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A085080 Smallest k such that n, k and n+k have the same prime signature (canonical form), or 0 if no such number exists. 0
0, 3, 2, 0, 2, 15, 0, 0, 0, 55, 2, 63, 0, 21, 6, 0, 2, 45, 0, 637, 14, 33, 0, 351, 0, 39, 0, 147, 2, 165, 0, 0, 6, 21, 22, 0, 0, 39, 26, 20237, 2, 231, 0, 325, 18, 39, 0, 4136875, 0, 18, 6, 423, 0, 135, 10, 1375, 34, 33, 2, 90, 0, 15, 12, 0, 21, 165, 0, 207, 22, 385, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) = 2 if n and n+2 form a twin prime pair.

a(n) = 0 if n is a perfect prime power or an odd prime such that n+2 is composite.

Here is a temporary list of integers <= 1000 for which a(n) is unknown (greater than a(48) or 0): 72, 200, 288, 432, 500, 648, 800, 864, 968, 972. - Michel Marcus, David A. Corneth, Mar 08 2019

a(96) = 1841996779; a(160) = 28521479; a(448) = 184390625; a(608) = 4633767. - Michel Marcus, Mar 08 2019

From David A. Corneth, Mar 08 2019: (Start)

By Fermat's Last Theorem, a(m^e) = 0 for e > 2 and positive integer m. For example, a(216) = a(6^3) = 0.

a(n) = 0 for squares < 1000, see worked example for n = 36 for the method.

a(192) = 30927921875, a(320) = 355182331, a(480) = 7771875, a(640) = 18243947439, a(832) = 194546043, a(896) = 2157109375, a(960) = 157546875. For the values to do, they are > 10^11 if a(n) > 0.

If n is even and a(n) > 0 and the exponent of 2 in the factorization of n is the largest in the prime signature then a(n) isn't necessarily odd. Ray Chandler found n = 392 as an example where a(n) = 108 is even. (End)

a(384) = 1281916327741, a(768) <= 1367088016014857. - Daniel Suteu, Mar 18 2019; confirmed by Michel Marcus, Mar 18 2019

a(768) = 85001950390625. - Ray Chandler, Mar 26 2019

LINKS

Table of n, a(n) for n=1..71.

EXAMPLE

a(12) = 63 as 12 + 63 = 75, 2^2*3 + 3^2*7 = 5^2*3, all have the prime signature p^2*q.

a(1) = 0, because the only possible value for k is then 1, giving n+k=2, with a different signature.

a(2) = 3, because 2, 3 and 2+3=5 have the same prime signature.

a(36) = 0, because if a(n) exists then k exists such that k^2 + 36 = m^2 where k^2, 36 and m^2 have the same prime signature. Rewriting 36 = m^2 - k^2 = (m - k)*(m + k) and then inspection over divisors of 36 gives no terms. Alternatively checking Pythagorean triplets gives the same result. - David A. Corneth, Mar 08 2019

MATHEMATICA

a[n_?PrimeQ] := If[PrimeQ[n + 2], 2, 0]; a[2] = 3; a[36] = 0; ps[n_] := Sort[ FactorInteger[n][[;; , 2]] ]; a[n_] := Module[{k = 2, f = FactorInteger[n]}, ps0 = Sort[f[[;; , 2]]]; If[Length[f] == 1, 0, While[ps[k] != ps0 || ps[n + k] != ps0, k++]; k]]; Array[a, 71] (* Amiram Eldar, Mar 07 2019 works for n <= 71 *)

PROG

(PARI) sigt(n) = vecsort(factor(n)[, 2]~);

a(n) = {

  if ((n==1) || (isprimepower(n) && !isprime(n)), return(0));

  if (isprimepower(n) && !isprime(n), return(0));

  if ((n!=2) && isprime(n), if (isprime(n+2), return(2), return(0)));

  if (n==36, return(0));

  my(k=2, v = sigt(n));

  while ((sigt(k) != v) || (sigt(n+k) != v), k++);

  k;

} \\ Michel Marcus, Mar 07 2019; works for n <= 71

CROSSREFS

Cf. A085072 (only n and n+k have same prime signature), A215199.

Sequence in context: A131728 A075115 A273528 * A260308 A079714 A286889

Adjacent sequences:  A085077 A085078 A085079 * A085081 A085082 A085083

KEYWORD

nonn,changed

AUTHOR

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 01 2003

EXTENSIONS

a(20)-a(47) from Max Alekseyev, Aug 12 2013

a(48)-a(71) from Amiram Eldar, Mar 05 2019

STATUS

approved

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Last modified April 24 00:02 EDT 2019. Contains 322404 sequences. (Running on oeis4.)