A128702 -id:A128702 - cp's OEIS frontend

A128699 Highly abundant numbers that are not superabundant, i.e., the complement of A004394 w.r.t. A002093.

3, 8, 10, 16, 18, 20, 30, 42, 72, 84, 90, 96, 108, 144, 168, 210, 216, 288, 300, 336, 420, 480, 504, 540, 600, 630, 660, 960, 1008, 1080, 1200, 1440, 1560, 1620, 1800, 1920, 1980, 2100, 2160, 2340, 2400, 2880, 3024, 3120, 3240, 3360, 3600, 3780, 3960, 4200

Offset: 1

Ant King, Mar 28 2007

nonn

In 1944, Alaoglu and Erdős conjectured that this sequence was infinite and this was proved to be true by Nicolas in 1969.

			The sequence of highly abundant numbers begins 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20 and the sequence of superabundant numbers begins 1, 2, 4, 6, 12, 24. Because 10 is the third number which is in the first sequence but not in the second, it follows that a(3)=10.

Amiram Eldar, Table of n, a(n) for n = 1..1000
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Jean-Louis Nicolas, Ordre maximal d'un élément du groupe Sn des permutations et "highly composite numbers" Bull. Soc. Math. France 97 (1969), pp. 129-191.
Eric Weisstein's World of Mathematics, Superabundant Number.
Wikipedia, Highly Abundant Numbers.

Cf. A000203, A002093, A004394, A128700, A128701, A128702.

habdata1=FoldList[Max,1,Table[DivisorSigma[1,n],{n,2,10000}]]; data1=Flatten[Position[habdata1,#,1,1]&/@Union[habdata1]];sabdata2=FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10000}]]; data2=Flatten[Position[sabdata2,#,1,1]&/@Union[sabdata2]];sabdata2=FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10000}]]; Complement[data1,data2]

The highly abundant numbers are those integers for which sigma(n) > sigma(m) for all m < n (A002093) and the superabundant numbers are those integers for which sigma(n)/n > sigma(m)/m for all m < n (A004394).

A128701 Highly abundant numbers that are not products of consecutive primes with nonincreasing exponents, i.e., that are not of the form n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2>=e_3>=...>=e_p.

Original entry on oeis.org

1, 3, 10, 18, 20, 42, 84, 90, 108, 168, 300, 336, 504, 540, 600, 630, 660, 1008, 1200, 1560, 1620, 1980, 2100, 2340, 2400, 3024, 3120, 3240, 3780, 3960, 4200, 4680, 5880, 6120, 6240, 7920, 8400, 8820

Offset: 1

Ant King, Mar 28 2007

nonn

This is the subsequence of those highly abundant numbers (A002093) that have a different canonical structure to the superabundant numbers (A004394), the colossally abundant numbers (A004490), the highly composite numbers (A002182) and the superior highly composite numbers (A002201).

			As 10 is the third highly abundant number that cannot be expressed as a product of consecutive primes with nonincreasing exponents, then a(3)=10.

Amiram Eldar, Table of n, a(n) for n = 1..8404
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Jeffrey C. Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis, arXiv:math/0008177 [math.NT], 2000-2001.
Jeffrey C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, American Mathematical Monthly 109 (2002), pp. 534-543.
Wikipedia, Highly Abundant Numbers.

Cf. A002093, A004394, A000203, A004490, A002182, A002201, A128699, A128700, A128702.

hadata1=FoldList[Max,1,Table[DivisorSigma[1,n],{n,2,10000}]]; data1=Flatten[Position[hadata1,#,1,1]&/@Union[hadata1]];primefactorlist[1]={1};primefactorlist[k_]:=First[Transpose[FactorInteger[k]]];exponentlist[1]={1};exponentlist[k_]:=Last[Transpose[FactorInteger[k]]];g[k_List]:=If[MemberQ[Table[k[[i]]<= k[[i-1]],{i,1,Length[k]}],False],False,True];h[k_]:=If[primefactorlist[k]==(Prime[ # ]&/@Range[Length[primefactorlist[k]]]),True,False];Select[data1,Or[ ! h[ # ],!g[exponentlist[ # ]]]&]
seq = {1}; sm = 0; Do[f = FactorInteger[n]; p = f[[;; , 1]]; e = f[[;; , 2]]; s = Times @@ ((p^(e + 1) - 1)/(p - 1)); If[s > sm, sm = s; m = Length[p]; If[p[[-1]] != Prime[m] || (m > 1 && ! AllTrue[Differences[e], # <= 0 &]), AppendTo[seq, n]]], {n, 2, 10^4}]; seq (* Amiram Eldar, Jun 18 2019 *)

The highly abundant numbers (A002093) are those values of n for which sigma(n)>sigma(m) for all mA000203(n).

cp's OEIS Frontend

A128699 Highly abundant numbers that are not superabundant, i.e., the complement of A004394 w.r.t. A002093.

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