A023199 -id:A023199 - cp's OEIS frontend

A358413 Smallest 3-abundant number (sigma(x) > 3x) which is not divisible by any of the first n primes.

180, 1018976683725, 5164037398437051798923642083026622326955987448536772329145127064375

Offset: 0

Jianing Song, Nov 14 2022

Data copied from the Hi.gher. Space link where Mercurial, the Spectre calculated the terms. We have a(0) = 2^2*3^2*5, a(1) = 3^3*5^2*7^2*11*13*17*19*23*29, and a(2) = 5^4*7^3*11^2*13^2*17*...*157 ~ 5.16404*10^66. a(3) = 7^3*11^3*13^2*17^2*19^2*23^2*29^2*31*...*569 ~ 2.54562*10^239 and a(4) = 11^3*13^3*17^2*...*47^2*53*...*1597 ~ 3.99515*10^688 are too large to display.

			a(1) = A119240(3) = 1018976683725 is the smallest 3-abundant odd number.
a(2) = A358412(3) = 5164037398437051798923642083026622326955987448536772329145127064375 is the smallest 3-abundant number that is coprime to 2 and 3.

Jianing Song, Table of n, a(n) for n = 0..4
Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.

Cf. A068403 (3-abundant numbers).
Smallest k-abundant number which is not divisible by any of the first n primes: A047802 (k=2), this sequence (k=3), A358414 (k=4).
Least p-rough number k such that sigma(k)/k >= n: A023199 (p=2), A119240 (p=3), A358412 (p=5), A358418 (p=7), A358419 (p=11).

A134716 a(n) = least number m such that sigma(m)/m > n, where sigma(m) = sum of divisors of m.

Original entry on oeis.org

1, 2, 12, 180, 27720, 122522400, 130429015516800, 1970992304700453905270400, 1897544233056092162003806758651798777216000, 4368924363354820808981210203132513655327781713900627249499856876120704000

Offset: 0

Pierre CAMI, Jan 27 2008, corrected Feb 26 2008 with the help of Emeric Deutsch

More terms can be obtained by expanding the expressions in the example lines. - N. J. A. Sloane, Feb 26 2008

			Values written as products of primorials:
   n a(n)
   0 1
   1 2
   2 2*3#
   3 3#*5# (= 180)
   4 2*3#*11#
   5 8*5#*17#
   6 16*3#*7#*29#
   7 8*3#*3#*7#*53#
   8 32*3#*3#*5#*11#*89#
   9 32*3#*3#*7#*17#*157#
  10 16*3#*3#*5#*7#*23#*271#
  11 16*3#*3#*5#*7#*29#*487#
  12 16*3#*3#*5#*7#*13#*31#*857#

Amiram Eldar, Table of n, a(n) for n = 0..13

Cf. A000203 (sigma), A023199 (least k with sigma(k) >= nk).

A358414 Smallest 4-abundant number (sigma(x) > 4x) which is not divisible by any of the first n primes.

Original entry on oeis.org

27720, 1853070540093840001956842537745897243375

Offset: 0

Jianing Song, Nov 14 2022

Data copied from the Hi.gher. Space link where Mercurial, the Spectre calculated the terms. We have a(0) = 2^3*3^2*5*7*11 and a(1) = 3^5*5^3*7^2*11^2*13*...*89 ~ 1.85307*10^39. a(2) = 5^5*7^4*11^3*13^3*17^2*19^2*23^2*29^2*31^2*37^2*41*...*853 ~ 1.83947*10^370, a(3) = 7^5*11^3*13^3*17^3*19^3*23^2*...*97^2*101*...*4561 ~ 1.11116*10^1986, and a(4) = 11^4*13^4*17^3*19^3*23^3*29^3*31^3*37^2*...*181^2*191*...*18493 ~ 2.99931*10^8063 are too large to display.

			a(1) = A119240(4) = 1853070540093840001956842537745897243375 is the smallest 4-abundant odd number.
a(2) = A358412(4) ~ 1.83947*10^370 is the smallest 4-abundant number that is coprime to 2 and 3.

Jianing Song, Table of n, a(n) for n = 0..2
Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.

Cf. A068404 (4-abundant numbers).
Smallest k-abundant number which is not divisible by any of the first n primes: A047802 (k=2), A358413 (k=3), this sequence (k=4).
Least p-rough number k such that sigma(k)/k >= n: A023199 (p=2), A119240 (p=3), A358412 (p=5), A358418 (p=7), A358419 (p=11).

A358418 Least number k coprime to 2, 3, and 5 such that sigma(k)/k >= n.

Original entry on oeis.org

1, 20169691981106018776756331

Offset: 1

Jianing Song, Nov 14 2022

Data copied from the Hi.gher. Space link where Mercurial, the Spectre calculated the terms. We have a(2) = 7^2*11^2*13*...*67 ~ 2.01697*10^25. a(3) = 7^3*11^3*13^2*17^2*19^2*23^2*29^2*31*...*569 ~ 2.54562*10^239 and a(4) = 7^5*11^3*13^3*17^3*19^3*23^2*...*97^2*101*...*4561 ~ 1.11116*10^1986 are too large to display.

			a(2) = A047802(3) = 20169691981106018776756331 is the smallest abundant number coprime to 2, 3, and 5.
Even if there is a number k coprime to 2, 3, and 5 with sigma(k)/k = 3, we have that k is a square since sigma(k) is odd. If omega(k) = m, then 3 = sigma(k)/k < Product_{i=4..m+3} (prime(i)/(prime(i)-1)) => m >= 97, and we have k >= prime(4)^2*...*prime(100)^2 ~ 2.46692*10^436 > A358413(3) ~ 2.54562*10^239. So a(3) = A358413(3).
Even if there is a number k coprime to 2, 3, and 5 with sigma(k)/k = 4, there can be at most 2 odd exponents in the prime factorization of k (see Theorem 2.1 of the Broughan and Zhou link). If omega(k) = m, then 4 = sigma(k)/k < Product_{i=4..m+3} (prime(i)/(prime(i)-1)) => m >= 606, and we have k >= prime(4)^2*...*prime(607)^2*prime(608)*prime(609) ~ 6.54355*10^3814 > A358414(3) ~ 1.11116*10^1986. So a(4) = A358414(3).

Jianing Song, Table of n, a(n) for n = 1..3
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, author’s version, Research Commons.
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, Journal of Number Theory 128 (2008) 1566-1575.
Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.

Smallest k-abundant number which is not divisible by any of the first n primes: A047802 (k=2), A358413 (k=3), A358414 (k=4).

Least p-rough number k such that sigma(k)/k >= n: A023199 (p=2), A119240 (p=3), A358412 (p=5), this sequence (p=7), A358419 (p=11).

A358419 Least number k coprime to 2, 3, 5, and 7 such that sigma(k)/k >= n.

Original entry on oeis.org

1, 49061132957714428902152118459264865645885092682687973

Offset: 1

Jianing Song, Nov 14 2022

Data copied from the Hi.gher. Space link where Mercurial, the Spectre calculated the terms. We have a(2) = 11^2*13^2*17*...137 ~ 4.90611*10^52. a(3) = 11^3*13^3*17^2*...*47^2*53*...*1597 ~ 3.99515*10^688 and a(4) = 11^4*13^4*17^3*19^3*23^3*29^3*31^3*37^2*...*181^2*191*...*18493 ~ 2.99931*10^8063 are too large to display.

			a(2) = A047802(4) = 49061132957714428902152118459264865645885092682687973 is the smallest abundant number coprime to 2, 3, 5, and 7.
Even if there is a number k coprime to 2, 3, 5, and 7 with sigma(k)/k = 3, we have that k is a square since sigma(k) is odd. If omega(k) = m, then 3 = sigma(k)/k < Product_{i=5..m+4} (prime(i)/(prime(i)-1)) => m >= 240, and we have k >= prime(5)^2*...*prime(244)^2 ~ 1.60834*10^1297 > A358413(4) ~ 3.99515*10^688. So a(3) = A358413(4).
Even if there is a number k coprime to 2, 3, 5, and 7 with sigma(k)/k = 4, there can be at most 2 odd exponents in the prime factorization of k (see Theorem 2.1 of the Broughan and Zhou link). If omega(k) = m, then 4 = sigma(k)/k < Product_{i=5..m+4} (prime(i)/(prime(i)-1)) => m >= 2096, and we have k >= prime(5)^2*...*prime(2098)^2*prime(2099)*prime(2100) ~ 6.21439*10^15801 > A358414(4) ~ 2.99931*10^8063. So a(4) = A358414(4).

Jianing Song, Table of n, a(n) for n = 1..3
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, author’s version, Research Commons.
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, Journal of Number Theory 128 (2008) 1566-1575.
Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.

Smallest k-abundant number which is not divisible by any of the first n primes: A047802 (k=2), A358413 (k=3), A358414 (k=4).

Least p-rough number k such that sigma(k)/k >= n: A023199 (p=2), A119240 (p=3), A358412 (p=5), A358418 (p=7), this sequence (p=11).

A204828 Numbers n with abundancy 3 <= sigma(n)/n < 4.

Original entry on oeis.org

120, 180, 240, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1080, 1200, 1260, 1320, 1344, 1440, 1512, 1560, 1584, 1620, 1680, 1800, 1848, 1872, 1890, 1920, 1980, 2016, 2040, 2100, 2160, 2184, 2280, 2340, 2352, 2376, 2400, 2520, 2640

Offset: 1

Jaroslav Krizek, Jan 22 2012

nonn

A subsequence of A023197 (numbers n such that sigma(n) >= 3n) which is in turn a subsequence of the abundant numbers A005101, i.e., numbers n with sigma(n)/n > 2.
Differs from A023197 from a(565) on: The first term of A023197 which is not in this sequence is A023197(565) = 27720 = A023198(1) = A023199(4), the least number with abundancy >= 4.
Numbers with abundancy sigma(n)/n < 2 are called deficient and listed in A005100. Numbers with sigma(n)/n in the interval [2,3) are listed in A204829. Numbers with sigma(n)/n in the interval [4,5) are listed in A230608. - M. F. Hasler, Dec 05 2013

			Number 180 is in the sequence because sigma(180)/180 = 546/180 = 3.0333...

Jaroslav Krizek, Table of n, a(n) for n = 1..3000
Eric Weisstein's World of Mathematics, Abundancy
Eric Weisstein's World of Mathematics, Abundant Number

Cf. A204829 (abundant numbers with abundancy 2 <= a < 3).

A072997 Smallest prime p such that Product_{primes q <= p} q+1 >= n*Product_{primes q <= p} q.

Original entry on oeis.org

2, 3, 13, 31, 89, 239, 617, 1571, 4007, 10141, 25673, 64853, 163367, 412007, 1037759, 2614369, 6584857, 16585291, 41764859, 105178831, 264877933, 667038311, 1679809291, 4230219377, 10652786759, 26826453991, 67555877849

Offset: 1

Benoit Cloitre, Aug 14 2002

For k > 2, the primorial number A034386(A072997(k)) = A002110(A072986(k)) is the least unitary k-abundant number, i.e., the least number m such that usigma(m) >= k*m, where usigma(m) = A034448(m) is the sum of unitary divisors of m. The sequence of these primorials is the unitary version of A023199. - Amiram Eldar, Aug 24 2018

Cf. A000040, A002110, A023199, A034386, A034448, A072986, A072997, A362151.

n=x=y=1; Do[x *= (Prime[s] + 1); y *= Prime[s]; If[x >= n*y, Print[Prime[s]]; n++ ], {s, 1, 10^6}] (* Ryan Propper, Jul 22 2005 *)

a(n)=if(n<0,0,s=1; while(prod(i=1,s, prime(i)+1)

It seems that lim_{n -> oo} a(n+1)/a(n) exists and is > 2.
a(n) = A000040(A072986(n)). - Amiram Eldar, Aug 24 2018
Limit_{n -> oo} a(n+1)/a(n) = exp(zeta(2)/exp(gamma)) = 2.518... (A362151). - Amiram Eldar, Aug 26 2025

7 more terms from Ryan Propper, Jul 22 2005
a(18)-a(22) added by Amiram Eldar, Aug 24 2018 from the data at A072986
a(23)-a(27) from Keith F. Lynch, Jan 13 2024

A073087 Least k such that sigma(k^k)>=n*k^k.

Original entry on oeis.org

1, 6, 30, 210, 30030, 223092870, 13082761331670030, 3217644767340672907899084554130, 1492182350939279320058875736615841068547583863326864530410

Offset: 1

Benoit Cloitre, Aug 18 2002

nonn

Does a(n) = the product of primes less than or equal to prime(n+1) = A002110(n+1)? Answer from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Sep 14 2005: No, this is not true.
Note that sigma(k^k) = prod (p^(k r + 1) - 1)/(p - 1). - Mitch Harris, Sep 14 2005
I have proved to my own satisfaction that for n >= 4, A073087(n) = p#, where p is the smallest prime satisfying p#/phi(p#) >= n. See link. - David W. Wilson, Sep 14 2005

David W. Wilson, Comments on this sequence

Cf. A023199.

a(n)=if(n<0,0,s=1; while(sigma(s^s)

a(n) = A091440(n)# = A002110(A112873(n)) for n >= 4.

More terms from David W. Wilson, Sep 15 2005

A230608 Numbers with abundancy 4 <= sigma(n)/n < 5.

Original entry on oeis.org

27720, 30240, 32760, 50400, 55440, 60480, 65520, 75600, 83160, 85680, 90720, 95760, 98280, 100800, 105840, 110880, 115920, 120120, 120960, 128520, 131040, 138600, 141120, 143640, 151200, 163800, 166320, 171360, 176400, 180180, 181440, 184800, 191520, 194040

Offset: 1

Jaroslav Krizek, Nov 29 2013

nonn

A subsequence of A023198 (numbers with abundancy >= 4). It differs from A023198 from a(31093) on: The term A023198(31093) = 122522400 = A023199(5) = A215264(1) is not in this sequence. It excludes all terms of A215264, but also the 5-perfect numbers A046060, which are neither in this sequence nor in A215264. [Corrected by M. F. Hasler, Dec 05 2013]
A108775(a(n)) = 4.
There are 31092 terms less than 122522399. - T. D. Noe, Dec 04 2013

			27720 is in sequence because sigma(27720) / 27720 = 112320 / 27720 = 4.0519....

Jaroslav Krizek, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Abundancy
Eric Weisstein's World of Mathematics, Abundant Number

Cf. A000203, A023198, A023199, A108775.
Cf. A005100 (deficient numbers with abundancy 1 <= a < 2),
Cf. A204829 (numbers with abundancy 2 <= a < 3),
Cf. A204828 (abundant numbers with abundancy 3 <= a < 4).
Cf. A215264 (abundant numbers with abundancy > 5).

Select[Range[200000], 4 <= DivisorSigma[1, #]/# < 5 &] (* T. D. Noe, Dec 04 2013 *)

Corrected and edited by M. F. Hasler, Dec 05 2013

A383920 Smallest m such that sigma(m) >= n*m/2.

Original entry on oeis.org

1, 2, 6, 24, 120, 1680, 27720, 720720, 122522400, 41902660800, 130429015516800, 3066842656354276800, 1970992304700453905270400, 168721307030313765796546413936000, 1897544233056092162003806758651798777216000, 8201519488959040182625924708238885435575055666675808000

Offset: 2

Michel Marcus, May 15 2025

nonn

For n>= 11, terms were computed with 2nd PARI program using the T. D. Noe algorithm.

			From _Michael De Vlieger_, May 22 2025: (Start)
Table of a(n), n = 2..10, showing prime power decomposition:
                                         Prime power
                                         factor exponent
                                             111
 n     m = a(n)    sigma(m)      n*m/2   2357137
------------------------------------------------
 2           1           1           1   0
 3           2           3           3   1
 4           6          12          12   11
 5          24          60          60   31
 6         120         360         360   311
 7        1680        5952        5880   4111
 8       27720      112320      110880   32111
 9      720720     3249792     3243240   421111
10   122522400   614210688   612612000   5221111 (End)

Michael De Vlieger, Table of n, a(n) for n = 2..27
T. D. Noe, An algorithm for finding the least k with sigma(k) >= nk

Cf. A000203, A023199, A317681.

Cf. A004490. Subsequence of A004394.

(* First, load function f from A025487, then *)
nn = 12; s = Union@ Flatten@ f[nn + 4]; m = Length[s];
Monitor[Reap[Do[k = 1; While[And[DivisorSigma[1, #] < n*#/2 &[ s[[k]] ], k < m], k++]; If[k == m, Break[], Sow[s[[k]] ] ], {n, 2, nn}] ][[-1, 1]], n] (* Michael De Vlieger, May 21 2025 *)

a(n) = my(k=1); while (sigma(k) < k*n/2, k++); k;

ab(x) = sigma(x)/x;
findpos(vca, val) = for (i=1, #vca -1, if ((sigma(vca[i])/vca[i] < val) && (sigma(vca[i+1])/vca[i+1] > val), return(i)););
a(n) = if (n==1, return(0)); if (n==2, return(1)); my(val = n/2, vca = readvec("c:/gp/bfiles/b004490.txt"), vsa = readvec("c:/gp/bfiles/b004394.txt"), wc = select(x->(ab(x) == val), vca)); if (#wc, return(wc[1])); my(ipos = findpos(vca, val), c1 = vca[ipos], c2 = vca[ipos+1], ws = select(x->((x>c1) && (x<=c2)), vsa)); for (i=1, #ws, if (ab(ws[i]) >= val, return(ws[i])););

cp's OEIS Frontend

A358413 Smallest 3-abundant number (sigma(x) > 3x) which is not divisible by any of the first n primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A134716 a(n) = least number m such that sigma(m)/m > n, where sigma(m) = sum of divisors of m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A358414 Smallest 4-abundant number (sigma(x) > 4x) which is not divisible by any of the first n primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A358418 Least number k coprime to 2, 3, and 5 such that sigma(k)/k >= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A358419 Least number k coprime to 2, 3, 5, and 7 such that sigma(k)/k >= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A204828 Numbers n with abundancy 3 <= sigma(n)/n < 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A072997 Smallest prime p such that Product_{primes q <= p} q+1 >= n*Product_{primes q <= p} q.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A073087 Least k such that sigma(k^k)>=n*k^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A230608 Numbers with abundancy 4 <= sigma(n)/n < 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs