A077804 -id:A077804 - cp's OEIS frontend

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

Tanya Khovanova, Dec 27 2006

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

			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.

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

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

Cf. A077804 (deficient oblong numbers).

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

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

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

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

More terms from Emeric Deutsch, Jan 01 2007

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

A191969 Numbers that are indices of deficient oblong numbers (A002378).

Original entry on oeis.org

1, 10, 13, 22, 37, 43, 46, 52, 58, 61, 67, 73, 82, 85, 94, 97, 106, 109, 118, 121, 130, 133, 136, 142, 145, 148, 151, 157, 163, 166, 172, 178, 181, 190, 193, 202, 205, 211, 214, 217, 226, 229, 232, 238, 241, 250, 253, 262, 268, 277, 283, 289, 292, 298, 301, 310, 313, 316, 322, 331, 334, 337, 346, 358, 361, 373, 382, 388, 394, 397

Offset: 1

Chris Fry, Jun 22 2011

Numbers k such that A002378(k) = k*(k+1) is deficient.
"In 1700, Charles de Neuveglise claimed the product of two consecutive integers n(n+1) with n>=3 is abundant." - Tattersall, p. 144.  In other words, de Neuveglise claimed that all oblong numbers greater than 6 are abundant.  In fact, up to A002378(1100), 17.6% of the oblong numbers are deficient.  The per-100 count of deficient oblong numbers from A002378(1) to A002378(1100) is 16, 19, 19, 16, 17, 20, 18, 17, 17, 15, 20. For most deficient oblong numbers A002378(k) in this range, either k or k+1 is prime, but this is not always the case, explaining why the density of deficient oblong numbers does not decrease in line with the primes.
All the terms are congruent to 1 or 4 mod 6, and there are no terms that are congruent to 0, 4, 15, or 19 mod 20. Therefore, the asymptotic density of this sequence is less than 4/15. The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 2, 16, 174, 1831, 18237, 182432, 1824453, 18241059, 182414767, 1824169736, ... . Apparently, the asymptotic density of this sequence equals 0.18241... . - Amiram Eldar, Mar 15 2024

			The third deficient oblong number is A002378(13) = 13*14 = 182: sigma(182) = 336 < 364 = 2*182.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Second Edition, Cambridge University Press, 2005.

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

Cf. A005101, A005100, A002378, A124672, A077804.

Select[Range[400], DivisorSigma[1, o = # (# + 1)] < 2 o &] (* Amiram Eldar, Jun 21 2019 *)

for(n=1, 400, o=n*(n+1); if(sigma(o)<2*o, print1(n, ", ")))

A002378(a(n)) = A077804(n). - Amiram Eldar, Mar 15 2024

cp's OEIS Frontend

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

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