A082467 -id:A082467 - cp's OEIS frontend

A104883 a(n) is the smallest number k such that A082467(k) = n.

4, 5, 8, 7, 24, 11, 54, 51, 22, 117, 222, 19, 114, 87, 46, 207, 216, 61, 258, 291, 128, 591, 336, 43, 306, 423, 136, 519, 492, 97, 888, 951, 146, 537, 318, 163, 1656, 561, 238, 699, 732, 191, 864, 1365, 286, 1353, 1674, 229, 1422, 1671, 802, 2451, 876, 283, 576, 2577

Offset: 1

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

nonn

This is simply the smallest inverse to A082467. Note that A082467 has offset 4. - N. J. A. Sloane, Jan 15 2020

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A082467.

Records are in A104884.

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, 2600}]; u = Table[0, {60}]; Do[a = t[[n]]; If[a < 61 && u[[a]] == 0, u[[a]] = n + 3], {n, 2597}]; u

a082467(n) = my(k=1); while(isprime(n-k)*isprime(n+k) == 0, k++); k;
a(n) = my(k=4); while (a082467(k) != n, k++); k; \\ Michel Marcus, Feb 12 2020

Definition corrected by Giovanni Resta. - N. J. A. Sloane, Jan 15 2020

A129301 Records in A082467.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 33, 36, 42, 60, 75, 84, 87, 90, 93, 102, 117, 120, 135, 138, 168, 180, 183, 210, 228, 300, 333, 369, 474, 486, 525, 621, 720, 792, 810, 846, 1086, 1281, 1305, 1515, 1590, 1617, 1722, 1794

Offset: 1

Klaus Brockhaus, Apr 08 2007

nonn

			As can be gathered from A082467, the first six records are A082467(4) = 1, A082467(5) = 2, A082467(7) = 4, A082467(11) = 6, A082467(19) = 12, A082467(43) = 24. Hence a(1) to a(6) are 1, 2, 4, 6, 12, 24.

T. D. Noe, Table of n, a(n) for n=1..69

Cf. A082467, A129302 (where records occur).

Cf. A025019

a(n) = A082467(A129302(n)). - Jason Kimberley, Sep 04 2011

A129302 Where records occur in A082467.

Original entry on oeis.org

4, 5, 7, 11, 19, 43, 97, 146, 163, 191, 223, 344, 457, 526, 877, 904, 1049, 1114, 1307, 1736, 1751, 1781, 2129, 2476, 3097, 3551, 5131, 8504, 10342, 10357, 18233, 24776, 40072, 68707, 99719, 125903, 174913, 181267, 371428, 827576, 936118, 1054141

Offset: 1

Klaus Brockhaus, Apr 08 2007

nonn

			The sixth record in A082467 is A129301(6) = 24 = A082467(43), hence a(6) = 43.

T. D. Noe, Table of n, a(n) for n=1..69

Cf. A082467, A129301 (records).

Cf. A025018

A231483 Smallest k > A082467(n) such that n-k and n+k are both prime.

Original entry on oeis.org

9, 8, 15, 18, 7, 12, 15, 10, 15, 18, 7, 28, 15, 10, 27, 12, 7, 24, 15, 8, 21, 18, 5, 30, 15, 8, 27, 24, 11, 18, 9, 10, 15, 30, 13, 18, 15, 10, 21, 30, 7, 42, 21, 10, 33, 18, 7, 30, 21, 10, 9, 18, 11, 30, 27, 8, 33, 24, 11, 48, 21, 8, 21, 30

Offset: 20

Pierre CAMI, Nov 09 2013

nonn

Starting at n=20 just because there are some n<20 without solution like for n=19, for the least k (A082467) it start at 4 as no solution for n<4.

Pierre CAMI, Table of n, a(n) for n = 20..10000

Cf. A082467.

a(n) = {k=1; while(isprime(n-k)*isprime(n+k) == 0, k++); k++; while(isprime(n-k)*isprime(n+k) == 0, k++); k; } \\ Michel Marcus, Nov 10 2013

A241686 a(1)=1, a(2)=2, a(3)=3; for n>=4, a(n) = n - A082467(n).

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 3, 5, 7, 7, 5, 11, 7, 11, 13, 13, 11, 17, 7, 17, 19, 13, 17, 19, 19, 23, 23, 19, 17, 29, 19, 23, 29, 31, 29, 31, 31, 29, 37, 37, 29, 41, 19, 41, 43, 31, 41, 43, 37, 47, 43, 43, 47, 47, 43, 53, 53, 43, 47, 59, 43, 53, 59, 61, 59, 61, 61, 53

Offset: 1

Vladimir Shevelev, Apr 27 2014

nonn

a(i)=i, i=1,2,3, since A082467 is natural extended as 0 for n=1,2,3. For n>=2, a(n) and 2*n-a(n) are both primes. Sequence {2*n - a(n)} begins 1,2,3,5,7,7,11,11,11,13,17,13,19,17,17,19,23,19,31,...

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A082467.

More terms from Peter J. C. Moses

A138567 Variant of A082467 if we regard 1 as a prime.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 4, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 12, 3, 2, 9, 6, 5, 6, 3, 4, 9, 12, 1, 12, 9, 4, 3, 6, 5, 6, 9, 2, 3, 12, 1, 24, 3, 2, 15, 6, 5, 12, 3, 8, 9, 6, 7, 12, 3, 4, 15, 12, 1, 18, 9, 4, 3, 6, 5, 6, 15, 2, 3, 12, 1, 6, 15, 4, 3, 6, 5, 18, 9, 2, 15, 24, 5, 12, 3, 14, 9, 18, 7, 12, 9, 4, 15, 6, 7, 30, 9

Offset: 1

J. S. Brilleaud (jeansebo(AT)yahoo.fr), May 18 2008

nonn

Edited by N. J. A. Sloane, May 18 2008

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.

Original entry on oeis.org

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

A078611 Radius of the shortest interval (of positive length) centered at prime(n) that has prime endpoints.

Original entry on oeis.org

2, 4, 6, 6, 6, 12, 6, 12, 12, 6, 12, 24, 6, 6, 12, 18, 6, 12, 6, 18, 24, 18, 30, 12, 6, 6, 30, 24, 24, 18, 30, 12, 18, 12, 6, 36, 30, 6, 12, 18, 42, 30, 30, 42, 12, 60, 30, 48, 6, 12, 30, 12, 6, 6, 12, 42, 6, 12, 54, 24, 24, 42, 36, 36, 18, 30, 36, 18, 6, 42, 30, 6, 30, 36, 30, 24, 18, 12

Offset: 3

Joseph L. Pe, Dec 09 2002

a(1) and a(2) are undefined. Alternatively, a(n) = least k, 1 < k < n, such that prime(n) + k and prime(n) - k are both prime. I conjecture that a(n) is defined for all n > 2. Equivalently, every prime > 3 is the average of two distinct primes.

a(n) embodies the difference between weak and strong Goldbach conjectures, and therefore between A047160 and A082467 which differ only for prime arguments (a(n)=A082467(prime(n)), while A047160(prime(n))=0). - Stanislav Sykora, Mar 14 2014

			prime(3) = 5 is the center of the interval [3,7] that has prime endpoints; this interval has radius = 7-5 = 2. Hence a(3) = 2. prime(5) = 11 is the center of the interval [5,17] that has prime endpoints; this interval has radius = 17-11 = 6. Hence a(5) = 6.

Stanislav Sykora, Table of n, a(n) for n = 3..40000

Cf. A047160, A082467. - Stanislav Sykora, Mar 14 2014

f[n_] := Module[{p, k}, p = Prime[n]; k = 1; While[(k < p) && (! PrimeQ[p - k] || ! PrimeQ[p + k]), k = k + 1]; k]; Table[f[i], {i, 3, 103}]

StrongGoldbachForPrimes(nmax)= {local(v,i,p,k);v=vector(nmax); for (i=1,nmax,p=prime(i);v[i] = -1; for (k=1,p-2,if (isprime(p-k)&&isprime(p+k),v[i]=k;break;););); return (v);} \\ Stanislav Sykora, Mar 14 2014

a(n) = A082467(A000040(n)). - Jason Kimberley, Jun 25 2012

A087712 a(1) = 1; if n = k-th prime, a(n) = k; otherwise write all prime factors of n in nondecreasing order, replace each prime with its rank, and concatenate the ranks.

Original entry on oeis.org

1, 1, 2, 11, 3, 12, 4, 111, 22, 13, 5, 112, 6, 14, 23, 1111, 7, 122, 8, 113, 24, 15, 9, 1112, 33, 16, 222, 114, 10, 123, 11, 11111, 25, 17, 34, 1122, 12, 18, 26, 1113, 13, 124, 14, 115, 223, 19, 15, 11112, 44, 133, 27, 116, 16, 1222, 35, 1114, 28, 110, 17, 1123, 18

Offset: 1

Eric Angelini, Feb 02 2009

Concatenations of consecutive entries of A112798. - R. J. Mathar, Feb 09 2009

The old entry with this A-number was a duplicate of A082467.

			n = 2 = first prime, a(2) = 1.
n = 3 = second prime, a(3) = 2.
n = 4 = 2*2 -> 1,1 -> 11, so a(4) = 11.
n = 6 = 2*3 -> 1,2 -> 12, so a(6) = 12.
n = 12 = 2*2*3 -> 1,1,2 -> 112, so a(12) = 112.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

See A098282 for lengths of trajectories. Cf. A077960, A156055.

Cf. A027746, A049084.

a087712 1 = 1
a087712 n = read $ concatMap (show . a049084) $ a027746_row n :: Integer
-- Reinhard Zumkeller, Oct 03 2012

# Maple program from R. J. Mathar, Feb 08 2009: (Start)
cat2 := proc(a,b) a*10^(max(1,ilog10(b)+1))+b ; end:
A049084 := proc(p) if isprime(p) then numtheory[pi](p) ; else 0 ; fi; end:
A087712 := proc(n) local pf,a,p,ex ; if isprime(n) then A049084(n) ; elif n = 1 then 1 ; else pf := ifactors(n)[2] ; a := 0 ; for p in pf do for ex from 1 to op(2,p) do a := cat2(a, A049084(op(1,p)) ) ; od: od: fi; end:
seq(A087712(n),n=1..140); # (End)
# (Maple program from David Applegate and N. J. A. Sloane, Feb 09 2009)
with(numtheory):
f := proc(n) local t1, v, r, x, j;
if (n = 1) then return 1; end if;
t1 := ifactors(n): v := 0;
for x in op(2,t1) do r := pi(x[1]):
for j from 1 to x[2] do
v := v * 10^length(r) + r;
end do; end do; v; end proc;

f[n_] := If[n == 1, 1, FromDigits@ Flatten[ IntegerDigits@# & /@ (PrimePi@# & /@ Flatten[ Table[ First@#, {Last@#}] & /@ FactorInteger@ n])]]; Array[f, 61] (* Robert G. Wilson v, Jun 06 2011 *)

from sympy import factorint, primepi
def a(n):
    if n == 1: return 1
    return int("".join(str(primepi(p))*e for p, e in factorint(n).items()))
print([a(n) for n in range(1, 62)]) # Michael S. Branicky, Oct 01 2024

More terms from R. J. Mathar (Feb 08 2009) and independently from David Applegate and N. J. A. Sloane, Feb 09 2009

A078496 Smallest prime p such that p>n and 2*n-p is also prime.

Original entry on oeis.org

5, 7, 7, 11, 11, 11, 13, 17, 13, 19, 17, 17, 19, 23, 19, 31, 23, 23, 31, 29, 29, 31, 29, 31, 37, 41, 31, 43, 41, 37, 37, 41, 41, 43, 47, 41, 43, 53, 43, 67, 47, 47, 61, 53, 53, 61, 53, 59, 61, 59, 61, 67, 59, 61, 73, 71, 61, 79, 71, 67, 67, 71, 71, 73, 83, 71, 73, 83, 73, 79

Offset: 4

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), Nov 26 2002

nonn

Suggested by Goldbach Conjecture.

Values of q from A143697. This follows from the factorization n^2-k^2 = (n-k)(n+k).

			a(11)=17.

P. CAMI, Table of n, a(n) for n = 4..60000

Cf. A143697, A078587.
a(n) = 2n - A078587(n).
Cf. A082467

Table[p=n+1; q=2n-p; While[q>0&&!(PrimeQ[p]&&PrimeQ[q]), p++; q-- ]; p, {n, 4, 100}]

a(n) = {my(p=nextprime(n+1)); while(!isprime(2*n-p), p = nextprime(p+1)); p;} \\ Michel Marcus, Oct 22 2016

n>3 integer; a(n)=min{p: p>n; p, 2*n-p are primes}.

Edited by N. J. A. Sloane, Jan 24 2009 at the suggestion of R. J. Mathar and T. D. Noe.

cp's OEIS Frontend

A104883 a(n) is the smallest number k such that A082467(k) = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A129301 Records in A082467.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A129302 Where records occur in A082467.

Views

Author

Keywords

Examples

Links

Crossrefs

A231483 Smallest k > A082467(n) such that n-k and n+k are both prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A241686 a(1)=1, a(2)=2, a(3)=3; for n>=4, a(n) = n - A082467(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A138567 Variant of A082467 if we regard 1 as a prime.

Views

Author

Keywords

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

UBASIC

Formula

Extensions

A078611 Radius of the shortest interval (of positive length) centered at prime(n) that has prime endpoints.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A087712 a(1) = 1; if n = k-th prime, a(n) = k; otherwise write all prime factors of n in nondecreasing order, replace each prime with its rank, and concatenate the ranks.

Views

Author