A211185 Numbers whose number of proper divisors equals the number of their anti-divisors.

1, 3, 9, 10, 14, 15, 21, 26, 28, 34, 51, 69, 75, 76, 88, 92, 99, 102, 104, 106, 110, 124, 134, 135, 136, 138, 141, 146, 164, 170, 231, 232, 236, 256, 258, 261, 268, 285, 290, 309, 321, 328, 386, 394, 405, 411, 424, 429, 441, 484, 490, 525, 531, 574, 580, 590, 602, 608, 614, 615, 620, 628, 639, 645, 651, 656, 658

Offset: 1

Author

Juri-Stepan Gerasimov, Feb 02 2013

Keywords

• nonn

Comments

See A066272 for definition of anti-divisor.

Numbers of divisors of n such that number of proper divisors of n equals the number of anti-divisors of n: 1, 2, 2, 3, 4, 4, 4, 4, 6, 4, 4, 4, 6, 6, 4, 4, 4, 12, 4, 6, 10, 4, 8, 8, 4, 12, 4, 6, 4, 12, 4, 4, 4,...

Primes p such that number of proper divisors of p - 1 equals the number of anti-divisors of p - 1 and number of proper divisors of p + 1 equals the number of anti-divisors of p + 1 : 2, 103, 137, 257,...

Numbers whose sum of proper divisors equals the sum of their anti-divisors: 1, 5, 41,...

Examples

			28 is here since it has 5 proper divisors {2, 4, 7, 14, 28} and 5 anti-divisors {3, 5, 8, 11, 19}.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A032741, A066272, A073694, A178029.

Programs

Maple

for n from 1 to 700 do
    if A032741(n) = A066272(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Feb 03 2013

PARI

is(n)=numdiv(2*n+1)+numdiv(2*n-1)+numdiv(n>>valuation(n, 2))-numdiv(n)==4 || n==1 \\ Charles R Greathouse IV, Feb 04 2013

Formula

{n: A032741(n) = A066272(n)}.

Extensions

Entries corrected by R. J. Mathar, Feb 03 2013