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

Author

Marius A. Burtea, Dec 10 2023

Keywords

• nonn

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.

Examples

			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).
		

Crossrefs

Fixed points of A367588.

Cf. A000005, A000203, A138636, A183063.

Programs

Magma

[k:k in [1..1000]|(DivisorSigma(1,k)-#Divisors(k)-#[d:d in Divisors(k)| IsEven(d)]) eq 2*k ];

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 *)

PARI

isok(k) = sumdiv(k, d, (d-1)\2) == k; \\ Michel Marcus, Dec 19 2023