A393635 Numbers k such that sigma(k) = psi(k) + tau(k) + Omega(k)^9.

699064, 1769499, 5461375, 71548019, 94596950, 131747525, 140332325, 397965113

Offset

1,1

Comments

a(9) > 10^10, if it exists. - Amiram Eldar, Apr 14 2026

From David A. Corneth, Apr 17 2026: (Start)

a(9)..a(12) <= 52505097576299, 417009509941855, 844463585427459, 1334476246021774. They are closely related to the first 4 terms that are of the form p^3*q. The current upper bounds on a(12) are of the form p*q^3 where p and q come from the first few terms.

If k = m*p, where k is a term, p is a prime and gcd(m, p) = 1, then p = (psi(m) - sigma(m) + 2*tau(m) + (Omega(m) + 1)^9)/(sigma(m) - psi(m)). This way, when testing m = 336973, we obtain p = 1181, which produces k = a(8) = 336973 * 1181. If no term is from A001694, then this method can be used to find all terms <= some bound. (End)

Links

Table of n, a(n) for n=1..8.

Example

699064 is a term since sigma(699064) = 1310760 = 1048608 + 8 + 4^9 = psi(699064) + tau(699064) + Omega(699064)^9.

Prog

(PARI) isok(k) = {my(f = factor(k)); sigma(f) == prod(i=1, #f~, (f[i, 1]+1) * f[i, 1]^(f[i, 2]-1)) + numdiv(f) + bigomega(f)^9; } \\ Amiram Eldar, Apr 13 2026

Crossrefs

Cf. A000005, A000203, A001222, A001615, A001694, A394658, A394738, A394751, A394816, A394925, A394926, A393481.

Sequence in context: A205439 A027829 A258129 * A204496 A332850 A378475

Adjacent sequences: A393632 A393633 A393634 * A393636 A393637 A393638

Keyword

nonn,hard,more

Author

S. I. Dimitrov, Apr 13 2026

Status

approved