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 must omit 3*d (because 3 | d) and get a(1) = 5*d = 30.
For d = 28 = 4*7, we get a(2) = 3*d = 84, a(3) = 5*d = 140, we omit 7*d,
  a(4) = 11*d = 308, a(5) = 13*d = 364, a(6) = 17*d = 476, a(7) = 19*d = 532,
  a(8) = 23*d = 644. So far all terms are in order of increasing size.
For d = 496 = 16*31, we get a(9) = 3*d = 1488 through a(21) = 47*d = 23312 (omitting 31*d), but the next larger term a(22) comes from the next perfect number, see below. Then we get a(23) = 53*d = 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) = 3*d = 24384, a(30) = 5*d = 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, ", "))) /* Note: This prints the terms in order of increasingly large perfect numbers, not in order of increasing terms: e.g., 243536, the last value for d = 496 = (2^5-1)*2^4, is printed before 24384, first term for d = 8128 = (2^7-1)*2^6. */
(PARI) A165772_upto(N=10^5)=select({
  is_A165772(n)=my(v=valuation(n, 2), P); isprime(v+1) && (n=divrem(n>>v, P=2^(v+1)-1))[2]==0 && n[1] < P<<v && n[1]!=P && isprime(n[1]) && isprime(P)
}, [1..N\2]*2) \\ Older code updated and extended by M. F. Hasler, Jul 30 2024

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