43338, 43336, 43324, 54024

Offset: 1

Felix Fröhlich, Jan 25 2018

The statement "a(n) >= 0 for n >= 1" is equivalent to Goldbach's conjecture (cf. Phong, Dongdong, 2004, Theorem (a)).

Records: 6, 10, 26, 90, 200, 282, 522, 942, 2566, 5268, 7280, 43356, 54024, ..., . - Robert G. Wilson v, Feb 28 2018

Robert G. Wilson v, Table of n, a(n) for n = 1..1205
Bui Minh Phong and Li Dongdong, Elementary problems which are equivalent to the Goldbach's Conjecture, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 33-37.
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A000040, A082467, A152522, A242165.

N:= 100: # to get a(1)..a(N)
P:= [seq(ithprime(i),i=1..N+1)]:
s:= proc(n,k0) local k;
  for k from max(k0,P[n]+1) by 2 do
    if andmap(not(isprime), map(t -> k - t, P[1..n])) then return k
  fi
od
end proc:
K[1]:= 6: A[1]:= 6:
for n from 2 to N do
  K[n]:= s(n,K[n-1]);
  A[n]:= K[n]- P[n+1]+3;
od:
seq(A[n],n=1..N); # Robert Israel, Mar 01 2018

f[n_] := Block[{k, x = 2, q = Prime@ Range@ n}, x += Mod[x, 2]; While[k = 1; While[k < n +1 && CompositeQ[x - q[[k]]], k++]; k < n +1, z = x += 2]; x - Prime[n +1] +3]; Array[f, 47] (* Robert G. Wilson v, Feb 26 2018 *)

s(n) = my(p=prime(n), x); if(p==2, x=4, x=p+1); while(1, forprime(q=1, p, if(ispseudoprime(x-q), break, if(q==p, return(x)))); x=x+2)
a(n) = s(n)-prime(n+1)+3

a(n) = A152522(n)-A000040(n+1)+3 for n > 0.

A177464 The smallest positive k such that the n-th Mersenne prime +-k are two primes.

Original entry on oeis.org

4, 12, 24, 30, 30, 66, 954, 1920, 30, 4116, 576, 214608

Offset: 2

Vladimir Joseph Stephan Orlovsky, May 09 2010

nonn

Smallest k>0 such that A000668(n)+k and A000668(n)-k are both prime.

			7+-4->primes, 31+-12->primes, 127+-24->primes, 8191+-30->primes, 131071+-30->primes, 524287+-66->primes..

Cf. A000668, A082467.

g[n_]:=2^Prime[n]-1; f[n_]:=Block[{k},If[OddQ[n],k=2,k=1];While[ !PrimeQ[n-k]||!PrimeQ[n+k],k+=2];k]; lst={};Do[If[PrimeQ[g[n]],AppendTo[lst,f[g[n]]]],{n,2,40}];lst

a(n) = A082467(A000668(n)). - R. J. Mathar, Jan 23 2011

A254886 a(n) = least k>0 such that n-k^2 and n+k^2 are both primes.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Feb 10 2015

nonn

If n is a square then a(n)=sqrt(n)-1 or 0.
Also if n is a square and a(n)=sqrt(n)-1 then sqrt(n) is a term in A178659.
First appearances of k for k=1..58 are at n = 4, 7, 14, 21, 36, 43, 90, 117, 86, 111, 210, 149, 768, 201, 236, 285, 468, 329, 366, 411, 446, 1137, 534, 647, 654, 807, 770, 885, 900, 911, 3090, 1665, 1192, 2415, 1296, 1313, 4212, 2163, 1600, 1671, 5448, 1769, 2040, 1941, 2054, 3207, 2214, 2333, 5340, 2601, 2792, 7725, 2814, 3095, 3054, 5913, 3442, 4377.
Among the first 10000 terms, the first missing values are 59, 79, 82, 83, 89, 91, 92, 94, 97, 98, 100.

Cf. A068501, A082467, A178659.

PARI
```
k=1;while(k^2Derek Orr, Feb 11 2015
```

cp's OEIS Frontend

A377320 a(n) is the smallest positive integer k such that n + k and n - k have the same number of prime factors.

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A377321 a(n) is the smallest positive integer k such that n + k and n - k have the same number of distinct prime factors.

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A104884 Records in A104883.

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A239146 Smallest k>0 such that n +/- k and n^2 +/- k are all prime. a(n) = 0 if no such number exists.

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A242165 Smallest k>=0, such that n+/-k are both Fermi-Dirac primes (A050376).

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A298736 a(n) = s(n) - prime(n+1)+3, where s(n) = smallest even number x > prime(n) such that the difference x-p is composite for all primes p <= prime(n).

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A177464 The smallest positive k such that the n-th Mersenne prime +-k are two primes.

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A254886 a(n) = least k>0 such that n-k^2 and n+k^2 are both primes.