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A063829 usigma(n) = 2n + d(n), where d(n) is the number of divisors of n. 1

%I

%S 150,294,726,1014,1428,1734,2166,3174,5046,5766,8214,10086,11094,

%T 13254,16854,20886,22326,26934,30246,31974,37446,41334,47526,56454,

%U 61206,63654,68694,71286,76614,96774,102966,112614,115926,133206,136806,147894,159414

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

%C 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_

%C 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_

%H Donovan Johnson, <a href="/A063829/b063829.txt">Table of n, a(n) for n = 1..5000</a>

%t 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-Fran├žois Alcover_, Jan 16 2013 *)

%o (PARI) us(n)=sumdiv(n,d, if(gcd(d,n/d)==1,d));

%o for(n=1,10^8, if(us(n)==2*n+numdiv(n),print1(n, ", ")))

%Y Cf. A002827, A034448.

%K nice,nonn

%O 1,1

%A _Jason Earls_, Aug 20 2001

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Last modified September 17 06:16 EDT 2021. Contains 347478 sequences. (Running on oeis4.)