A364990 Coreful triperfect numbers: numbers k such that csigma(k) = 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

3600, 25200, 28224, 39600, 46800, 61200, 68400, 82800, 104400, 111600, 133200, 141120, 147600, 154800, 169200, 190800, 212400, 219600, 241200, 255600, 262800, 277200, 284400, 298800, 310464, 320400, 327600, 349200, 363600, 366912, 370800, 385200, 392400, 406800

Offset: 1

Author

Amiram Eldar, Aug 15 2023

Keywords

• nonn

Comments

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947) (the term "coreful divisor" was used by Hardy and Subbarao, 1983).

If k is a term, then also m*k is, for any squarefree m coprime to k. Thus there are infinitely many coreful triperfect numbers, and all of them can be generated from the sequence of primitive terms, which is the subsequence of powerful terms of this sequence. This sequence is c(n) = A064549(A005820(n)), i.e., the triperfect numbers (A005820), multiplied by their squarefree kernel (A007947): 3600, 28224, 1071645696, 1651818858099200, 532098668445696, 317519577357516800, ...

The asymptotic density of this sequence is Sum_{i>=1} beta(c(i))/c(i), where beta(n) = (6/Pi^2) * Product_{p|n} (p/(p+1)) = 0.0000797856... . If there are only 6 triperfect numbers, then the exact value of this density is 18575679807276818039685539/(23589576231586703755916083200 * Pi^2).

Examples

			3600 is in the sequence since its coreful divisors are {30, 60, 90, 120, 150, 180, 240, 300, 360, 450, 600, 720, 900, 1200, 1800, 3600}, whose sum is 10800 = 3 * 3600.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)

Crossrefs

Cf. A005820, A007947, A057723, A064549, A307958.

Programs

Mathematica

f[p_, e_] := (p^(e+1)-1)/(p-1)-1; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[500000], s[#] == 3*# &]

PARI

s(n) = {my(f = factor(n)); prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);}
is(n) = s(n) == 3*n;