A102084 -id:A102084 - cp's OEIS frontend

A047160 For n >= 2, a(n) = smallest number m >= 0 such that n-m and n+m are both primes, or -1 if no such m exists.

0, 0, 1, 0, 1, 0, 3, 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, 9, 2, 3, 0, 1, 0, 3, 2, 15, 0, 5, 12, 3, 8, 9, 0, 7, 12, 3, 4, 15, 0, 1, 0, 9, 4, 3, 6, 5, 0, 15, 2, 3, 0, 1, 0, 15, 4, 3, 6, 5, 0, 9, 2, 15, 0, 5, 12, 3, 14, 9, 0, 7, 12, 9, 4, 15, 6, 7, 0, 9, 2, 3

Offset: 2

Lior Manor

I have confirmed there are no -1 entries through integers to 4.29*10^9 using PARI. - Bill McEachen, Jul 07 2008
From Daniel Forgues, Jul 02 2009: (Start)
Goldbach's Conjecture: for all n >= 2, there are primes (distinct or not) p and q s.t. p+q = 2n. The primes p and q must be equidistant (distance m >= 0) from n: p = n-m and q = n+m, hence p+q = (n-m)+(n+m) = 2n.
Equivalent to Goldbach's Conjecture: for all n >= 2, there are primes p and q equidistant (distance >= 0) from n, where p and q are n when n is prime.
If this conjecture is true, then a(n) will never be set to -1.
Twin Primes Conjecture: there is an infinity of twin primes.
If this conjecture is true, then a(n) will be 1 infinitely often (for which each twin primes pair is (n-1, n+1)).
Since there is an infinity of primes, a(n) = 0 infinitely often (for which n is prime).
(End)
If n is composite, then n and a(n) are coprime, because otherwise n + a(n) would be composite. - Jason Kimberley, Sep 03 2011
From Jianglin Luo, Sep 22 2023: (Start)
a(n) < primepi(n)+sigma(n,0);
a(n) < primepi(primepi(n)+n);
a(n) < primepi(n), for n>344;
a(n) = o(primepi(n)), as n->+oo. (End)
If -1 < a(n) < n-3, then a(n) is divisible by 3 if and only if n is not divisible by 3, and odd if and only if n is even. - Robert Israel, Oct 05 2023

			16-3=13 and 16+3=19 are primes, so a(16)=3.

T. D. Noe, Table of n, a(n) for n = 2..10000
Jason Kimberley, Symmetrical plot of A047160

Cf. A001031, A002092, A002372, A002373, A002374, A002375, A014092, A025583, A035026, A047949, A071406, A082467, A102084, A103147, A112823, A155764, A155765, A177461, A078611, A010051, A045917, A325142.

a047160 n = if null ms then -1 else head ms
            where ms = [m | m <- [0 .. n - 1],
                            a010051' (n - m) == 1, a010051' (n + m) == 1]
-- Reinhard Zumkeller, Aug 10 2014

A047160:=func;[A047160(n):n in[2..100]]; // Jason Kimberley, Sep 02 2011

Table[k = 0; While[k < n && (! PrimeQ[n - k] || ! PrimeQ[n + k]), k++]; If[k == n, -1, k], {n, 2, 100}]
smm[n_]:=Module[{m=0},While[AnyTrue[n+{m,-m},CompositeQ],m++];m]; Array[smm,100,2] (* Harvey P. Dale, Nov 16 2024 *)

a(n)=forprime(p=n,2*n, if(isprime(2*n-p), return(p-n))); -1 \\ Charles R Greathouse IV, Jun 23 2017

10 N=2// 20 M=0// 30 if and{prmdiv(N-M)=N-M,prmdiv(N+M)=N+M} then print M;:goto 50// 40 inc M:goto 30// 50 inc N: if N>130 then stop// 60 goto 20

a(n) = n - A112823(n).

a(n) = A082467(n) * A005171(n), for n > 3. - Jason Kimberley, Jun 25 2012

More terms from Patrick De Geest, May 15 1999
Deleted a comment. - T. D. Noe, Jan 22 2009
Comment corrected and definition edited by Daniel Forgues, Jul 08 2009

A099303 Greatest integer x such that x' = n, or 0 if there is no such x, where x' is the arithmetic derivative of x.

Original entry on oeis.org

0, 0, 4, 6, 9, 10, 15, 14, 25, 0, 35, 22, 49, 26, 55, 0, 77, 34, 91, 38, 121, 0, 143, 46, 169, 27, 187, 0, 221, 58, 247, 62, 289, 0, 323, 0, 361, 74, 391, 42, 437, 82, 403, 86, 529, 0, 551, 94, 589, 63, 667, 0, 713, 106, 703, 0, 841, 70, 899, 118, 961, 122, 943, 0, 1073, 0

Offset: 2

T. D. Noe, Oct 12 2004

nonn

This is the largest member of the set I(n) in the paper by Ufnarovski and Ahlander. They show that a(n) <= (n/2)^2.

Because this sequence is quite different for even and odd n, it is bisected into A102084 and A189762. The upper bound for odd n appears to be (n/3)^(3/2), which is attained when n = 3p^2 for primes p>5. - T. D. Noe, Apr 27 2011

See A003415

T. D. Noe, Table of n, a(n) for n=2..1000

Cf. A003415 (arithmetic derivative of n), A099302 (number of solutions to x' = n), A098699 (least x such that x' = n), A098700 (n such that x' = n has no integer solution).

dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; d1=Table[dn[n], {n, 40000}]; Table[x=Max[Flatten[Position[d1, n]]]; If[x>-Infinity, x, 0], {n, 2, 400}]

from sympy import factorint
def A099303(n):
    for m in range(n**2>>2,0,-1):
        if sum((m*e//p for p,e in factorint(m).items())) == n:
            return m
    return 0 # Chai Wah Wu, Sep 12 2022

A073046 Write 2n = p+q (p,q prime), pq minimal; then a(n) = p*q.

Original entry on oeis.org

4, 9, 15, 21, 35, 33, 39, 65, 51, 57, 95, 69, 115, 161, 87, 93, 155, 217, 111, 185, 123, 129, 215, 141, 235, 329, 159, 265, 371, 177, 183, 305, 427, 201, 335, 213, 219, 365, 511, 237, 395, 249, 415, 581, 267, 445, 623, 1501, 291, 485, 303, 309, 515, 321, 327

Offset: 2

Werner D. Sand, Aug 31 2002

nonn

Least semiprime whose sum of prime factors equals 2*n.

Assuming Goldbach's conjecture, a(n) exists for all n >= 2. - David James Sycamore, Jan 08 2019

			n=13: 2n=26; 26 = 23 + 3 = 19 + 7 = 13 + 13; 23*3 = minimal => p*q = 23*3 = 69.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A000040, A061397.

Cf. A102084, A193315, A288814.

a073046 n = head $ dropWhile (== 0) $
                   zipWith (*) prims $ map (a061397 . (2*n -)) prims
   where prims = takeWhile (<= n) a000040_list
-- Reinhard Zumkeller, Aug 28 2011

Array[Block[{p = 2, q}, While[! PrimeQ@ Set[q, 2 # - p], p = NextPrime[p]]; p q] &, 55, 2] (* Michael De Vlieger, Aug 02 2020 *)

For all n except 3, a(n) = A288814(2*n). - David James Sycamore, Jan 08 2019

Corrected by Ray Chandler, Jun 11 2005

A189762 Greatest integer x such that x' = 2n+1, or 0 if there is no such x, where x' is the arithmetic derivative (A003415).

Original entry on oeis.org

0, 6, 10, 14, 0, 22, 26, 0, 34, 38, 0, 46, 27, 0, 58, 62, 0, 0, 74, 42, 82, 86, 0, 94, 63, 0, 106, 0, 70, 118, 122, 0, 0, 134, 105, 142, 146, 98, 0, 158, 0, 166, 117, 0, 178, 0, 175, 0, 194, 130, 202, 206, 0, 214, 218, 154, 226, 0, 245, 138, 171, 0, 0, 254

Offset: 1

T. D. Noe, Apr 27 2011

nonn

Bisection of A099303. In contrast to the sequence for even numbers, A102084, there appear to be an infinite number of zeros in this sequence (see A098700). The density of the zeros appears to be 1/3. Quite Often a(n) = 4n-2. For odd number 2n+1, an upper bound on the largest anti-derivative x appears to ((2n+1)/3)^(3/2).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A003415, A099303, A102084 (another bisection of A099303).

dn[0] = 0; dn[1] = 0; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; nn = 100; d = Array[dn, (nn/2)^2]; Table[pos = Position[d, n]; If[pos == {}, 0, pos[[-1, 1]]], {n, 3, nn, 2}]

A189764 Greatest integer x such that x' = 2n and x is not a semiprime, or 0 if there is no such x, where x' is the arithmetic derivative (A003415).

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 0, 12, 0, 0, 0, 20, 0, 0, 0, 28, 0, 0, 0, 0, 0, 24, 0, 44, 0, 0, 0, 52, 0, 36, 0, 0, 0, 40, 0, 68, 0, 0, 0, 76, 0, 0, 0, 0, 0, 60, 0, 92, 0, 0, 0, 0, 0, 81, 0, 48, 0, 0, 0, 116, 0, 84, 0, 124, 0, 0, 0, 0, 0, 100, 0, 0, 0, 0, 0, 148, 0, 72

Offset: 1

T. D. Noe, Apr 27 2011

nonn

As mentioned in A102084, the anti-derivatives of even numbers are overwhelmingly semiprimes of the form n^2 - k^2. This sequence excludes those semiprimes. The upper bound of a(n) appears to be (n/2)^(4/3), which is attained when 2n = 4p^3 for primes p>3.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A003415, A102084, A189763 (n such that a(n)>0).

dn[0] = 0; dn[1] = 0; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; nn = 200; d = Array[dn, (nn/2)^2]; Table[s1 = Flatten[Position[d, n]]; s2 = Select[s1, ! IntegerQ[Sqrt[(n/2)^2 - #]] &]; If[s2 == {}, 0, s2[[-1]]], {n, 2, nn, 2}]

A193315 Write 2n=j+q (j,q positive noncomposite numbers); jq maximal; then a(n)=jq.

Original entry on oeis.org

1, 4, 9, 15, 25, 35, 49, 55, 77, 91, 121, 143, 169, 187, 221, 247, 289, 323, 361, 391, 437, 403, 529, 551, 589, 667, 713, 703, 841, 899, 961, 943, 1073, 1147, 1189, 1271, 1369, 1363, 1517, 1591, 1681, 1763, 1849, 1927, 2021, 1891, 2209, 2279, 2257, 2491, 2537, 2623, 2809, 2867, 2881

Offset: 1

Juri-Stepan Gerasimov, Aug 26 2011

nonn

a(n) = A102084(n) for n > 0. [Reinhard Zumkeller, Aug 28 2011]

			At n=6, 2n=12; 12 = 1 + 11 = 7 + 5; 7*5 = maximal => j*q = 7*5 = 35.

Cf. A073046, A008578, A061397.

a193315 1 = 1
a193315 n = maximum $ zipWith (*) prims $ map (a061397 . (2*n -)) prims
   where prims = takeWhile (<= n) a008578_list
-- Reinhard Zumkeller, Aug 28 2011

isA008578 := proc(n) if n = 1 then true ; elif isprime(n) then true; else false; end if; end proc:
A193315 := proc(n) local mx,j,q ; mx := 0 ; for j from 1 to 2*n-1 do if isA008578(j) then q := 2*n-j ; if isA008578(q) then mx := max(mx,j*q) ; end if ; end if; end do: mx ; end proc:
seq(A193315(n),n=1..60) ; # R. J. Mathar, Aug 28 2011

def is_A008578(n): return n == 1 or is_prime(n)
def A193315(n): return max((j*(2*n-j)) for j in [1]+prime_range(n+1) if is_A008578(2*n-j))
[A193315(i) for i in range(1,15)]
# D. S. McNeil, Aug 27 2011

A350455 T(n,k) is the k-th semiprime whose sum of prime factors equals 2n, triangle T(n,k), n>=2, 1<=k<=A045917(n), read by rows.

Original entry on oeis.org

4, 9, 15, 21, 25, 35, 33, 49, 39, 55, 65, 77, 51, 91, 57, 85, 121, 95, 119, 143, 69, 133, 169, 115, 187, 161, 209, 221, 87, 247, 93, 145, 253, 289, 155, 203, 299, 323, 217, 361, 111, 319, 391, 185, 341, 377, 437, 123, 259, 403, 129, 205, 493, 529, 215, 287, 407

Offset: 2

Alois P. Heinz, Dec 31 2021

Assuming Goldbach's conjecture, no row is empty.

			Triangle T(n,k) begins:
    4;
    9;
   15;
   21,  25;
   35     ;
   33,  49;
   39,  55;
   65,  77;
   51,  91;
   57,  85, 121;
   95, 119, 143;
   69, 133, 169;
  115, 187     ;
  161, 209, 221;
   87, 247     ;
   93, 145, 253, 289;
  155, 203, 299, 323;
  ...

Alois P. Heinz, Rows n = 2..1000, flattened
Wikipedia, Goldbach's conjecture

Column k=1 gives A073046.
Last elements of rows give A102084.
Row sums give A228553.
Row products give A337568.
Row lengths give A045917.
Cf. A000040, A001358, A046315, A350419.

T:= n-> seq(`if`(andmap(isprime, [h, 2*n-h]), h*(2*n-h), [][]), h=2..n):
seq(T(n), n=2..30);

cp's OEIS Frontend

A047160 For n >= 2, a(n) = smallest number m >= 0 such that n-m and n+m are both primes, or -1 if no such m exists.

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A099303 Greatest integer x such that x' = n, or 0 if there is no such x, where x' is the arithmetic derivative of x.

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A073046 Write 2*n = p+q (p,q prime), p*q minimal; then a(n) = p*q.

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A189762 Greatest integer x such that x' = 2n+1, or 0 if there is no such x, where x' is the arithmetic derivative (A003415).

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A189764 Greatest integer x such that x' = 2n and x is not a semiprime, or 0 if there is no such x, where x' is the arithmetic derivative (A003415).

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A193315 Write 2n=j+q (j,q positive noncomposite numbers); j*q maximal; then a(n)=j*q.

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A350455 T(n,k) is the k-th semiprime whose sum of prime factors equals 2n, triangle T(n,k), n>=2, 1<=k<=A045917(n), read by rows.

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Maple

A073046 Write 2n = p+q (p,q prime), pq minimal; then a(n) = p*q.

A193315 Write 2n=j+q (j,q positive noncomposite numbers); jq maximal; then a(n)=jq.