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A066731 f-triperfect numbers, where f(n) = sigma(n) and f-triperfect numbers are defined similarly to f-perfect numbers in A066218. 0

%I

%S 4680,60192,179424,1737000,2578968,3441888,5604192,6008184,6331104,

%T 302459850,320457888,477229032,589459104,731925000,766073448,928765600

%N f-triperfect numbers, where f(n) = sigma(n) and f-triperfect numbers are defined similarly to f-perfect numbers in A066218.

%C n is f-triperfect iff sum_{k divides n} f(k) = 3*f(n). f-triperfect numbers appear to be rare for many f. For the usual f(n) = n, there are at least 6 f-triperfect numbers known < 3.2 x 10^10, the first being 120. However, there are no f-triperfect numbers < 10^6 for f(n) = n+1 and f(n) = n-1 (if one ignores the trivial n = 1).

%C Also numbers n such that 2*sigma(n) is equal to the sum of the divisors of their aliquot parts. - _Paolo P. Lava_, Jul 13 2016

%D Wells, D. Curious and Interesting Numbers, Revised Edition. Penguin Books, 1997. (See the entry on "120".)

%e Divisors of 4680 = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 18, 20, 24, 26, 30, 36, 39, 40, 45, 52, 60, 65, 72, 78, 90, 104, 117, 120, 130, 156, 180, 195, 234, 260, 312, 360, 390, 468, 520, 585, 780, 936, 1170, 1560, 2340, 4680}; f applied to these yield {1, 3, 4, 7, 6, 12, 15, 13, 18, 28, 14, 24, 39, 42, 60, 42, 72, 91, 56, 90, 78, 98, 168, 84, 195, 168, 234, 210, 182, 360, 252, 392, 546, 336, 546, 588, 840, 1170, 1008, 1274, 1260, 1092, 2352, 2730, 3276, 5040, 7644, 16380}, which sum to 49140 = 3 * 16380 = 3 * f(4680). Hence 4680 is a term of the sequence.

%t f[x_] := DivisorSigma[1, x]; Do[If[ Apply[ Plus, Map[ f, Divisors[ n ] ] ] == 3*f[n], Print[n]], {n, 1, 10^7}]

%Y Cf. A066218.

%K nonn

%O 1,1

%A _Joseph L. Pe_, Jan 15 2002

%E a(5)-a(16) from _Giovanni Resta_, Jul 13 2016

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Last modified May 23 03:02 EDT 2019. Contains 323507 sequences. (Running on oeis4.)