A004394 -id:A004394 - cp's OEIS frontend

A002093 Highly abundant numbers: numbers k such that sigma(k) > sigma(m) for all m < k.

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 210, 216, 240, 288, 300, 336, 360, 420, 480, 504, 540, 600, 630, 660, 720, 840, 960, 1008, 1080, 1200, 1260, 1440, 1560, 1620, 1680, 1800, 1920, 1980, 2100

Offset: 1

N. J. A. Sloane

Where record values of sigma(n) occur.
Also record values of A070172: A070172(i) < a(n) for 1 <= i < A085443(n), a(n) = A070172(A085443(n)). - Reinhard Zumkeller, Jun 30 2003
Numbers k such that sum of the even divisors of 2*k is a record. - Arkadiusz Wesolowski, Jul 12 2012
Conjecture: (a) Every highly abundant number > 10 is practical (A005153). (b) For every integer k there exists A such that k divides a(n) for all n > A. Daniel Fischer proved that every highly abundant number greater than 3, 20, 630 is divisible by 2, 6, 12 respectively. The first conjecture has been verified for the first 10000 terms. - Jaycob Coleman, Oct 16 2013
Conjecture: For each term k: (1) Let p be the largest prime less than k (if one exists) and let q be the smallest prime greater than k; then k-p is either 1 or a prime, and q-k is either 1 or a prime. (2) The closest prime number p < k located to a distance d = k-p > 1 is also always at a prime distance. These would mean that the even highly abundant numbers greater than 2 always have at least a Goldbach pair of primes. h=p+d. Both observations verified for the first 10000 terms. - David Morales Marciel, Jan 04 2016
Pillai used the term "highly abundant numbers of the r-th order" for numbers with record values of the sum of the reciprocals of the r-th powers of their divisors. Thus highly abundant numbers of the 1st order are actually the superabundant numbers (A004394). - Amiram Eldar, Jun 30 2019

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata.
D. Fischer, Proof 2, 6, 12 divides a(n) greater than 3, 20, 630 resp.
S. S. 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. S. Pillai, Highly abundant numbers, Bulletin of the Calcutta Mathematical Society, Vol. 35, No. 1 (1943), pp. 141-156.
N. J. A. Sloane, Transforms (The RECORDS transform returns both the high-water marks and the places where they occur).
Wikipedia, Highly abundant number

Cf. A034091, A000203, A004394, A005153.
The record values are in A034885.
Cf. A193988, A193989 (records for sigma_2 and sigma_3).

N:= 100: # to get a(1) to a(N)
best:= 0: count:= 0:
for n from 1 while count < N do
  s:= numtheory:-sigma(n);
  if s > best then
    best:= s;
    count:= count+1;
    A[count]:= n;
  fi
od:
seq(A[i],i=1..N);# Robert Israel, Jan 20 2016

a={}; k=0; Do[s=DivisorSigma[1,n]; If[s>k, AppendTo[a,n]; k=s], {n,3000}]; a (* Vladimir Joseph Stephan Orlovsky, Jul 25 2008 *)
DeleteDuplicates[Table[{n,DivisorSigma[1,n]},{n,100}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]] (* Harvey P. Dale, May 14 2022 *)

for(n=1,1000,if(sum(i=1,n-1,sign(sigma(n)-sigma(i))) == n-1,print1(n,",")))

Better description from N. J. A. Sloane, Apr 15 1997

More terms from Jud McCranie, Jul 04 2000

A004490 Colossally abundant numbers: m for which there is a positive exponent epsilon such that sigma(m)/m^{1 + epsilon} >= sigma(k)/k^{1 + epsilon} for all k > 1, so that m attains the maximum value of sigma(m)/m^{1 + epsilon}.

Original entry on oeis.org

2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 160626866400, 321253732800, 9316358251200, 288807105787200, 2021649740510400, 6064949221531200, 224403121196654400

Offset: 1

N. J. A. Sloane, Jan 22 2001

nonn

S. Ramanujan, Highly composite numbers, Proc. London Math. Soc., 14 (1915), 347-407. Reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, pp. 78-129. See esp. pp. 87, 115.

Amiram Eldar, Table of n, a(n) for n = 1..382 (terms 1..150 from T. D. Noe)
Hirotaka Akatsuka, Maximal order for divisor functions and zeros of the Riemann zeta-function, arXiv:2411.19259 [math.NT], 2024. See p. 4.
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata
Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.
Kevin Broughan, A Variety of Abundant Numbers, in Equivalents of the Riemann Hypothesis. Cambridge University Press, 2017, pp. 144-164.
Geoffrey 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.
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv preprint arXiv:1112.6010 [math.NT], 2011. - From _N. J. A. Sloane_, Apr 14 2012
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.
S. Nazardonyavi and S. Yakubovich, Extremely Abundant Numbers and the Riemann Hypothesis, Journal of Integer Sequences, 17 (2014), Article 14.2.8.
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.
T. Schwabhäuser, Preventing Exceptions to Robin's Inequality, arXiv preprint arXiv:1308.3678 [math.NT], 2013.
M. Waldschmidt, From highly composite numbers to transcendental number theory, 2013.
Eric Weisstein's World of Mathematics, Colossally Abundant Number.

A subsequence of A004394 (superabundant numbers).
Cf. A000203, A002201, A073751.
Cf. A002093 (highly abundant numbers), A002182, A005101 (abundant numbers), A006038, A189228 (superabundant numbers that are not colossally abundant).

a(n) = Product_{k=1..n} A073751(k). - Jeppe Stig Nielsen, Nov 28 2021

A023199 a(n) is the least k with sigma(k) >= n*k.

Original entry on oeis.org

1, 6, 120, 27720, 122522400, 130429015516800, 1970992304700453905270400, 1897544233056092162003806758651798777216000, 4368924363354820808981210203132513655327781713900627249499856876120704000

Offset: 1

David W. Wilson

Following a suggestion from Ed Pegg Jr, the sequence can be written in a more readable form as: 1!, 3!, 5!, 11# * 3! * 2, 17# * 5! * 2, 29# * 7! * 4, 53# * 7! * 12, 89# * 11! * 2, 157# * 17# * 8! * 6, 271# * 23# * 10!, 487# * 29# * 10!, 857# * 37# * 11! * 42, 1487# * 53# * 15! * 2, ..., where p# = primorial(p) = A034386.
From T. D. Noe, Jul 06 2005: (Start)
Let c(p) be the smallest colossally-abundant number having the prime factor p. See A073751 for info about computing these numbers.
Then the terms of this sequence can be expressed as
  a(2) = c(3)
  a(3) = c(5) * 2
  a(4) = c(11) / 2
  a(5) = c(17) / 3
  a(6) = c(29) * 14
  a(7) = c(53)
  a(8) = c(89) * 4
  a(9) = c(157) * 34
  a(10) = c(271) * 23
  a(11) = c(487) / 2
  a(12) = c(857) / 2
  a(13) = c(1487) * 212
  a(14) = c(2621) * 710
  a(15) = c(4567) * 2/21
  a(16) = c(8011) / 2
  a(17) = c(13999) * 1630. (End)
Initially, each term is divisible by the previous one. Is there a reason this should always be true? - Santi Spadaro, Aug 13 2002
The conjecture a(n)|a(n+1) holds out to n=10. - Devin Kilminster (devin(AT)maths.uwa.edu.au), Mar 10 2003
The conjecture a(n)|a(n+1) fails for n=15. - T. D. Noe, Jul 08 2005
We have a(n) = min{A007539(n), A134716(n)}, and clearly A007539(n) != A134716(n) for every n. For what values of n is the former less than the latter? - Jeppe Stig Nielsen, Jun 16 2015

Amiram Eldar, Table of n, a(n) for n = 1..13
Walter Nissen, Abundancy : Some Resources, 2008-2010.
T. D. Noe, An algorithm for finding the least k with sigma(k) >= nk, 2005-2009.

A subsequence of A004394.
The dominating primes are in A108402.
Cf. A000203 (sigma), A007539, A034386, A073751, A134716.

a(n) = my(k=1); while (sigma(k)/k < n, k++); k; \\ Michel Marcus, Oct 07 2019

More terms from Walter Nissen, Apr 15 1997
Further terms from Devin Kilminster (devin(AT)maths.uwa.edu.au), Mar 10 2003
The term a(10) = 271#23#10! was apparently found independently by Bodo Zinser and Don Reble, circa Jul 05 2005
The next term, a(11) = 487#29#10!, was corrected by Don Reble, Jul 06 2005
a(12) = 857#37#11!42 from Don Reble, Jul 06 2005
a(13) = 1487#53#15!2 found by T. D. Noe and confirmed by Don Reble, Jul 07 2005
a(14)-a(17) found by T. D. Noe and rechecked by him Oct 11 2005
a(15) corrected. The conjecture still fails at n=15. - T. D. Noe, Oct 13 2009

A007539 a(n) = first n-fold perfect (or n-multiperfect) number.

Original entry on oeis.org

1, 6, 120, 30240, 14182439040, 154345556085770649600, 141310897947438348259849402738485523264343544818565120000

Offset: 1

N. J. A. Sloane

On the Riemann Hypothesis, a(n) > exp(exp(n / e^gamma)) for n > 3. Unconditionally, a(n) > exp(exp(0.9976n / e^gamma)) for n > 3, where the constant is related to A004394(1000000). - Charles R Greathouse IV, Sep 06 2012
Each of the terms 1, 6, 120, 30240 divides all larger terms <= a(8). See A227765, A227766, ..., A227769. - Jonathan Sondow, Jul 30 2013
Is a(n) < a(n+1)? - Jeppe Stig Nielsen, Jun 16 2015
Equivalently, a(n) is the smallest number k such that sigma(k)/k = n. - Derek Orr, Jun 19 2015
The number of divisors of these terms are: 1, 4, 16, 96, 1920, 110592, 1751777280, 63121588161085440. - Michel Marcus, Jun 20 2015
Given n, let S_n be the sequence of integers k that satisfy numerator(sigma(k)/k) = n. Then a(n) is a member of S_n. In fact a(n) = S_n(i), where the successive values of i are 1, 1, 2, 2, 4, 2, (23, 6, 31, 12, ...), where the terms in parentheses need to be confirmed. - Michel Marcus, Nov 22 2015
The first four terms are the only multiperfect numbers in A025487 among the 1600 initial terms of A007691. Can it be proved that these are the only ones among the whole A007691? See also A349747. - Antti Karttunen, Dec 04 2021

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 22.
A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 138.
R. K. Guy, Unsolved Problems in Number Theory, B2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..8
Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Jamie Bishop, Abigail Bozarth, Rebekah Kuss, and Benjamin Peet, The Abundancy Index and Feebly Amicable Numbers, arXiv:2104.11366 [math.NT], 2021.
C. K. Caldwell, The Prime Glossary, multiply perfect
F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Achim Flammenkamp, The Multiply Perfect Numbers Page
Fred Helenius, Link to Glossary and Lists
G. P. Michon, Multiperfect and hemiperfect numbers
Walter Nissen, Abundancy: Some Resources

Cf. A000396, A005820, A007691, A025487, A027687, A046060, A046061, A072002, A227765, A227766, A227767, A227768, A227769, A349747.

Table[k = 1; While[DivisorSigma[1, k]/k != n, k++]; k, {n, 4}] (* Michael De Vlieger, Jun 20 2015 *)

a(n)=k=1;while((sigma(k)/k)!=n,k++);k
vector(4,n,a(n)) \\ Derek Orr, Jun 19 2015

More terms sent by Robert G. Wilson v, Nov 30 2000

A067698 Positive integers such that sigma(n) >= exp(gamma) * n * log(log(n)).

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, 5040

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

Previous name was: Numbers with relatively many and large divisors.
n is in the sequence iff sigma(n) >= exp(gamma) * n * log(log(n)), where gamma = Euler-Mascheroni constant and sigma(n) = sum of divisors of n.
Robin has shown that 5040 is the last element in the sequence iff the Riemann hypothesis is true. Moreover the sequence is infinite if the Riemann hypothesis is false. Gronwall's theorem says that
  lim sup_{n -> infinity} sigma(n)/(n*log(log(n))) = exp(gamma).
a(28) > 10^(10^13.11485), if the Riemann hypothesis is false (Morrill and Platt, 2021). Briggs (2006) found the lower bound 10^(10^10). - Amiram Eldar, Jul 23 2025

			9 is in the sequence since sigma(9) = 13 > 12.6184... = exp(gamma) * 9 * log(log(9)).

Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 15, No. 2 (2006), p. 251-256.
Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33.
Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, On SA, CA, and GA numbers, The Ramanujan Journal, Vol. 29 (2012), pp. 359-384; arXiv preprint, arXiv:1112.6010 [math.NT], 2011-2012.
Jeffrey C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, The American Mathematical Monthly, Vol. 109, No. 6 (2002), pp. 534-543; alternative copy; arXiv preprint, arXiv:math/0008177 [math.NT], 2000-2001.
Thomas Morrill and David John Platt, Robin's inequality for 20-free integers, Integers, Vol. 21 (2021), #A28.
Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
Eric Weisstein's World of Mathematics, Gronwall's Theorem.
Eric Weisstein's World of Mathematics, Robin's Theorem.

Cf. A057641 (based on Lagarias's extension of Robin's result).

Cf. A000203, A004394, A073004, A091901, A189686, A196229.

with (numtheory): expgam := exp(evalf(gamma)): for i from 2 to 6000 do: a := sigma (i): b := expgam*i*evalf(ln(ln(i))): if a >= b then print (i, a, b): fi: od:

fQ[n_] := DivisorSigma[1, n] > n*Exp@ EulerGamma*Log@ Log@n; lst = {}; Do[ If[ fQ[n], AppendTo[lst, n]], {n,2,10^4}]; lst (* Robert G. Wilson v, May 16 2003 *)
Select[Range[2,5050], Exp[EulerGamma] # Log[Log[#]]-DivisorSigma[1,#]<0 &] (* Ant King, Feb 28 2013 *)

is(n)=sigma(n) >= exp(Euler) * n * log(log(n)) \\ Charles R Greathouse IV, Feb 08 2017

from sympy import divisor_sigma, EulerGamma, E, log
print([n for n in range(2, 5041) if divisor_sigma(n) >= (E**EulerGamma * n * log(log(n)))]) # Karl-Heinz Hofmann, Apr 22 2022

Edited by N. J. A. Sloane at the suggestion of Max Alekseyev, Jul 17 2007

New name from Jud McCranie, Aug 14 2017

A071927 Barely abundant numbers: abundant n such that sigma(n)/n < sigma(m)/m for all abundant numbers m

Original entry on oeis.org

12, 18, 20, 70, 88, 104, 464, 650, 1888, 1952, 4030, 5830, 8925, 17816, 26742, 26778, 26886, 26898, 26958, 27042, 27078, 27102, 27114, 27138, 27282, 27294, 27366, 27402, 27498, 27546, 27582, 27618, 27726, 27822, 27834, 27858, 27894, 27906

Offset: 1

Joe McCauley (mccauley(AT)davesworld.net), Jun 14 2002

nonn

The 103 prime numbers in the range 4457 to 5351, multiplied by 6, produce 103 terms of the series and likewise for the 33774 primes in the range 924493 through 1396393. There are likely to be similar long runs of a range of prime numbers multiplied by 6 further in the sequence. One could eliminate these by adding the requirement that n be primitive abundant, whose only additional effect would be to eliminate the first two terms of the sequence.
The inverse of this sequence, barely deficient numbers, includes all powers of 2 since their proper divisors always add up to one less than themselves. No other number through 2^24 has this attribute.
The sequence begins 12, 18, 20, 70, 88, 104, 464, 650, 1888, 1952, 4030, 5830, 8925, 17816, 26742, [101 terms omitted], 32106, 32128, 77744, 91388, 128768, 130304, 442365, 521728, 522752, 1848964, 5546958, [33772 terms omitted], 8378358, 8378368, 8382464, ...

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A004394.

r = 3; Do[ s = DivisorSigma[1, n]/n; If[ s > 2 && s < r, Print[n]; r = s], {n, 1, 32200}] (* Robert G. Wilson v, Jun 18 2002 *)

lista(nn) = {abk = 3; for (n = 1, nn, ab = sigma(n)/n; if ((ab > 2) && (ab < abk), print1(n, ", "); abk = ab););} \\ Michel Marcus, Jun 23 2015

More terms from Robert G. Wilson v, Jun 18 2002

A065218 Consider the subsets of proper divisors of a number that sum to the number. These are numbers that set a record number of such subsets.

Original entry on oeis.org

1, 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, 831600, 1081080, 1441440

Offset: 1

Jud McCranie, Oct 21 2001

nonn

Indices of records in A065205 and A033630. The corresponding records (number of subsets) are in A065219.

This sequence is not a subset of A002182: 831600 belongs to this sequence but not A002182.

			Proper divisors of 12 are {1, 2, 3, 4, 6}. Two subsets of this sum to 12: {2, 4, 6} and {1, 2, 3, 6} - more than any smaller number, so 12 is in the sequence.

Michael De Vlieger, Correlation of A065218 and A065219.

Cf. A065219, A064771, A063205, A005835, A002182, A004394.

With[{s = Table[-1 + SeriesCoefficient[Series[Times @@ ((1 + z^#) & /@ Divisors[n]), {z, 0, n}], n], {n, 2520}]}, FirstPosition[s, #][[1]] & /@ Union@ FoldList[Max, s]] (* Michael De Vlieger, Oct 10 2017 *)

More terms from Franklin T. Adams-Watters, Nov 27 2006
Edited and extended by Max Alekseyev, May 29 2009
Offset changed by Andrey Zabolotskiy, Oct 10 2017

A095848 Deeply composite numbers: numbers n where sigma_k(n) increases to a record for all sufficiently low (i.e., negative) values of k.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 48, 60, 120, 240, 360, 420, 840, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 360360, 720720, 1441440, 2162160, 3603600, 7207200, 10810800, 12252240, 24504480, 36756720, 61261200, 122522400

Offset: 1

Matthew Vandermast, Jun 09 2004

nonn

Sigma_k(n) > sigma_k(m) for all m < n (where the function sigma_k(n) is the sum of the k-th powers of all divisors of n) for all or almost all negative values of k.
This sequence is infinite, because it includes every term in A051451. This follows from the formula for a(n), and the fact that A051451 consists of the distinct terms of A003418. - Hal M. Switkay, Mar 22 2021
From Hal M. Switkay, Aug 27 2023: (Start)
There is a formula defining members of this sequence for all n.
Let the extended natural numbers N+ = {1, 2, 3, ..., oo}, with the ordering 1 < 2 < 3 < ... < oo.
For every natural number k, let Div+(k) be an infinitely long vector of extended natural numbers, starting with the divisors of k in increasing order, followed by infinitely many coordinates equal to oo. For example:
  Div+(6) = (1, 2, 3, 6, oo, oo, oo, ...)
  Div+(7) = (1, 7, oo, oo, oo, ...)
Then for all natural numbers n, a(n) = k if and only if k is the smallest natural number such that Div+(k) lexicographically precedes Div+(a(i)), for 1 <= i < n.
(End)

			The list of the divisors of a(6)=24, {1,2,3,4,6,8,12,24}, lexicographically precedes the list for the previous term in the sequence (in this case, {1,2,3,4,6,12}, the list for a(5)=12). Therefore 24 belongs in the sequence.
36 does not satisfy this requirement, as {1,2,3,4,6,9,...} comes after {1,2,3,4,6,8,...} in lexicographic order. Since 8^k/9^k increases without limit as k decreases, sigma(36)_k < sigma(24)_k for almost all negative values of k; therefore 36 does not belong in the sequence.

T. D. Noe, Table of n, a(n) for n = 1..448 (terms < 10^100)
Wikipedia, Table of divisors.

Cf. A004394, A095849.

Cf. A003418, A051451.

For n >= 4, a(n) is the smallest integer > a(n-1) such that the list of its divisors precedes the list of a(n-1)'s divisors in lexicographic order.

A306802 Position of highly composite numbers in the sequence of products of primorials.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 12, 13, 17, 20, 24, 27, 34, 36, 43, 47, 55, 67, 77, 84, 95, 102, 107, 112, 129, 133, 138, 154, 166, 183, 198, 211, 220, 245, 252, 261, 264, 294, 314, 348, 369, 390, 406, 446, 457, 476, 500, 533, 555, 582, 634, 652, 676, 726, 756, 822

Offset: 1

Michael De Vlieger, Mar 12 2019

nonn

Indices of A002182 in A025487. All terms in A002182 are products of terms in A002110; A025487 lists products of terms in A002110.

The first 28 terms of this sequence and those of A293635 are identical since the smallest 28 terms of A002182 and A004394 are the same.

			The number 120 is 10th in the sequence of highly composite numbers, since it sets a record for the divisor counting function. The index of this number in A025487 is 17.

Michael De Vlieger, Table of n, a(n) for n = 1..511

Cf. A002182, A025487, A098719, A293635.

Block[{P = Product[Prime@ i, {i, 8}], s, t, u}, s = Array[DivisorSigma[0, #] &, P]; t = Array[If[# == 1, {0}, Sort[FactorInteger[#][[All, -1]], Greater]] &, P]; u = Values[PositionIndex@ t][[All, 1]]; Map[FirstPosition[u, #][[1]] &, FirstPosition[s, #][[1]] & /@ Union@ FoldList[Max, s]] ]

A087315 a(n) = Product_{k=1..n} prime(k)^prime(n-k+1).

Original entry on oeis.org

1, 4, 72, 21600, 190512000, 580909190400000, 428616352408083840000000, 859278392084450410309036800000000000, 2097197194438629126172451944256706311040000000000000

Offset: 0

Amarnath Murthy, Sep 03 2003

nonn

			a(3) = 2^5*3^3*5^2 = 21600.

G. C. Greubel, Table of n, a(n) for n = 0..23

Cf. A006939, A002110, A025487, A004394, A002093, A002182, A055932, A025487, A089247, A071364, A097320, A046523, A056166, A114129, A053810.

[1] cat [(&*[NthPrime(k)^(NthPrime(n-k+1)): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

seq(product(ithprime(k)^ithprime(n-k+1), k=1..n), n=0..10);

Table[Product[Prime[k]^Prime[n - k + 1], {k, 1, n}], {n, 0, 10}] (* G. C. Greubel, Oct 14 2018 *)

for(n=0, 10, print1(prod(k=1,n, prime(k)^prime(n-k+1)), ", ")) \\ G. C. Greubel, Oct 14 2018

[prod(nth_prime(i)^nth_prime(k-i+1) for i in (1..k)) for k in (0..10)] # Giuseppe Coppoletta, Nov 03 2014

More terms from Jorge Coveiro, Dec 22 2004

Corrected by David Wasserman, May 02 2005

cp's OEIS Frontend

A002093 Highly abundant numbers: numbers k such that sigma(k) > sigma(m) for all m < k.

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A004490 Colossally abundant numbers: m for which there is a positive exponent epsilon such that sigma(m)/m^{1 + epsilon} >= sigma(k)/k^{1 + epsilon} for all k > 1, so that m attains the maximum value of sigma(m)/m^{1 + epsilon}.

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A023199 a(n) is the least k with sigma(k) >= n*k.

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A007539 a(n) = first n-fold perfect (or n-multiperfect) number.

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A067698 Positive integers such that sigma(n) >= exp(gamma) * n * log(log(n)).

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A071927 Barely abundant numbers: abundant n such that sigma(n)/n < sigma(m)/m for all abundant numbers m

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A065218 Consider the subsets of proper divisors of a number that sum to the number. These are numbers that set a record number of such subsets.

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