A392691 Numbers k such that sigma(k) = 2*k + 3*phi(k).

60, 168, 1530, 3672, 21648, 37536, 40608, 48024, 83904, 87648, 352704, 354384, 560664, 7285968, 7816608, 11877408, 19186848, 90799104, 235465056, 551812608, 766099296, 5813035008

Offset

1,1

Comments

Also numbers k with abundance 3*phi(k) = A200905(k).

From David A. Corneth, Jan 21 2026: (Start)

If k = m * p where p is prime and gcd(m, p) = 1 then p = (sigma(m) + 3*phi(m)) / (2*m + 3*phi(m) - sigma(m)).

a(23)..a(26) are at most 56298786588, 151289579664, 337081763664, 372049379328 respectively. (End)

Links

Table of n, a(n) for n=1..22.

Example

n=60 has sigma(60) = 168, phi(60) = 16, 2*60 + 3*16 = 168.
n=168 has sigma(168) = 480, phi(168) = 48, 2*168 + 3*48 = 480.

Mathematica

q[k_] := DivisorSigma[1, k] == 2*k + 3*EulerPhi[k]; Select[Range[400000], q] (* Amiram Eldar, Jan 21 2026 *)

Prog

(PARI) isok(n) = sigma(n) == 2*n + 3*eulerphi(n);

Crossrefs

If we generalize to numbers with abundance k*phi(x), then a(n) is the case of k=3, and we have:

Cf. A008578 (k=-1), A000396 (k=0), A088826 (k=1), A099650 (k=2), A392692 (k=4), A392693 (k=5).

Cf. A000010 (phi), A000203 (sigma), A005843, A068412, A200905.

Cf. A047802.

Sequence in context: A119630 A216480 A257146 * A291549 A259946 A249911

Adjacent sequences: A392688 A392689 A392690 * A392692 A392693 A392694

Keyword

nonn,hard,more

Author

Aloe Poliszuk, Jan 20 2026

Status

approved