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A258840 a(n) is the least integer k such that there are n values of i <= k for which gpf(i^2 + 1) = gpf(k^2 + 1), where gpf(x) is the greatest prime factor of x. 0
1, 3, 7, 38, 47, 157, 302, 327, 515, 616, 697, 798, 818, 1303, 2818, 3141, 3323, 5648, 6962, 9193, 9872, 13213, 13747, 15445, 16271, 17149, 18263, 20491, 20727, 24389, 26915, 29078, 31867, 37848, 38007, 40182, 41508, 43328, 46349, 55025, 62258, 63133, 66893 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A014442(n) gives the largest prime factor of n^2 + 1.

The primes of the sequence are 3, 7, 47, 157, 1303, 3323, 46349, ...

The corresponding sequence Gpf(a(n)^2+1) is 2, 5, 5, 17, 17, 29, 37, 37, 101, 101, 101, 101, 101, 101, 101, 101, 101, 97, 97, 97, 97, 401, 349, 389, 557, 557, 557, 557, 557, 421, 421, 421, 557, ... and it is interesting to observe the frequency of repetitions for the numbers 5, 17, 37, 97, 101, 557, ...

LINKS

Table of n, a(n) for n=1..43.

EXAMPLE

a(3) = 7 because gpf(7^2 + 1) = gpf(3^2 + 1) = gpf(2^2 + 1) = 5 => 3 occurrences.

a(4) = 38 because gpf(38^2 + 1) = gpf(21^2 + 1) = gpf(13^2 + 1) = gpf(4^2 + 1) = 17 => 4 occurrences.

MAPLE

with(numtheory):nn:=70000:T:=array(1..nn):k:=0:kk:=1:

for m from 1 to nn do:

x:=factorset(m^2+1):n1:=nops(x):p:=x[n1]:k:=k+1:T[k]:=p:

od:

for n from 1 to 43 do:jj:=0:for k from kk to nn while(jj=0) do:

  q:=T[k]:ii:=0:jj:=0:

    for i from 1 to k do:

      if T[i]=q then ii:=ii+1:

      else

      fi:

    od:if ii=n then jj:=1:kk:=k:

    printf ( "%d %d \n", n, k):else fi:

  od:od:

PROG

(PARI) gpf(n) = my(f=factor(n^2+1)); f[#f~, 1];

nboc(k) = my(gpfk = gpf(k)); sum(i=1, k, gpf(i) == gpfk);

a(n) = my(k = 1); while (nbo(k) != n, k++); k; \\ Michel Marcus, Jun 12 2015

CROSSREFS

Cf. A014442, A242012.

Sequence in context: A162926 A042895 A173561 * A209029 A169741 A282427

Adjacent sequences:  A258837 A258838 A258839 * A258841 A258842 A258843

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jun 12 2015

STATUS

approved

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Last modified March 25 12:11 EDT 2019. Contains 321470 sequences. (Running on oeis4.)