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A152212
Numbers k such that sigma_0(k) divides (k*sigma_1(k) - sigma_2(k)).
1
1, 3, 5, 7, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 44, 47, 49, 51, 53, 55, 57, 59, 60, 61, 64, 65, 67, 68, 69, 71, 73, 75, 77, 79, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 101, 103, 105, 107, 108, 109, 111, 113, 115, 116
OFFSET
1,2
COMMENTS
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], ...
(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).
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
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
LINKS
FORMULA
k : A000005(k) | k*A000203(k) - A001157(k).
MATHEMATICA
aQ[n_] := Divisible[n * DivisorSigma[1, n] - DivisorSigma[2, n], DivisorSigma[0, n]]; Select[Range[100], aQ] (* Amiram Eldar, Aug 06 2019 *)
PROG
(Magma) [n: n in [1..101]| IsIntegral(((n*DivisorSigma(1, n)-DivisorSigma(2, n)) /#Divisors(n)))]; // Marius A. Burtea, Sep 07 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ctibor O. Zizka, Nov 29 2008
EXTENSIONS
a(34) = 64 and a(44) = 81 inserted by Amiram Eldar, Aug 06 2019
Edited by Jon E. Schoenfield, Mar 09 2024
STATUS
approved