A244447 -id:A244447 - cp's OEIS frontend

A246626 a(1)=1, a(n+1) is the smallest number m such that A244447(a(n)) < A244447(m).

1, 3, 4, 44, 84, 304, 1236, 7144, 7628, 11502, 12732

Offset: 1

Jahangeer Kholdi and Farideh Firoozbakht, Sep 01 2014

A246627 is the related subsequence of A244447 that gives A244447(a(n)).

			a(4)=44 because 44 is the smallest number m such that
A244447(a(3)) < A244447(m), where a(3)=4, A244447(4)=37350 and A244447(44)=40878.

Cf. A244447, A246627.

A246627 a(n) = A244447(A246626(n)).

Original entry on oeis.org

11, 13, 37350, 40878, 77565, 2419530, 5311238, 5893650, 6809166, 12387704, 98376110

Offset: 1

Jahangeer Kholdi and Farideh Firoozbakht, Sep 01 2014

			A244447(A246626(1)) = A244447(1) = 11.
A244447(A246626(4)) = A244447(44) = 40878.
A244447(A246626(6)) = A244447(304) = 2419530.

Cf. A244447, A246626.

A087711 a(n) = smallest number k such that both k-n and k+n are primes.

Original entry on oeis.org

2, 4, 5, 8, 7, 8, 11, 10, 11, 14, 13, 18, 17, 16, 17, 22, 21, 20, 23, 22, 23, 26, 25, 30, 29, 28, 33, 32, 31, 32, 37, 36, 35, 38, 37, 38, 43, 42, 41, 44, 43, 48, 47, 46, 57, 52, 51, 50, 53, 52, 53, 56, 55, 56, 59, 58, 75, 70, 69, 72, 67, 66, 65, 68, 67, 72, 71, 70, 71, 80, 81, 78

Offset: 0

Zak Seidov, Sep 28 2003

Let b(n), c(n) and d(n) be respectively, smallest number m such that phi(m-n) + sigma(m+n) = 2n, smallest number m such that phi(m+n) + sigma(m-n) = 2n and smallest number m such that phi(m-n) + sigma(m+n) = phi(m+n) + sigma(m-n), we conjecture that for each positive integer n, a(n)=b(n)=c(n)=d(n). Namely we conjecture that for each positive integer n, a(n) < A244446(n), a(n) < A244447(n) and a(n) < A244448(n). - Jahangeer Kholdi and Farideh Firoozbakht, Sep 05 2014

			n=10: k=13 because 13-10 and 13+10 are both prime and 13 is the smallest k such that k +/- 10 are both prime
4-1=3, prime, 4+1=5, prime; 5-2=3, 5+2=7; 8-3=5, 8+3=11; 9-4=5, 9+4=13, ...

Zak Seidov, Table of n, a(n) for n = 0..1000

Cf. A087695, A087696, A087697, A087678, A087679, A087680, A087681, A087682, A087683.
Cf. A082467. See A137169 for another version.
Cf. A244446, A244447, A244448.
Cf. A020483.

distance:=function(n); k:=n+2; while not IsPrime(k-n) or not IsPrime(k+n) do k:=k+1; end while; return k; end function; [ distance(n): n in [1..71] ]; /* Klaus Brockhaus, Apr 08 2007 */

Primes:= select(isprime,{seq(2*i+1,i=1..10^3)}):
a[0]:= 2:
for n from 1 do
  Q:= Primes intersect map(t -> t-2*n,Primes);
  if nops(Q) = 0 then break fi;
  a[n]:= min(Q) + n;
od:
seq(a[i],i=0..n-1); # Robert Israel, Sep 08 2014

s = ""; k = 0; For[i = 3, i < 22^2, If[PrimeQ[i - k] && PrimeQ[i + k], s = s <> ToString[i] <> ","; k++ ]; i++ ]; Print[s] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2008 *)
snk[n_]:=Module[{k=n+1},While[!PrimeQ[k+n]||!PrimeQ[k-n],k++];k]; Array[ snk,80,0] (* Harvey P. Dale, Dec 13 2020 *)

a(n)=my(k);while(!isprime(k-n) || !isprime(k+n),k++);return(k) \\ Edward Jiang, Sep 05 2014

a(n) = A020483(n)+n for n >= 1. - Robert Israel, Sep 08 2014

Entries checked by Klaus Brockhaus, Apr 08 2007

A244448 a(n) is the smallest integer m such that m-n is composite and phi(m-n) + sigma(m+n) = phi(m+n) + sigma(m-n).

Original entry on oeis.org

4, 153, 442, 213, 179, 120, 46, 37, 47, 264, 145416, 1101, 107, 79, 71, 78, 716, 637, 98, 249, 71, 126, 13258, 1243, 119, 163, 119, 131, 140497, 381, 191, 156, 101, 169, 1574, 315, 151, 193, 167, 2158, 148, 104, 202, 289, 1969, 882, 2572, 428, 251, 357, 314, 283

Offset: 0

Jahangeer Kholdi and Farideh Firoozbakht, Aug 30 2014

nonn

For each n, a(n) > n and like a(n)-n, a(n)+n is also composite.
If both numbers p and p + 2n are primes then x = p+n is a solution to the equation phi(x-n) + sigma(x+n) = phi(x+n) + sigma(x-n). But for these many solutions x, both numbers x-n and x+n are primes.
a(n) is the smallest integer m such that m-n is composite and A051612(m+n) = A051612(m-n) where A051612(n) = sigma(n) - phi(n). - Michel Marcus, Mar 20 2020

			a(1)=153 because 153-1 is composite, phi(153-1)+sigma(153+1) = phi(153+1)+sigma(153-1) and there is no such number less than 153.

Michel Marcus, Table of n, a(n) for n = 0..4501 (terms 0..1000 from Jinyuan Wang)

Cf. A000010 (phi), A000203 (sigma), A051612 (sigma - phi).

Cf. A244446, A244447, A246628.

a[0]=4;a[n_]:=a[n]=(For[m=n+1,PrimeQ[m-n]||EulerPhi[m-n]+DivisorSigma[1,m+n]!=EulerPhi[m+n]+DivisorSigma[1,m-n],m++];m);
Table[a[n],{n,0,70}]

a(n) = {my(m=n+4); while(isprime(m-n) || eulerphi(m+n)+sigma(m-n)!=eulerphi(m-n)+sigma(m+n), m++); m; }
vector(100,n,a(n)) \\ Derek Orr, Aug 30 2014

A244446 a(n) is the smallest integer m such that m-n is composite and phi(m-n) + sigma(m+n) = 2*m.

Original entry on oeis.org

25, 323, 48, 34, 53, 471, 58, 78, 84, 76, 71, 122, 64, 144, 162, 118, 74, 188, 106, 258, 156, 2512, 68, 254, 94, 107, 132, 2326, 876, 536, 154, 182, 268, 468, 98, 2061, 106, 408, 264, 286, 258, 1520900, 423, 618, 276, 648, 579, 518, 204, 708, 196, 370, 164, 1088, 300, 1518, 412, 3616, 158, 1226

Offset: 1

Jahangeer Kholdi and Farideh Firoozbakht, Aug 30 2014

nonn

For each n, a(n)>n and like a(n)-n, a(n)+n is also composite.

If both numbers p & p+2n are primes the x=p+n is a solution for the equation phi(x-n)+sigma(x+n)=2x. But for these many solutions x, both x-n & x+n are primes.

			a(1)=25 because 25-1 is composite, phi(25-1)+sigma(25+1)=2*25 and there is no such number less than 25.

Cf. A000010, A000203, A244447, A244448.

a[n_]:=(For[m=n+1, PrimeQ[m-n]||EulerPhi[m-n]+DivisorSigma[1,m+n]!=2m,m++];m);Table[a[n],{n,60}]

a(n)=m=n+4;while(isprime(m-n)||eulerphi(m-n)+sigma(m+n)!=2*m,m++);m
vector(100,n,a(n)) \\ Derek Orr, Aug 30 2014

cp's OEIS Frontend

A246626 a(1)=1, a(n+1) is the smallest number m such that A244447(a(n)) < A244447(m).

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A246627 a(n) = A244447(A246626(n)).

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A087711 a(n) = smallest number k such that both k-n and k+n are primes.

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A244448 a(n) is the smallest integer m such that m-n is composite and phi(m-n) + sigma(m+n) = phi(m+n) + sigma(m-n).

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A244446 a(n) is the smallest integer m such that m-n is composite and phi(m-n) + sigma(m+n) = 2*m.

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