A089992 -id:A089992

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A089158

Second prime factor, if it exists, of Mersenne numbers.

+10
3

89, 178481, 1103, 616318177, 164511353, 9719, 4513, 69431, 3203431780337, 761838257287, 48544121, 2298041, 202029703, 57912614113275649087721, 13842607235828485645766393, 341117531003194129, 3976656429941438590393

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..183

Chris Caldwell, Mersenne Primes: History, Theorems and Lists.

EXAMPLE

The 5th Mersenne number 2^11 - 1 = 23*89 and 89 is the second prime divisor.

The 9th Mersenne number 2^23 - 1 = 47*178481 and 178481 is the second prime divisor.

Notice 23, 89 congruent to 1 mod 11 and 47, 178481 congruent to 1 mod 23.

PROG

(PARI) mersenne(b, n, d) = { c=0; forprime(x=2, n, c++; y = b^x-1; f=factor(y); v=component(f, 1); ln = length(v); if(ln>=d, print1(v[d]", ")); ) }

CROSSREFS

Cf. A001348, A016047, A065341, A089159, A089992, A135978.

KEYWORD

nonn

AUTHOR

Cino Hilliard, Dec 06 2003

STATUS

approved

A089993

Penultimate prime divisor of numbers that are not powers of primes (A024619).

+10
2

2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 3, 2, 5, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 5, 2, 3, 2, 3, 2, 3, 5, 3, 2, 3, 5, 2, 2, 3, 2, 7, 3, 2, 2, 3, 5, 2, 3, 2, 3, 7, 2, 3, 2, 5, 2, 2, 3, 2, 3, 2, 5, 2, 2, 5, 3, 2, 3, 5, 2, 3, 2, 7, 3, 2, 3, 2, 3, 3, 5, 3, 7, 2, 3, 2, 3, 5, 3, 2, 11, 2, 5, 2, 3, 2, 3, 2, 3, 7, 5

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A076820(A024619(n)). - Amiram Eldar, Apr 12 2021

MATHEMATICA

Select[Table[If[Length[(f = FactorInteger[n])] > 1, f[[-2, 1]], 1], {n, 1, 150}], # > 1 &]

PROG

(PARI) f(n) = a=factor(n); v=a[, 1]; ln=length(v); if(ln>1, return(v[ln-1]));

g(m) = for(x=2, m, if(f(x)>0, print1(f(x)", ")))

CROSSREFS

Cf. A024619, A076820, A089992.

KEYWORD

nonn

AUTHOR

Cino Hilliard, Jan 14 2004

EXTENSIONS

Offset corrected by Amiram Eldar, Apr 12 2021

STATUS

approved

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