A379112 Kalita-Saikia numbers: Numbers k such that the number of prime factors (with multiplicity) of sigma(k) is equal to the number of distinct prime factors of k.

1, 2, 4, 9, 16, 18, 25, 36, 50, 64, 100, 144, 225, 289, 400, 450, 576, 578, 729, 900, 1156, 1458, 1600, 1681, 2401, 2601, 2916, 3362, 3481, 3600, 4096, 4624, 4802, 5041, 5202, 6724, 6962, 7225, 7921, 9604, 10082, 10201, 10404, 11664, 13924, 14400, 14450, 15129, 15625, 15842, 17161, 18225, 18496, 20164, 20402, 21609

Offset

1,2

Comments

Numbers k such that for every prime power factor p^e||k, sigma(p^e) = ((p^(1+e)-1)/(p-1)) is a prime, i.e, every p^e is in A023194. Here e is the max. exponent such that p^e divides k.

If x and y are terms and gcd(x,y) = 1, then x*y is also a term.

These are called "Kalita-Saikia numbers" in the 2025 paper by Beri and Zelinsky. - Antti Karttunen, Jul 07 2025

Links

David A. Corneth, Table of n, a(n) for n = 1..10247 (first 1630 terms from Antti Karttunen, terms <= 4*10^10)

Satvik Beri and Joshua Zelinsky, On near superperfect numbers, the Goormaghtigh conjecture, and Mertens' theorem, arXiv:2505.08160 [math.NT], 2025. See pp. 4-5.

Index entries for sequences related to sigma(n).

Formula

{k such that A001222(A000203(k)) = A001221(k)}.

Mathematica

Select[Range[21609], PrimeOmega[DivisorSigma[1, #]]==PrimeNu[#]&] (* James C. McMahon, Dec 17 2024 *)

Prog

(PARI) is_A379112 = A379111;

Crossrefs

Cf. A000203, A001221, A001222, A023194 (subsequence), A058063, A379111 (characteristic function).

Subsequence of A028982.

Cf. also A336359, A336547.

Sequence in context: A230868 A014290 A337343 * A015730 A073858 A006474

Adjacent sequences: A379109 A379110 A379111 * A379113 A379114 A379115

Keyword

nonn

Author

Antti Karttunen, Dec 17 2024

Extensions

Added "Kalita-Saikia numbers" to the name. - Antti Karttunen, Jul 07 2025

Status

approved