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
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