A303695 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.

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, 41, 30, 37, 29, 40, 31, 34, 47, 53, 47, 60, 37, 59, 55, 61, 41, 54, 43, 57, 53, 71, 47, 53, 56, 57, 71, 67, 53, 74, 71, 83

Offset

1,2

Comments

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

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.

No other such terms up to 10^7. - Michel Marcus, Jul 05 2018

Links

Ivan Neretin, Table of n, a(n) for n = 1..10000

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000203, A001222.

Sequence in context: A319680 A350337 A133477 * A345321 A350339 A039653

Adjacent sequences: A303692 A303693 A303694 * A303696 A303697 A303698

Keyword

nonn

Author

Lechoslaw Ratajczak, Jul 03 2018

Extensions

Name edited by Michel Marcus, Jul 05 2018

Status

approved