A096058 -id:A096058 - cp's OEIS frontend

A096057 a(1) = 1, a(n) = least prime divisor of b(n), where b(1) = 1, b(n) = n*b(n-1) + 1 = A002627(n).

1, 3, 2, 41, 2, 1237, 2, 29, 2, 43, 2, 5, 2, 5, 2, 35951249665217, 2, 28001, 2, 1409, 2, 5, 2, 5, 2, 47, 2, 661, 2, 13, 2, 5, 2, 5, 2, 13, 2, 13, 2, 73, 2, 5, 2, 5, 2, 71, 2, 2437159, 2, 31, 2, 5, 2, 5, 2, 13, 2, 3020497643, 2, 23, 2, 5, 2, 5, 2, 5672529813439, 2, 15336863

Offset: 1

Amarnath Murthy, Jun 17 2004

nonn

			a(4) = 41 because b(3) = 3*b(2)+1 = 3*3+1 = 10 and 4*10+1 = 41, which is prime.
b(n) = 1, 3, 10, 41, ... with least prime divisors a(n) = 1, 3, 2, 41, ....

Amiram Eldar, Table of n, a(n) for n = 1..97

Cf. A096058.

Cf. A020639, A002627, A096058.

nxt[{n_,a_}]:={n+1,a(n+1)+1}; FactorInteger[#][[1,1]]&/@NestList[nxt,{1,1},40][[All,2]] (* The program generates the first 41 terms of the sequence. *) (* Harvey P. Dale, Jun 21 2022 *)

a(n) = A020639(A002627(n)).

Corrected and extended by Ray G. Opao, Aug 02 2004
Edited by Jonathan Sondow, Jan 09 2005
More terms from Michel Marcus, Mar 28 2020
More terms from Giovanni Resta, Mar 28 2020

A102469 Largest prime factor of numerator of Sum_{k=0...n} 1/k!, with a(0) = 1.

Original entry on oeis.org

1, 2, 5, 2, 13, 163, 103, 137, 863, 98641, 10687, 31469, 1540901, 522787, 5441, 226871807, 13619, 1276861, 414026539, 2124467, 12670743557, 838025081381, 44659157, 323895443, 337310723185584470837549, 54352957, 11301647941785046703319941, 102505951982728548829

Offset: 0

Jonathan Sondow, Jan 09 2005

nonn

It appears that a(n) = A102468(n) (Smarandache number of the same numerator) except when n = 3. The largest prime factor of the corresponding denominator is A007917(n) for n > 1. Omitting the 0th term in the sum, it appears that the largest prime factor and the Kempner number A002034, of the numerator of Sum_{k=1...n} 1/k! are both equal to A096058(n).

			Sum_{k=0...3} 1/k! = 8/3 and 2 is the largest prime factor 8, so a(3) = 2.

Daniel Suteu, Table of n, a(n) for n = 0..74
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Index entries for sequences related to factorial numbers.

Cf. A102468, A096058, A006530, A061354, A000522, A007917.

FactorInteger[#][[-1,1]]&/@Numerator[Accumulate[1/Range[0,30]!]] (* Harvey P. Dale, Nov 14 2012 *)

a(n) = if(n==0, return(1)); vecmax(factor(numerator(sum(k=0, n, 1/k!)))[,1]); \\ Daniel Suteu, Jun 09 2022

a(n) = A006530(A061354(n)).

A102468 a(n)! is the smallest factorial divisible by the numerator of Sum_{k=0...n} 1/k!, with a(0) = 1.

Original entry on oeis.org

1, 2, 5, 4, 13, 163, 103, 137, 863, 98641, 10687, 31469, 1540901, 522787, 5441, 226871807, 13619, 1276861, 414026539, 2124467, 12670743557, 838025081381, 44659157, 323895443, 337310723185584470837549, 54352957

Offset: 0

Jonathan Sondow, Jan 09 2005

nonn

It appears that a(n) = A102469(n) (largest prime factor of the same numerator) except when n = 3. The smallest factorial divisible by the corresponding denominator is n!. Omitting the 0th term in the sum, it appears that the Kempner number (A002034) and the largest prime factor, of the numerator of Sum_{k=1...n} 1/k! are both equal to A096058(n).

The Mathematica program given below was used to generate the sequence. If the numerator of Sum_{k=0...n}(1/k!) is squarefree, the program prints the value of the numerator's largest prime factor, which must equal a(n). Otherwise, the program prints the complete factorization of the numerator so a(n) can be determined by inspection. - Ryan Propper, Jul 31 2005

			Sum_{k=0...3} 1/k! = 8/3 and 4! is the smallest factorial divisible by 8, so a(3) = 4.

A. J. Kempner, Miscellanea, Amer. Math. Monthly, 25 (1918), 201-210 [ See Section II, "Concerning the smallest integer m! divisible by a given integer n". ]
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007, 2010.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Eric Weisstein's World of Mathematics, Smarandache Function.
Index entries for sequences related to factorial numbers.

Cf. A102469, A000522, A002034, A061354, A096058, A007917.

Do[l = FactorInteger[Numerator[Sum[1/k!, {k, 0, n}]]]; If[Length[l] == Plus @@ Last /@ l, Print[Max[First /@ l]], Print[l]], {n, 1, 30}] (* Ryan Propper, Jul 31 2005 *)
nmax = 30; Clear[a]; Do[f = FactorInteger[ Numerator[ Sum[1/k!, {k, 0, n}] ] ]; a[n] = If[Length[f] == Total[f[[All, 2]] ], Max[f[[All, 1]] ], f[[-1, 1]] ], {n, 0, nmax}]; a[3] = 4; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Sep 16 2015, adapted from Ryan Propper's script *)

a(n) = {j = 1; s = numerator(sum(k=0, n, 1/k!)); while (j! % s, j++); j;} \\ Michel Marcus, Sep 16 2015

a(n) = A002034(A061354(n)).

More terms from Ryan Propper, Jul 31 2005

cp's OEIS Frontend

A096057 a(1) = 1, a(n) = least prime divisor of b(n), where b(1) = 1, b(n) = n*b(n-1) + 1 = A002627(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A102469 Largest prime factor of numerator of Sum_{k=0...n} 1/k!, with a(0) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A102468 a(n)! is the smallest factorial divisible by the numerator of Sum_{k=0...n} 1/k!, with a(0) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions