A063829 usigma(n) = 2n + d(n), where d(n) is the number of divisors of n.

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

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]] (* Jean-François 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