A152212 - OEIS

%I #31 Mar 10 2024 04:19:14

%S 1,3,5,7,11,12,13,15,16,17,19,20,21,23,27,29,31,33,35,37,39,41,43,44,

%T 47,49,51,53,55,57,59,60,61,64,65,67,68,69,71,73,75,77,79,81,83,84,85,

%U 87,89,91,92,93,95,97,101,103,105,107,108,109,111,113,115,116

%N Numbers k such that sigma_0(k) divides (k*sigma_1(k) - sigma_2(k)).

%C All known blocks of consecutive integers that are terms have length 1, 2, or 3. All known blocks of length 1 are odd numbers. Blocks of length 2 are relatively rare. All known blocks of length 3 begin with an odd number. Examples: [1], [3], [5], [7], [11,12,13], [15,16,17], [19,20,21], [23], [27], [29], [31], [33], [35], [37], [39], [41], [43,44], ...

%C (k*sigma_1(k) - sigma_2(k))/sigma_0(k) is a square for k in {1, 3, 15, 19, 20, 39, 73, 81, 123, 163, 267, 692, 883, ...} (not in the OEIS).

%C 476 is the smallest even term k such that neither k-1 nor k+1 is a term. Up to 4000000, the sequence contains no k == 2 (mod 4), which if it holds up implies that every run of 3 starts with an odd number (in fact, k == 3 (mod 4)), and that there are no runs of 4. It appears, based on congruence arguments mod powers of 2, that no k == 2 (mod 4) is possible, but I am not sure that I can actually prove this. -_ Franklin T. Adams-Watters_, Nov 08 2009

%C For every k == 2 (mod 4) up to 10^8, the largest power of 2 that divides sigma_0(k) is exactly twice the largest power of 2 that divides (k*sigma_1(k) - sigma_2(k)), so k is not a term. Can it be proved that this holds for all k == 2 (mod 4)? - _Jon E. Schoenfield_, Mar 09 2024

%H Amiram Eldar, <a href="/A152212/b152212.txt">Table of n, a(n) for n = 1..10000</a>

%F k : A000005(k) | k*A000203(k) - A001157(k).

%t aQ[n_] := Divisible[n * DivisorSigma[1,n] - DivisorSigma[2, n], DivisorSigma[0, n]]; Select[Range[100], aQ] (* _Amiram Eldar_, Aug 06 2019 *)

%o (Magma) [n: n in [1..101]| IsIntegral(((n*DivisorSigma(1,n)-DivisorSigma(2,n)) /#Divisors(n)))]; // _Marius A. Burtea_, Sep 07 2019

%Y Cf. A000005, A000203, A001157.

%Y Cf. A064987. - _Franklin T. Adams-Watters_, Nov 08 2009

%K easy,nonn

%O 1,2

%A _Ctibor O. Zizka_, Nov 29 2008

%E a(34) = 64 and a(44) = 81 inserted by _Amiram Eldar_, Aug 06 2019

%E Edited by _Jon E. Schoenfield_, Mar 09 2024