

A063829


usigma(n) = 2n + d(n), where d(n) is the number of divisors of n.


1



150, 294, 726, 1014, 1428, 1734, 2166, 3174, 5046, 5766, 8214, 10086, 11094, 13254, 16854, 20886, 22326, 26934, 30246, 31974, 37446, 41334, 47526, 56454, 61206, 63654, 68694, 71286, 76614, 96774, 102966, 112614, 115926, 133206, 136806, 147894, 159414
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OFFSET

1,1


COMMENTS

The sequence includes all numbers of the form 6 * p^2 with p a prime >= 5. All of the terms above are of this form, except for 1428. There are similar subsequences corresponding to each of the five known unitary perfect numbers (A002827), namely 60 * p^9 (p>=7), 90 * p^14 (p>=7), 87360 * p^1559 (p=11 or p>=17) and 146361946186458562560000 * p^3009086064688703999 (p>=17 and not equal to 19, 37, 79, 109, 157, or 313). It is not known if there are other terms in the sequence besides these and 1428.  Dean Hickerson
The term 33872160 was found later: it is not of the form a * p^e where a is a unitary perfect number and p is a prime not dividing a.  Jason Earls


LINKS

Donovan Johnson, Table of n, a(n) for n = 1..5000


MATHEMATICA

usigma[n_] := Sum[d*Boole[GCD[d, n/d] == 1], {d, Divisors[n]}]; Reap[For[n = 1, n < 140000, n++, If[usigma[n] == 2 n + DivisorSigma[0, n], Sow[n]]]][[2, 1]] (* JeanFrançois Alcover, Jan 16 2013 *)


PROG

(PARI) us(n)=sumdiv(n, d, if(gcd(d, n/d)==1, d));
for(n=1, 10^8, if(us(n)==2*n+numdiv(n), print1(n, ", ")))


CROSSREFS

Cf. A002827, A034448.
Sequence in context: A207044 A292705 A335145 * A291959 A210643 A211550
Adjacent sequences: A063826 A063827 A063828 * A063830 A063831 A063832


KEYWORD

nice,nonn


AUTHOR

Jason Earls, Aug 20 2001


STATUS

approved



