A166981 -id:A166981 - cp's OEIS frontend

A338786 Numbers in A166981 that are neither superior highly composite nor colossally abundant.

1, 4, 24, 36, 48, 180, 240, 720, 840, 1260, 1680, 10080, 15120, 25200, 27720, 110880, 166320, 277200, 332640, 554400, 665280, 2162160, 3603600, 7207200, 8648640, 10810800, 36756720, 61261200, 73513440, 122522400, 147026880, 183783600, 698377680, 735134400, 1102701600

Offset: 1

Michael De Vlieger, Nov 09 2020

These are numbers both highly composite and superabundant but neither superior highly composite nor colossally abundant.
This sequence, A224078, A304234, and A304235 are mutually exclusive subsets that comprise A166981.
Superset A166981 has 449 terms; this sequence has 358, A224078 has 20, A304234 has 39, and A304235 has 32.

			1 is in the sequence since it is the empty product, setting records for both the number of divisors and the sum of divisors, and it is neither also superior highly composite nor colossally abundant.
2 is not in the sequence since it is both colossally abundant and superior highly composite.
4 is in the sequence since it sets a record for the divisor counting and divisor sum functions, yet it is neither superior highly composite nor colossally abundant.
20951330400 is not in the sequence since it is colossally abundant though it is an HCN and SA. etc.

Michael De Vlieger, Table of n, a(n) for n = 1..358
Michael De Vlieger, Annotated plot of a(n) at (x,y) = (a(n)/A002110(A001221(a(n)), A002110(A001221(a(n)))
Eric Weisstein's World of Mathematics, Highly Composite Number
Eric Weisstein's World of Mathematics, Colossally Abundant Number
Eric Weisstein's World of Mathematics, Superabundant Number
Eric Weisstein's World of Mathematics, Superior Highly Composite Number

Cf. A189228, A002201, A002182, A004394, A004490, A166981, A224078, A304234, A304235.

Complement[Import["https://oeis.org/A166981/b166981.txt", "Data"][[1 ;; 449, -1]], Union[FoldList[Times, Import["https://oeis.org/A073751/b073751.txt", "Data"][[1 ;; 120, -1]] ], FoldList[Times, Import["https://oeis.org/A000705/b000705.txt", "Data"][[1 ;; 120, -1]] ] ] ] (* Program reads OEIS b-files Michael De Vlieger, Nov 09 2020 *)

Complement of (the union of A002182 and A004394) and (the union of A002201 and A004490).

A166735 Superabundant numbers (A004394) that are not highly composite (A002182).

Original entry on oeis.org

1163962800, 4658179125600, 13974537376800, 144403552893600, 433210658680800, 10685862914126400, 21371725828252800, 32057588742379200, 37400520199442400, 64115177484758400, 1533421328177138400

Offset: 1

T. D. Noe, Oct 20 2009

nonn

Alaoglu and Erdos mention the first term in footnote 14.

Because the "shapes" of superabundant and highly composite numbers are different, there is a last superabundant number that is also highly composite. In factored form, that 154-digit number is N = A004394(1023) = A002182(2567) = 2^10 3^6 5^4 7^3 11^3 13^2 17^2 19^2 23^2 29 31 37...347. In other words, this sequence contains all superabundant numbers greater than N. - T. D. Noe, Oct 26 2009

T. D. Noe, Table of n, a(n) for n = 1..574
L. Alaoglu and P. Erdos, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata
Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. Mentions this sequence.
S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 - 409.
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.

Cf. A166981 (intersection of SA and HC numbers). - T. D. Noe, Oct 26 2009

Cf. A189228 (SA numbers that are not CA).

a(574+i) = A004394(1023+i) for i>0.

A095849 Numbers j where sigma_k(j) increases to a record for all real values of k.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 48, 60, 120, 240, 360, 840, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 720720, 1441440, 2162160, 3603600, 7207200, 10810800, 36756720, 61261200, 122522400, 183783600, 698377680

Offset: 1

Matthew Vandermast, Jun 09 2004

nonn

For any value of k, sigma_k(j) > sigma_k(m) for all m < j, where the function sigma_k(j) is the sum of the k-th powers of all divisors of j.

Conjecture: a number is in this sequence if and only if it is in both A002182 and A095848. - J. Lowell, Jun 21 2008

T. D. Noe, Table of n, a(n) for n = 1..74 (complete) [I assume this is only conjectured to be complete. - _N. J. A. Sloane_, Jan 02 2019]

Cf. A002093 (highly abundant numbers), A002182 (highly composite numbers) and A004394 (superabundant numbers), consisting of numbers that establish records for sigma_k(j) where k equals 1, 0 and -1 respectively. See also A095848.

Cf. also A166981 (numbers that establish records for both k=0 and k=-1).

Extended by T. D. Noe, Apr 22 2010
Corrected by T. D. Noe and Matthew Vandermast, Oct 04 2010
Removed keyword "fini", since it appears that as yet there is no proof. - N. J. A. Sloane, Sep 17 2022

A304234 Superior highly composite numbers that are superabundant but not colossally abundant.

Original entry on oeis.org

13967553600, 2248776129600, 65214507758400, 195643523275200, 12129898443062400, 448806242393308800, 18401055938125660800, 185942670254759802384000, 9854961523502269526352000, 1162885459773267804109536000, 780296143507862696557498656000

Offset: 1

Michael De Vlieger, May 08 2018

Numbers m in A002201 that are also in A004394 but not A004490.
Subset of A166981. Numbers in this sequence are in neither A224078 nor A304235.
There are 39 terms in this sequence.
The smallest term is 2^5 * 3^2 * 5 * A002110(8) or the product of A002110(k) with k = {1,1,1,2,3,8}.
The largest is 2^10 * 3^6 * 5^3 * 7^2 * 11 * 13 * 17 * 19 * 23 * A002110(65) or the product of A002110(k) with k = {1,1,1,1,2,2,2,3,4,9,65}, a 144 digit decimal number.

Michael De Vlieger, Table of n, a(n) for n = 1..39
Michael De Vlieger, Superior highly composite numbers m that are also superabundant but not colossally abundant
Michael De Vlieger, Color coded plot of m A002182 and A004394 at (x,y) where A301414(x) * A002110(y) = m, terms in this sequence are colored dark blue.
Michael De Vlieger, Annotated plot of a(n) for 1 <= n <= 39 at (x,y) = (a(n)/A002110(A001221(a(n)), A002110(A001221(a(n)))

Cf. A002110, A002182, A002201, A004394, A004490, A166981, A224078, A304235.

(* First, download b-files at A002201, A004394, and A004490 *)
f[w_] := Times @@ Flatten@ {Complement[#1, Union[#2, #3]], Product[Prime@ i, {i, PrimePi@ #}] & /@ #2, Factorial /@ #3} & @@ ToExpression@ {StringSplit[w, _?(! DigitQ@ # &)], StringCases[w, (x : DigitCharacter ..) ~~ "#" :> x], StringCases[w, (x : DigitCharacter ..) ~~ "!" :> x]};
With[{s = Import["b002201.txt", "Data"][[All, -1]], t = Select[Map[Which[StringTake[#, 1] == {"#"}, f@ Last@ StringSplit@ Last@ #, StringTake[#, 1] == {}, Nothing, True, ToExpression@ StringSplit[#][[1, -1]]] &, Drop[Import["b004394.txt", "Data"], 3] ], IntegerQ@ First@ # &][[All, -1]], u = Import["b004490.txt", "Data"][[All, -1]]}, Select[Intersection[s, t], FreeQ[u, #] &]]

A304235 Colossally abundant numbers that are highly composite, but not superior highly composite.

Original entry on oeis.org

160626866400, 9316358251200, 288807105787200, 2021649740510400, 224403121196654400, 9200527969062830400, 395622702669701707200, 1970992304700453905270400, 35468006523084668025340848000, 135483209545341953934626770390608000

Offset: 1

Michael De Vlieger, May 08 2018

Numbers m in A004490 that are also in A002182 but not A002201.
Subset of A166981. Numbers in this sequence are in neither A224078 nor A304234.
There are 32 terms in this sequence.
The smallest term is 2^4 * 3^2 * 5 * A002110(9) or the product of k = {1,1,2,3,9} in A002110.
The largest term is 2^9 * 3^5 * 5^3 * 7^2 * 11 * 13 * 17 * 19 * 23 * A002110(66) or the product of A002110(k) with k = {1,1,1,1,2,2,3,4,9,66}, a 146 digit decimal number.

Michael De Vlieger, Table of n, a(n) for n = 1..32
Michael De Vlieger, Colossally abundant m that are also highly composite but not superior highly composite
Michael De Vlieger, Color coded plot of m A002182 and A004394 at (x,y) where A301414(x) * A002110(y) = m, terms in this sequence are colored dark red.
Michael De Vlieger, Annotated plot of a(n) for 1 <= n <= 32 at (x,y) = (a(n)/A002110(A001221(a(n)), A002110(A001221(a(n)))

Cf. A002110, A002182, A002201, A004394, A004490, A166981, A224078, A304234.

(* First, download b-files at A002182, A002201, and A004490 *)
With[{s = Import["b004490.txt", "Data"][[All, -1]], t = Import["b002182.txt", "Data"][[All, -1]], u = Import["b002201.txt", "Data"][[All, -1]]}, Select[Intersection[s, t], FreeQ[u, #] &]]

A333953 Recursively superabundant numbers: numbers m such that A330575(m)/m > A330575(k)/k for all k < m.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 240, 288, 360, 480, 576, 720, 1152, 1440, 2160, 2880, 4320, 5760, 8640, 11520, 17280, 25920, 30240, 34560, 51840, 60480, 69120, 103680, 120960, 172800, 181440, 207360, 241920, 345600, 362880, 414720, 483840, 725760

Offset: 1

Amiram Eldar, Apr 11 2020

nonn

Fink (2019) defined this sequence. He asked whether 720 is the largest term that is also superabundant number (A004394).
He noted that up to 10^6 all the recursively superabundant numbers are also recursively highly composite numbers (A333952), except for 181440 (the next term which is not recursively highly composite is 2177280). He asked whether there are a finite number of numbers that are both recursively highly composite and recursively superabundant (in analogy to A166981).
From David A. Corneth, Apr 13 2020: (Start)
The 76 terms in the b-file are products of primorials (Cf. A025487) and 7-smooth numbers  (Cf. A002473). All terms are in A025487.
Proof: As A330575(n) = Sum_{d|n} A074206(d) * n/d we have A330575(n) / n = Sum_{d|n} A074206(d)/d which is maximal for some prime signature when n is a product of primorials.
Assuming terms below 10^17 are 13-smooth gives the 213 11-smooth numbers in the Corneth a-file. (End)

David A. Corneth, Table of n, a(n) for n = 1..213 (first 76 terms from Amiram Eldar)
Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. See section 5.

Cf. A002473, A004394, A166981, A025487, A330575, A333952.

s[1] = 1; s[n_] := s[n] = n + DivisorSum[n, s[#] &, # < n &]; seq={}; rm = 0; Do[r1 = s[n]/n; If[r1 > rm, rm = r1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A189686 Superabundant numbers (A004394) satisfying the reverse of Robin's inequality (A091901).

Original entry on oeis.org

2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 2520, 5040

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, May 30 2011

nonn

5040 is the last element in the sequence if and only if the Riemann Hypothesis is true. (See Akbary and Friggstad in A004394.)

G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33 (see Table 1).
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.

Cf. A004394, A091901, A067698, A166981, A077006.

kmax = 10^4;
A004394 = Join[{1}, Reap[For[r = 1; k = 2, k <= kmax, k = k + 2, s = DivisorSigma[-1, k]; If[s > r, r = s; Sow[k]]]][[2, 1]]];
A067698 = Select[Range[2, kmax], DivisorSigma[1, #] > Exp[EulerGamma] # Log[Log[#]]&];
Intersection[A004394, A067698] (* Jean-François Alcover, Jan 28 2019 *)

is(n)=sigma(n) >= exp(Euler) * n * log(log(n)); \\ A067698
lista(nn) = my(r=1, t); forstep(n=2, nn, 2, t=sigma(n, -1); if(t>r && is(n), r=t; print1(n, ", "))); \\ Michel Marcus, Jan 28 2019; adapted from A004394

Equals A004394 intersect A067698.

Erroneous terms 1260 and 1680 removed by Jean-François Alcover, Jan 28 2019

A308913 Highly composite numbers (A002182) that are not superabundant numbers (A004394).

Original entry on oeis.org

7560, 20160, 45360, 50400, 83160, 221760, 498960, 1081080, 2882880, 6486480, 14414400, 17297280, 32432400, 43243200, 110270160, 245044800, 294053760, 551350800, 2095133040, 2205403200, 4655851200, 5587021440, 10475665200, 64250746560, 73329656400, 97772875200

Offset: 1

Amiram Eldar, Jun 30 2019

nonn

Pillai noted in 1941 that 7560 is the first term of this sequence. He also asked for the opposite sequence and wondered whether its first term (A166735(1) = 1163962800) is within the reach of modern computation.

Since the sequence of superabundant numbers that are also highly composite, A166981, is finite, this sequence contains all the highly composite numbers above A002182(2567) = A004394(1023).

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. Sivasankaranarayana Pillai, On numbers analogous to highly composite numbers of Ramanujan, Rajah Sir Annamalai Chettiar Commemoration Volume, ed. Dr. B. V. Narayanaswamy Naidu, Annamalai University, 1941, pp. 697-704.
S. Sivasankaranarayana Pillai, Highly Composite Numbers of the t th Order, J. Indian Math. Soc., Vol. 8 (1944), pp. 61-74.

Cf. A002182, A004394, A166735, A166981, A181309.

seq = {}; dm = 0; sm = 0; Do[d = DivisorSigma[0, n]; s = DivisorSigma[1, n]; If[d > dm, dm = d]; If[s > s, sm = s, AppendTo[seq, n]], {n, 1, 3000000}]; seq

a(2118+i) = A002182(2567+i) for i > 0.

A340840 Union of the highly composite and superabundant numbers.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880

Offset: 1

Michael De Vlieger, Jan 27 2021

nonn

Numbers m that set records in A000005 and numbers k that set records for the ratio A000203(k)/k, sorted, with duplicates removed.
All terms are in A025487, since all terms in A002182 and A004394 are products of primorials P in A002110.
For numbers that are highly composite but not superabundant, see A308913; for numbers that are superabundant but not highly composite, see A166735. - Jon E. Schoenfield, Jun 14 2021

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated scatterplot of a(n) = A301414(x) * A002110(y) at (x, y) showing the intersection A166981 of A002182 and A004394, and the intersection A224078 of A002201 and A004490.

Cf. A000005, A000203, A027750, A002110.
Supersets: A025487, A055932.
Subsets: A002182, A002201, A004394, A004490, A166735, A166981, A168263, A189228, A224078, A304234, A304235, A308913, A333655, A338786.

(* Load the function f[] at A025487, then: *) Block[{t = Union@ Flatten@ f[15], a = {}, b = {}, d = 0, s = 0}, Do[(If[#2 > d, d = #2; AppendTo[a, #1]]; If[#3/#1 > s, s = #3/#1; AppendTo[b, #1]]) & @@ Flatten@ {t[[i]], DivisorSigma[{0, 1}, t[[i]]]}, {i, Length@ t}]; Union[a, b]]

A212169 List of highly composite numbers (A002182) with an exponent in its prime factorization that is at least as great as the number of positive exponents; intersection of A002182 and A212165.

Original entry on oeis.org

1, 2, 4, 12, 24, 36, 48, 120, 240, 360, 720, 1680, 5040, 10080, 15120, 20160, 25200, 45360, 50400, 110880, 221760, 332640, 554400, 665280, 2882880, 8648640, 14414400, 17297280, 43243200, 294053760

Offset: 1

Matthew Vandermast, May 22 2012

Sequence can be used to find the largest highly composite number in subsequences of A212165 (of which there are several in the database).
Ramanujan showed that, in the canonical prime factorization of a highly composite number with largest prime factor prime(n), the largest exponent cannot exceed 2*log_2(prime(n+1)). (See formula 54 on page 15 of the Ramanujan paper.) This limit is less than n for all n >= 9 (and prime(n) >= 23).
1. Direct calculation verifies this for 9 <= n <= 11.
2. Nagura proved that, for any integer m >= 25, there is always a prime between m and 1.2*m.  Let n = 11, at which point prime(11) = 31 and log_2(prime(n+1)) = log 37/log 2 = 5.209453.... Since log 1.2/log 2 is only 0.263034..., it follows that n must increase by at least 3k before 2*log_2(prime(n+1)) can increase by 2k, for all values of k. Therefore, 2*log_2(prime(n+1)) can never catch up to prime(n) for n > 11.
665280 = 2^6*3^3*5*7*11 is the largest highly composite number whose prime factorization contains an exponent that is strictly greater than the number of positive exponents in that factorization (including the implied 1's).

			A002182(62) = 294053760 = 2^7*3^3*5*7*11*13*17 has 7 positive exponents in its prime factorization, including 5 implied 1's. The maximal exponent in its prime factorization is also 7. Therefore, 294053760 is a term of this sequence.

S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

A. Flammenkamp, List of the first 1200 highly composite numbers
J. Nagura, On the interval containing at least one prime number, Proc. Japan Acad., 28 (1952), 177-181.
S. Ramanujan, Highly Composite Numbers (p. 15) (pages 11-12 introduce some of the notation in formula 54)

A212165 also includes all terms in A006939, A066120, A087980, A130091, A138534, A141586, A166475, A181555, A181813-A181814, A181818, A181823-A181825, A182763.

Other finite subsequences of A002182 include A072938, A095921, A106037, A136253, A162935, A166981, A168263.

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] >= Length[f]]; a = 0; t = {}; Do[b = DivisorSigma[0, n]; If[b > a, a = b; If[okQ[n], AppendTo[t, n]]], {n, 10^6}]; t (* T. D. Noe, May 24 2012 *)

cp's OEIS Frontend

A338786 Numbers in A166981 that are neither superior highly composite nor colossally abundant.

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A166735 Superabundant numbers (A004394) that are not highly composite (A002182).

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A095849 Numbers j where sigma_k(j) increases to a record for all real values of k.

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A304234 Superior highly composite numbers that are superabundant but not colossally abundant.

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A304235 Colossally abundant numbers that are highly composite, but not superior highly composite.

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A333953 Recursively superabundant numbers: numbers m such that A330575(m)/m > A330575(k)/k for all k < m.

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A189686 Superabundant numbers (A004394) satisfying the reverse of Robin's inequality (A091901).

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A308913 Highly composite numbers (A002182) that are not superabundant numbers (A004394).

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A340840 Union of the highly composite and superabundant numbers.

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