A128699 -id:A128699 - cp's OEIS frontend

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

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).

A128702 Highly abundant numbers (A002093) that are not Harshad numbers (A005349).

Original entry on oeis.org

16, 96, 168, 47880, 85680, 95760, 388080, 458640, 526680, 609840, 637560, 776160, 887040, 917280, 942480, 1219680, 1244880, 1607760, 1774080, 2439360, 3880800, 5266800, 5569200, 6098400, 7761600, 9424800, 12196800, 17907120, 20900880

Offset: 1

Ant King, Mar 28 2007

All superabundant numbers (A004394), colossally abundant numbers (A004490), highly composite numbers (A002182) and superior highly composite numbers (A002201) are Harshad numbers. However, this is not true of the highly abundant numbers (A002093) and there are 32 exceptions in the 394 highly abundant numbers less than 50 million.

The previous comment is erroneous. The first superabundant number that is not a Harshad number is A004394(105) = 149602080797769600. The first highly composite number that is not a Harshad number is A002182(61) = 245044800. For all exceptions I found, the sum of digits is a power of 3. Although the first 60000 terms of the colossally abundant numbers and the superior highly composite numbers are Harshad numbers, I am not aware of a proof that all terms are Harshad numbers. There may be large counterexamples. [T. D. Noe, Oct 27 2009]

			The third highly abundant number that is not a Harshad number is 168. So a(3)=168.

Giovanni Resta, Table of n, a(n) for n = 1..400
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Wikipedia, Highly Abundant Numbers.
Wikipedia, Harshad Number.

Cf. A002093, A005349, A000203, A004394, A004490, A002182, A002201, A128699, A128700, A128701.

hadata1=FoldList[Max,1,Table[DivisorSigma[1,n],{n,2,10^6}]]; data1=Flatten[Position[hadata1,#,1,1]&/@Union[hadata1]];HarshadQ[k_]:=If[IntegerQ[ k/(Plus @@ IntegerDigits[ k ])],True,False];Select[data1,!HarshadQ[ # ] &]

The highly abundant numbers (A002093) are those values of n for which sigma(n)>sigma(m) for all mA000203(n). Harshad numbers (A005349) are divisible by the sum of their digits.

a(16)-a(29) from Donovan Johnson, May 09 2009

cp's OEIS Frontend

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.

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