%I #43 Jul 11 2019 00:15:58

%S 1,6,28,120,259,496,8128,18990,667296,1858939,2097414,2383279,4843717,

%T 33550336,150588313,186695863,188908297

%N Numbers n such that sigma(n) can be obtained as the base-3 carryless product of 2n and some k.

%C Numbers n that satisfy A000203(n) = A325820(2n, k) for some k.

%C Numbers n such that polynomial p divides polynomial q over GF(3), where p and q are obtained from the base-3 representations of 2n and sigma(n). (See the examples).

%C Conjecture: If we select only those n of these for which sigma(n) >= 2n, then we get a subsequence which contains only even terms: 6, 28, 120, 496, 8128, 18990, 667296, 2097414, 33550336, etc. If this is true, then there are no odd perfect numbers. See also conjectures in A325638 and A325639.

%H <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>

%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%e 2*120 has ternary representation (A007089) 22220_3, thus it encodes polynomial 2*x^4 + 2*x^3 + 2*x^2 + 2*x, while sigma(120) = 360 = 111100_3, encodes polynomial x^5 + x^4 + x^3 + x^2 which is a multiple of the former as it is equal to 2x(x^4 + x^3 + x^2 + x) when polynomial multiplication is done over GF(3). Thus 120 is included in this sequence.

%e 2*259 = 201012_3 encodes polynomial 2*x^5 + x^3 + x + 2, while sigma(259) = 304 = 102021_3 encodes polynomial x^5 + 2*x^3 + 2*x + 1 = 2(2*x^5 + x^3 + x + 2), thus 259 is included.

%e 2*18990 = 1221002200_3 encodes polynomial x^9 + 2*x^8 + 2*x^7 + x^6 + 2*x^3 + 2*x^2, while sigma(18990) = 49608 = 2112001100_3 encodes polynomial 2*x^9 + x^8 + x^7 + 2*x^6 + x^3 + x^2 = 2(x^9 + 2*x^8 + 2*x^7 + x^6 + 2*x^3), thus 18990 is included.

%e 2*667296 = 2111210201100_3 encodes polynomial 2*x^12 + x^11 + x^10 + x^9 + 2*x^8 + x^7 + 2*x^5 + x^3 + x^2, while sigma(667296) = 2175264 = 11002111220100_3 encodes polynomial x^13 + x^12 + 2*x^9 + x^8 + x^7 + x^6 + 2*x^5 + 2*x^4 + x^2 = (2*x + 1)(2*x^12 + x^11 + x^10 + x^9 + 2*x^8 + x^7 + 2*x^5 + x^3 + x^2) [when polynomial multiplication is done over GF(3)], thus 667296 is included.

%o (PARI) isA325808(n) = { my(p=Pol(digits(n+n,3))*Mod(1, 3), q=Pol(digits(sigma(n),3))*Mod(1, 3)); !(q%p); };

%Y Cf. A000203, A325638, A325820, A325825, A325827.

%Y Cf. A000396 (a subsequence).

%K nonn,more,base

%O 1,2

%A _Antti Karttunen_, May 22 2019