A124672 Oblong (promic) abundant numbers = abundant numbers of the form k(k+1).

12, 20, 30, 42, 56, 72, 90, 132, 156, 210, 240, 272, 306, 342, 380, 420, 462, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1482, 1560, 1640, 1722, 1806, 1980, 2070, 2256, 2352, 2450, 2550, 2652, 2862, 2970, 3080, 3192, 3306

Offset

1,1

Comments

Promic numbers are highly divisible, so most of them are abundant.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

If k > 2 is 0 or 2 mod 3, then k*(k+1) is in this sequence; the bounds n^2 < a(n) < (9/4)*n^2 + 6n + 4 can be derived from this. Probably a(n) ~ kn^2 with k near 1.496. - Charles R Greathouse IV, Mar 16 2022

Example

56 is in the sequence because 56=7*8 and the sum of its divisors 1+2+4+7+8+14+28+56=120 > 2*56.

Maple

with(numtheory): a:=proc(k) if sigma(k*(k+1))>2*k*(k+1) then k*(k+1) else fi end: seq(a(k), k=1..75); # Emeric Deutsch, Jan 01 2007
isA005101 := proc(n) if numtheory[sigma](n) > 2*n then RETURN(true) ; else RETURN(false) ; fi ; end : for k from 1 to 80 do if isA005101(k*(k+1)) then printf("%d, ", k*(k+1)) ; fi ; od ; # R. J. Mathar, Jan 07 2007

Mathematica

s = {}; Do[ob = n*(n + 1); If[DivisorSigma[1, ob] > 2*ob, AppendTo[s, ob]], {n, 1, 100}]; s (* Amiram Eldar, Jun 07 2019 *)

Prog

(PARI) helper(n)=my(k=sqrtint(n)); if(k*(k+1)>n, k, k+1)
list(lim)=my(v=List(), last=4/3, cur); forfactored(n=4, helper(lim\1), cur=sigma(n, -1); if(cur*last>2, listput(v, (n[1]-1)*n[1])); last=cur); Vec(v) \\ Charles R Greathouse IV, Mar 16 2022

Crossrefs

Intersection of A002378 (oblong numbers) and A005101 (abundant numbers).

Cf. A077804 (deficient oblong numbers).

Sequence in context: A210968 A107277 A256883 * A289980 A109547 A334418

Adjacent sequences: A124669 A124670 A124671 * A124673 A124674 A124675

Keyword

nonn,easy

Author

Tanya Khovanova, Dec 27 2006

Extensions

More terms from Emeric Deutsch, Jan 01 2007

More terms from R. J. Mathar, Jan 07 2007

Status

approved