A394060 a(n+1) = a(n) + 2*d(a(n)) if s(a(n)) <= a(n), else 2*a(n) - s(a(n)), with a(1) = 1; d(k) is the number of divisors of k A000005 and s(k) is the aliquot sum of k A001065.

1, 3, 7, 11, 15, 23, 27, 35, 43, 47, 51, 59, 63, 75, 87, 95, 103, 107, 111, 119, 127, 131, 135, 151, 155, 163, 167, 171, 183, 191, 195, 211, 215, 223, 227, 231, 247, 255, 271, 275, 287, 295, 303, 311, 315, 339, 347, 351, 367, 371, 379, 383, 387, 399, 415, 423

Offset

1,2

Comments

The sequence becomes periodic starting from a(898)

Starting with a(1) = 2 instead of 1 yields the prefix 2, 6, 14, 22, 30, 18, 15, after which the sequence merges with the current terms starting from a(8) = 23 (which corresponds to a(6) of the main sequence).

Links

Adam Karol Drażyk, Table of n, a(n) for n = 1..2000

Formula

a(n+1) = if s(a(n)) <= a(n) then a(n) + 2*d(a(n)) else 2*a(n) - s(a(n)), where s(k) is A001065(k) and d(k) is A000005(k).

Example

Starting with a(1) = 1.
Sum of proper divisors of 1 is less than 1, so 1 + 2 * d(1) = 1 + 2 * 0 = 3. So a(2) = 3.
Sum of proper divisors of 3 is less than 3, so 3 + 2 * d(3) = 3 + 2 * 2 = 7. So a(3) = 7
Sum of proper divisors of 7 is less than 7, so 7 + 2 * d(7) = 7 + 2 * 2 = 11. So a(4) = 11.
Sum of proper divisors of 11 is less than 11, so 11 + 2 * d(1) = 1 + 2 * 2 = 15. So a(5) = 15.
Sum of proper divisors of 15 is less than 15, so 15 + 2 * d(15) = 15 + 2 * 4 = 23. So a(6) = 23.

Mathematica

NestWhileList[
 If[DivisorSigma[1, #] <= 2*#, # + 2*DivisorSigma[0, #],
   3*# - DivisorSigma[1, #]] &, 1, Unequal, All]

Crossrefs

Cf. A000005, A001065, A392092 (different initial values).

Sequence in context: A036994 A243115 A380358 * A279106 A172306 A309274

Adjacent sequences: A394057 A394058 A394059 * A394061 A394062 A394063

Keyword

nonn

Author

Adam Karol Drażyk, Mar 09 2026

Status

approved