A303695 - OEIS

%I #54 Jul 04 2019 03:10:44

%S 1,2,3,6,5,11,7,10,12,17,11,17,13,23,23,22,17,23,19,27,31,35,23,39,30,

%T 41,30,37,29,40,31,34,47,53,47,60,37,59,55,61,41,54,43,57,53,71,47,53,

%U 56,57,71,67,53,74,71,83

%N a(n) is the sum of divisors d of n such that bigomega(d) is a divisor of bigomega(n) where bigomega(x) is A001222(x), the number of prime divisors of x counted with multiplicity.

%C a(n) < sigma(n) (for n>1).

%C The least nonprime solution m of the equation m^2 mod (a(m^2) - m^2) = 0 is 18. What is the next nonprime solution of this equation? If it exists, it is greater than 10^6.

%C No other such terms up to 10^7. - _Michel Marcus_, Jul 05 2018

%H Ivan Neretin, <a href="/A303695/b303695.txt">Table of n, a(n) for n = 1..10000</a>

%e a(24) = 39 because: 24 = (2^3)*(3^1) (bigomega(24) = 4) and (2^0)*(3^1) = 3 ((0+1)|4), (2^1)*(3^0) = 2 ((1+0)|4), (2^1)*(3^1) = 6 ((1+1)|4), (2^2)*(3^0) = 4 ((2+0)|4), (2^3)*(3^1) = 24 ((3+1)|4), hence 3+2+6+4+24 = 39

%t {1}~Join~Table[Total@Select[Divisors[n], PrimeOmega[#] > 0 && Divisible[PrimeOmega[n], PrimeOmega[#]] &], {n, 2, 56}] (* _Ivan Neretin_, Jun 20 2019 *)

%o (PARI) a(n) = if (n==1, 1, my(b=bigomega(n)); sumdiv(n, d, if (d != 1, d*((b % bigomega(d)) == 0)))); \\ _Michel Marcus_, Jul 03 2018

%Y Cf. A000203, A001222.

%K nonn

%O 1,2

%A _Lechoslaw Ratajczak_, Jul 03 2018

%E Name edited by _Michel Marcus_, Jul 05 2018