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.

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

1,2

Comments

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).

Links

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.

Formula

The highly abundant numbers (A002093) are those values of n for which sigma(n)>sigma(m) for all m<n, where sigma(n)= A000203(n).

Example

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.

Mathematica

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 *)

Crossrefs

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

Sequence in context: A043293 A208894 A178644 * A030390 A063220 A063234

Adjacent sequences: A128698 A128699 A128700 * A128702 A128703 A128704

Keyword

nonn

Author

Ant King, Mar 28 2007

Status

approved