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A198391 Numbers m such that 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 m (with multiplicity).

%I A198391 #34 May 30 2025 10:01:05
%S A198391 2,15,20,272,476,19024,47425,65792,125172,216900,539280,1222976,
%T A198391 1372736,2770496,3494336,5321808,5844528,6177168,7032528,8885808,
%U A198391 20670768,60727876,69081344,82724356,95579136,544382208,907440192,1657497600,4295032832,5048574976
%N A198391 Numbers m such that 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 m (with multiplicity).
%C A198391 The numbers of the sequence are solutions of the differential equation m’=(k-a)*m+b, which can be written as A003415(m)=(k-a)*m+A003958(m), where k is the number of prime factors of m, and a is the integer Sum_{i=1..k} (1-1/p_i) + Product_{i=1..k} (1-1/p_i).
%C A198391 If k = a we have m’ = b or A003415(m) = A003958(m). 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.
%C A198391 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.
%H A198391 Giovanni Resta, <a href="/A198391/b198391.txt">Table of n, a(n) for n = 1..48</a> (terms < 10^12)
%H A198391 J. M. Borwein and E. Wong, <a href="https://citeseerx.ist.psu.edu/document?doi=eefc789579b51c34a09658b91858447e89714d7c">A survey of results relating to Giuga’s conjecture on primality</a>, May 8, 1995.
%H A198391 Romeo Mestrovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mestrovic/mes4.html">On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers</a>, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
%e A198391 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.
%p A198391 isA198391 := proc(n)
%p A198391     p := ifactors(n)[2] ;
%p A198391     add(op(2,d)-op(2,d)/op(1,d),d=p) + mul((1-1/op(1,d))^op(2,d),d=p) ;
%p A198391     type(%,'integer') ;
%p A198391 end proc:
%p A198391 for n from 2 to 20000000 do
%p A198391     if isA198391(n) then
%p A198391         printf("%d,\n",n);
%p A198391     end if;
%p A198391 end do: # _R. J. Mathar_, Nov 26 2011
%t A198391 Select[Range[2, 10^5], IntegerQ[(Plus @@ # + Times @@ #) &@ (1 - 1/ Flatten[ Table[#1, {#2}] & @@@ FactorInteger@#])] &] (* _Giovanni Resta_, May 23 2016 *)
%Y A198391 Cf. A003415, A003958, A007850, A199767.
%K A198391 nonn
%O A198391 1,1
%A A198391 _Paolo P. Lava_, Oct 24 2011
%E A198391 Missing a(23) and a(26)-a(30) from _Giovanni Resta_, May 23 2016