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A063829
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usigma(n) = 2n + d(n), where d(n) is the number of divisors of n.
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1
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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
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1,1
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COMMENTS
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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
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(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, ", ")))
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CROSSREFS
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KEYWORD
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nice,nonn
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AUTHOR
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STATUS
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approved
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