A367449 Numbers k for which there are exactly k pairs (i, j), 1 <= i < j < k, such that i + j is a divisor of k.

30, 42, 54, 66, 78, 102, 114, 138, 174, 186, 208, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1312, 1338, 1362, 1374

Offset

1,1

Comments

Numbers k >= 1 for which A367588(k) = Sum_{d|k} floor((d-1)/2) = k;

Numbers k >= 1 for which A000203(k) - A000005(k) - A183063(k) = 2*k.

The sequence is infinite because all numbers of the form m = 6*p, p >= 5 prime (A138636), are terms.

Indeed: sigma(6*p) - tau(6(p) - A183063(6*p) = 3*4*(p + 1) - 8 - 4 = 12*p = 2*m.

If m = 2^k*p, p = 2^(k + 1) - 4*k - 3 prime number, then m is a term. Indeed: sigma(m) - tau(m) - A183063(m) = (2^(k + 1) - 1)*(p + 1) - 2*(k + 1) - 2*k = 2*m.

Links

Table of n, a(n) for n=1..51.

Example

30 is a term since it has exactly 30 pairs (i,j): (1, 2), (2, 3), (1, 4), (2, 4), (1, 5), (4, 6), (3, 7), (2, 8), (7, 8), (1,9), (6, 9), (5, 10), (4, 11), (3, 12), (2, 13), (1, 14), (14, 16), (13, 17),(12, 18), (11, 19), (10, 20), (9, 21), (8, 22), (7, 23), (6, 24), (5, 25), (4,26), (3, 27), (2, 28), (1, 29).

Maple

filter:= proc(n) uses numtheory;
  sigma(n) - tau(n) - `if`(n::even, tau(n/2), 0) = 2*n
end proc:
select(filter, [$1..10000]); # Robert Israel, Dec 12 2023

Mathematica

f1[p_, e_] := e+1; f1[2, e_] := 2*e+1; f2[p_, e_] := (p^(e+1)-1)/(p-1); s[1] = 0; s[n_] := Module[{fct = FactorInteger[n]}, Times @@ f2 @@@ fct - Times @@ f1 @@@ fct]; Select[Range[1400], s[#] == 2*# &] (* Amiram Eldar, Dec 16 2023 *)

Prog

(Magma) [k:k in [1..1000]|(DivisorSigma(1, k)-#Divisors(k)-#[d:d in Divisors(k)| IsEven(d)]) eq 2*k ];
(PARI) isok(k) = sumdiv(k, d, (d-1)\2) == k; \\ Michel Marcus, Dec 19 2023

Crossrefs

Fixed points of A367588.

Cf. A000005, A000203, A138636, A183063.

Sequence in context: A074696 A127663 A008885 * A097036 A090790 A090800

Adjacent sequences: A367446 A367447 A367448 * A367450 A367451 A367452

Keyword

nonn

Author

Marius A. Burtea, Dec 10 2023

Status

approved