A165772 Numbers d*p where d is a perfect number and p<d a prime not dividing d.

30, 84, 140, 308, 364, 476, 532, 644, 1488, 2480, 3472, 5456, 6448, 8432, 9424, 11408, 14384, 18352, 20336, 21328, 23312, 24384, 26288, 29264, 30256, 33232, 35216, 36208, 39184, 40640, 41168, 44144, 48112, 50096, 51088, 53072, 54064, 56048

Offset

1,1

Comments

A subsequence of A109321, and thus admirable numbers (A111592, solutions to sigma(x)-2x = 2d with d being a proper divisor of x): If d is a perfect number (A000396), then for any prime p<d coprime to d, sigma(dp)-2dp = 2d (thus dp is in A111592) and d > sqrt(dp).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

For d=6=2*3, we get a(1)=5d=30.
For d=28=4*7, we get a(2)=3d=84, a(3)=5d=140, omitting 7d, a(4)=11d=308, a(5)=13d=364, a(6)=17d=476, a(7)=19d=532, a(8)=23d=644.
For d=496=16*31, we get a(9)=3d=1488 through a(21)=47d=23312 (omitting 31d), a(23)=53d=26288 through a(29)=39184, a(31)=41168 through a(38)=56048 and a(40)=62992.
For d=8128=64*127, we get a(22)=3d=24384, a(30)=5d=40640, a(39)=56896, a(41)=89408, and all following terms up to 3*4096*8191.

Mathematica

f[p_] := (2^p - 1)*2^(p - 1); evenPerf[n_] := f[MersennePrimeExponent[n]]; sp[p_, max_] := With[{pn = f[p]}, pn * Select[Complement[Range[3, Min[pn - 1, max/pn]], {2^p - 1}], PrimeQ]];
seq[max_] := Module[{s = {}, k = 1}, While[(pn = evenPerf[k]) < max/3, s = Join[s, sp[MersennePrimeExponent[k], max]]; k++]; Union[s]]; seq[60000] (* Amiram Eldar, Aug 05 2023, assuming that there are no odd perfect numbers below max *)

Prog

(PARI) forprime(q=1, 9, isprime(2^q-1)|next; print("\n/* q="q", d=", d=(2^q-1)<<(q-1), " */"); forprime(p=3, d-1, d%p | next; print1(d*p, ", ")))
is_A165772(n)={ my(q=valuation(n, 2)); q | return; n=divrem(n>>q, 2^q-1); n[2]==0 && n[1] < (2^q-1)<<q && isprime(n[1]) }

Crossrefs

Cf. A000396, A109321, A111592.

Sequence in context: A044549 A353176 A155461 * A277980 A241025 A326309

Adjacent sequences: A165769 A165770 A165771 * A165773 A165774 A165775

Keyword

nonn

Author

M. F. Hasler, Oct 11 2009

Status

approved