A124672 - OEIS

%I #15 Mar 16 2022 20:59:13

%S 12,20,30,42,56,72,90,132,156,210,240,272,306,342,380,420,462,552,600,

%T 650,702,756,812,870,930,992,1056,1122,1190,1260,1332,1482,1560,1640,

%U 1722,1806,1980,2070,2256,2352,2450,2550,2652,2862,2970,3080,3192,3306

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

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

%H Amiram Eldar, <a href="/A124672/b124672.txt">Table of n, a(n) for n = 1..10000</a>

%F 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

%e 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.

%p 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

%p 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

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

%o (PARI) helper(n)=my(k=sqrtint(n)); if(k*(k+1)>n, k, k+1)

%o 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

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

%Y Cf. A077804 (deficient oblong numbers).

%K nonn,easy

%O 1,1

%A _Tanya Khovanova_, Dec 27 2006

%E More terms from _Emeric Deutsch_, Jan 01 2007

%E More terms from _R. J. Mathar_, Jan 07 2007