

A198391


Numbers n for which sum(i=1..k) (11/p_i) + product(i=1..k) (11/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
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OFFSET

1,1


COMMENTS

The numbers of the sequence are solutions of the differential equation n’=(ka)n+b, which can be written as A003415(n)=(ka)*n+A003958(n), where k is the number of prime factors of n, and a is the integer sum(i=1..k) (11/p_i) +Prod(1=1..k) (11/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*535+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 557271+1 =25920.
The numbers of the sequence satisfy also sum(i=1..k) (1+1/p_i)  product(i=1..k)(11/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. 11/2 +11/2 +11/3 +11/3 +11/3 +11/19 +11/61 =5715/1159 is the sum over the 11/p_i. (11/2) *(11/2) *(11/3) *(11/3) *(11/3) *(11/19) *(11/61) =80/1159 is the product of the 11/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((11/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



