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A198391 Numbers n for which sum(i=1..k) (1-1/p_i) + product(i=1..k) (1-1/p_i) is an integer, where p_i are the k prime factors of n (with multiplicity). 7
2, 15, 20, 272, 476, 19024, 47425, 65792, 125172, 216900, 539280, 1222976, 1372736, 2770496, 3494336, 5321808, 5844528, 6177168, 7032528, 8885808, 20670768, 60727876, 69081344, 82724356, 95579136, 544382208, 907440192, 1657497600, 4295032832, 5048574976 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The numbers of the sequence are solutions of the differential equation  n’=(k-a)n+b, which can be written as A003415(n)=(k-a)*n+A003958(n), where k is the number of prime factors of n, and a is the integer sum(i=1..k) (1-1/p_i) +Prod(1=1..k) (1-1/p_i).

If k=a we have n’=b or A003415(n)= A003958(n). For instance 15 has prime factors 3, 5; its arithmetic derivative is 15’=8 and b=3*5-3-5+1=8. The term 47425 has prime factors 5, 5, 7, 271. Its arithmetic derivative is  47425’= 25920 and b= 5*5*7*271 -5*5*7 -5*5*271 -5*7*271 -5*7*271 +5*5 +5*7 +5*271 +5*7 +5*271 +7*271 -5-5-7-271+1 =25920.

The numbers of the sequence satisfy also sum(i=1..k) (1+1/p_i) - product(i=1..k)(1-1/p_i) = some integer.

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..48 (terms < 10^12)

J. M. Borwein and E. Wong, A survey of results relating to Giuga’s conjecture on primality, May 8, 1995.

R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

EXAMPLE

125172 has prime factors 2, 2, 3, 3, 3, 19, 61. 1-1/2 +1-1/2 +1-1/3 +1-1/3 +1-1/3 +1-1/19 +1-1/61 =5715/1159 is the sum over the 1-1/p_i. (1-1/2) *(1-1/2) *(1-1/3) *(1-1/3) *(1-1/3) *(1-1/19) *(1-1/61) =80/1159 is the product of the 1-1/p_i. The sum over sum and product is 5715/1159 +80/1159 =5, an integer.

MAPLE

isA198391 := proc(n)

    p := ifactors(n)[2] ;

    add(op(2, d)-op(2, d)/op(1, d), d=p) + mul((1-1/op(1, d))^op(2, d), d=p) ;

    type(%, 'integer') ;

end proc:

for n from 2 to 20000000 do

    if isA198391(n) then

        printf("%d, \n", n);

    end if;

end do: # R. J. Mathar, Nov 26 2011

MATHEMATICA

Select[Range[2, 10^5], IntegerQ[(Plus @@ # + Times @@ #) &@ (1 - 1/ Flatten[ Table[#1, {#2}] & @@@ FactorInteger@#])] &] (* Giovanni Resta, May 23 2016 *)

CROSSREFS

Cf. A003415, A003958, A007850, A199767.

Sequence in context: A281660 A244324 A143660 * A278889 A075722 A169597

Adjacent sequences:  A198388 A198389 A198390 * A198392 A198393 A198394

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Oct 24 2011

EXTENSIONS

Missing a(23) and a(26)-a(30) from Giovanni Resta, May 23 2016

STATUS

approved

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Last modified December 18 23:46 EST 2018. Contains 318245 sequences. (Running on oeis4.)