

A175607


Largest number k such that the greatest prime factor of k^21 is prime(n).


42



3, 17, 161, 8749, 19601, 246401, 672281, 23718421, 10285001, 354365441, 3222617399, 9447152318, 127855050751, 842277599279, 2218993446251, 2907159732049, 41257182408961, 63774701665793, 25640240468751, 238178082107393, 4573663454608289, 19182937474703818751, 34903240221563713, 332110803172167361, 99913980938200001
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OFFSET

1,1


COMMENTS

For any prime p, there are finitely many k such that k^21 has p as its largest prime factor.
For every prime p, is there some k where the greatest prime factor of k^21 is p? Answer from Artur Jasinski, Oct 22 2010: Yes.
As mentioned by Luca and Najman, this problem is closely related to the one in A002071.
The terms give an upper bound with a method for the simultaneous computation of logarithms of small primes, see the fxtbook link.  Joerg Arndt, Jul 03 2012


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..25
Joerg Arndt, Matters Computational (The Fxtbook), section 32.4, pp.632633.
Florian Luca and Filip Najman, "On the largest prime factor of x^21", Mathematics of Computation 80:273 (2011), pp. 429435. (Paper has errata that was posted on the MOC website.)
Filip Najman, Home Page (gives all 16167 numbers n such that n^21 has no prime factor greater than 97)


PROG

(PARI) /* up to term for p=97 */
/* S[] is the list computed by Filip Najman (16223 elements) */
S=[2, 3, 4, ... , 332110803172167361, 19182937474703818751];
lpf(n)={ vecmax(factor(n)[, 1]) } /* largest prime factor */
{ forprime (p=2, 97,
t = 0;
for (n=1, #S, if ( lpf(S[n]^21)==p, t=n ) );
print1(S[t], ", ");
); }
/* Joerg Arndt, Jul 03 2012 */


CROSSREFS

Cf. A214093 (largest primes p such that the greatest prime factor of p^21 is prime(n)).
Cf. A076605 (largest prime divisor of n^21).
Cf. A285283 (equivalent for n^2+1).  Tomohiro Yamada, Apr 22 2017
Cf. A006530, A005563.  M. F. Hasler, Jun 13 2018
Sequence in context: A066211 A163884 A221410 * A052143 A268758 A069856
Adjacent sequences: A175604 A175605 A175606 * A175608 A175609 A175610


KEYWORD

nice,nonn,hard


AUTHOR

Charles R Greathouse IV, Jul 23 2010


EXTENSIONS

More terms (using Filip Najman's list) by Joerg Arndt, Jul 03 2012


STATUS

approved



