A165772 - OEIS

%I #16 Jul 31 2024 11:43:07

%S 30,84,140,308,364,476,532,644,1488,2480,3472,5456,6448,8432,9424,

%T 11408,14384,18352,20336,21328,23312,24384,26288,29264,30256,33232,

%U 35216,36208,39184,40640,41168,44144,48112,50096,51088,53072,54064,56048

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

%C 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).

%H Amiram Eldar, <a href="/A165772/b165772.txt">Table of n, a(n) for n = 1..10000</a>

%e For d = 6 = 2*3, we must omit 3*d (because 3 | d) and get a(1) = 5*d = 30.

%e For d = 28 = 4*7, we get a(2) = 3*d = 84, a(3) = 5*d = 140, we omit 7*d,

%e   a(4) = 11*d = 308, a(5) = 13*d = 364, a(6) = 17*d = 476, a(7) = 19*d = 532,

%e   a(8) = 23*d = 644. So far all terms are in order of increasing size.

%e 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.

%e 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.

%t 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]];

%t 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 *)

%o (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. */

%o (PARI) A165772_upto(N=10^5)=select({

%o   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)

%o }, [1..N\2]*2) \\ Older code updated and extended by _M. F. Hasler_, Jul 30 2024

%Y Cf. A000396, A109321, A111592.

%K nonn

%O 1,1

%A _M. F. Hasler_, Oct 11 2009