A047802 -id:A047802 - cp's OEIS frontend

A337202 a(n) = 2*A246277(A047802(n)).

12, 120, 19399380, 195534950863140268380, 1678409980907129617069656971232406858049983380, 1193774258350145889842491509271710921616406416330926349273223856572483463433620

Offset: 0

Antti Karttunen, Aug 19 2020

nonn

Question: Are there any duplicate terms, not necessarily consecutive? That is, are there two or more terms of A047802 that occur in the same column of array A246278?

Antti Karttunen, Table of n, a(n) for n = 0..13 (Prepared from the b-file of A047802 provided by Jeppe Stig Nielsen)
Index entries for sequences computed from indices in prime factorization

Cf. A047802, A246277, A246278, A336389.

A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
A337202(n) = 2*A246277(A047802(n));

For all n >= 0, a(n) >= A336389(1+n).

A005231 Odd abundant numbers (odd numbers m whose sum of divisors exceeds 2m).

Original entry on oeis.org

945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8925, 9135, 9555, 9765, 10395, 11025, 11655, 12285, 12705, 12915, 13545, 14175, 14805, 15015, 15435, 16065, 16695, 17325, 17955

Offset: 1

N. J. A. Sloane

nonn

While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number.
Schiffman notes that 945+630k is in this sequence for all k < 52. Most of the first initial terms are of the form. Among the 1996 terms below 10^6, 1164 terms are of that form, and only 26 terms are not divisible by 5, cf. A064001. - M. F. Hasler, Jul 16 2016
From M. F. Hasler, Jul 28 2016: (Start)
Any multiple of an abundant number is again abundant, see A006038 for primitive terms, i.e., those which are not a multiple of an earlier term.
An odd abundant number must have at least 3 distinct prime factors, and 5 prime factors when counted with multiplicity (A001222), whence a(1) = 3^3*5*7. To see this, write the relative abundancy A(N) = sigma(N)/N = sigma[-1](N) as A(Product p_i^e_i) = Product (p_i-1/p_i^e_i)/(p_i-1) < Product p_i/(p_i-1).
See A115414 for terms not divisible by 3, A064001 for terms not divisible by 5, A112640 for terms coprime to 5*7, and A047802 for other generalizations.
As of today, we don't know of a difference between this set S of odd abundant numbers and the set S' of odd semiperfect numbers: Elements of S' \ S would be perfect (A000396), and elements of S \ S' would be weird (A006037), but no odd weird or perfect number is known. (End)
For any term m in this sequence, A064989(m) is also an abundant number (in A005101), and for any term x in A115414, A064989(x) is in this sequence. If there are no odd perfect numbers, then applying A064989 to these terms and sorting into ascending order gives A337386. - Antti Karttunen, Aug 28 2020
There exist infinitely many terms m such that 2*m+1 is also a term. An example of such a term is given by m = 985571808130707987847768908867571007187. - Max Alekseyev, Nov 16 2023

W. Dunham, Euler: The Master of Us All, The Mathematical Association of America Inc., Washington, D.C., 1999, p. 13.
R. K. Guy, Unsolved Problems in Number Theory, B2.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 128.

Metin Sariyar, Table of n, a(n) for n = 1..32000 (terms 1..1000 from T. D. Noe)
Jill Britton, Perfect Number Analyzer.
L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
Mitsuo Kobayashi, Paul Pollack and Carl Pomerance, On the distribution of sociable numbers, Journal of Number Theory, Vol. 129, No. 8 (2009), pp. 1990-2009. See Theorem 10 on p. 2007.
Victor Meally, Letter to N. J. A. Sloane, no date.
Walter Nissen, Abundancy : Some Resources.
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 10.
Jay L. Schiffman, Odd Abundant Numbers, Mathematical Spectrum, Volume 37, Number 2 (January 2005), pp 73-75.
Jay L. Schiffman and Christopher S. Simons, More Odd Abundant Sequences, Volume 38, Number 1 (September 2005), pp. 7-8.

Cf. A000203, A005835, A006038, A115414, A064001, A112640, A122036, A136446, A005101, A173490, A039725, A064989, A337386.

A005231 := proc(n) option remember ; local a ; if n = 1 then 945 ; else for a from procname(n-1)+2 by 2 do if numtheory[sigma](a) > 2*a then return a; end if; end do: end if; end proc: # R. J. Mathar, Mar 20 2011

fQ[n_] := DivisorSigma[1, n] > 2n; Select[1 + 2Range@ 9000, fQ] (* Robert G. Wilson v, Mar 20 2011 *)

je=[]; forstep(n=1,15000,2, if(sigma(n)>2*n, je=concat(je,n))); je

is_A005231(n)={bittest(n,0)&&sigma(n)>2*n} \\ M. F. Hasler, Jul 28 2016

list(lim)=my(v=List()); forfactored(n=945,lim\1, if(n[2][1,1]>2 && sigma(n,-1)>2, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Apr 21 2022

a(n) ~ k*n for some constant k (perhaps around 500). - Charles R Greathouse IV, Apr 21 2022

482.8 < k < 489.8 (based on density bounds by Kobayashi et al., 2009). - Amiram Eldar, Jul 17 2022

More terms from James Sellers

A336835 Number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2

Offset: 1

Antti Karttunen, Aug 07 2020

nonn

It holds that a(n) <= A336836(n) for all n, because sigma(n) <= A003961(n) for all n (see A286385 for a proof).
The first 3 occurs at n = 19399380, the first 4 at n = 195534950863140268380. See A336389.
If x and y are relatively prime (i.e., gcd(x,y) = 1), then a(x*y) >= max(a(x),a(y)). Compare to a similar comment in A336915.

			For n = 120, sigma(120) = 360 >= 2*120, thus 120 is not deficient, and we get the next number by applying the prime shift, A003961(120) = 945, and sigma(945) = 1920 >= 945*2, so neither 945 is deficient, so we prime shift once again, and A003961(945) = 9625, which is deficient, as sigma(9625) = 14976 < 2*9625. Thus after two iteration steps we encounter a deficient number, and therefore a(120) = 2.

Cf. A003961, A005100, A005940, A023196, A046523, A047802, A071364, A108951, A286385, A294934, A336834, A337474, A337475.
Cf. A336389 (position of the first occurrence of a term >= n).
Differs from A294936 for the first time at n=120.
Cf. also A246271, A252459, A336836 and A336915 for similar iterations.

Array[-1 + Length@ NestWhileList[If[# == 1, 1, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]] &, #, DivisorSigma[1, #] >= 2 # &] &, 120] (* Michael De Vlieger, Aug 27 2020 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A336835(n) = { my(i=0); while(sigma(n) >= (n+n), i++; n = A003961(n)); (i); };

If A294934(n) = 1, a(n) = 0, otherwise a(n) = 1 + a(A003961(n)).
From Antti Karttunen, Aug 21-Sep 01 2020: (Start)
For all n >= 1,
a(A046523(n)) >= a(n).
a(A071364(n)) >= a(n).
a(A108951(n)) = A337474(n).
a(A025487(n)) = A337475(n).
(End)

A336389 The least positive integer k for which A336835(k) >= n, where A336835(k) is the number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=k.

Original entry on oeis.org

1, 6, 120, 19399380, 195534950863140268380, 538938984694949877040715541221415046162838700, 216487559804430601784907786655491617909711008142914104790481010259258659171900

Offset: 0

Antti Karttunen, Aug 07 2020

For n > 0, the least k such that for at least n-1 iterations of map x -> A003961(x), starting from x=k, x stays nondeficient. In other words, from each a(n) starts a chain of at least n nondeficient numbers (A023196) obtained by successive prime shifts, e.g, for a(3) we have: 19399380 -> 334639305 -> 5391411025, where -> stands for applying A003961, the prime shift towards larger primes.
After 1 all other terms here are even, because if an odd number k is nondeficient, then A064989(k) is nondeficient also, where A064989 is the prime shift towards smaller primes. Moreover, because A047802 is defined for every n >= 0, also this sequence is.
From Peter Munn, Aug 13 2020 (Start)
Upper bounds for a(4) and a(5) are:
a(4) <= 195534950863140268380 = A064989(A064989(A064989(20169691981106018776756331))) = A337202(3).
a(5) <= 538938984694949877040715541221415046162838700 = A064989^4((A047802(4)*17*19)/137).
(End)
From David A. Corneth, Aug 21 2020: (Start)
Subsequence of A025487.
Let prime(n)# be the n-th primorial number, A002110(n) = A034386(prime(n)). Then:
a(6) <= 191# * 7#;
a(7) <= 311# * 5#;
a(8) <= 457# * 5#.
(End)
That each term occurs in A025487 follows because (1), the abundancy index of prime(i)^e is larger than that of prime(i+1)^e, that is, sigma(prime(i)^e)/prime(i)^e > sigma(prime(i+1)^e)/prime(i+1)^e, and (2) because the abundancy index of p^(e+d) * q^e is larger than that of p^e * q^(e+d), where p and q are distinct primes, p < q, and e, d > 0. Thus, for any n, we can first find a "prime-factorization compressed version" of it, A071364(n), and then sort the exponents to the non-ascending order with A046523 (and actually, A046523(A071364(n)) = A046523(n), so we need to apply just A046523), to get a term x of A025487, that certainly have the abundancy index >= n [and this inequivalence stays same for their successive prime shifts as well, the abundancy index of A003961(x) being at least that of A003961(n), etc.], and as A046523(n) <= n for all n, it is guaranteed that the least k for which A336835(k) >= n are found from A025487, which is the range of A046523.

Cf. A003961, A005100, A023196, A025487, A047802, A108951, A246277, A329900, A336834, A336835, A337202, A337205, A337477, A337478.

From term a(2) = 120 onward a subsequence of A337386.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A336835(n) = { my(i=0); while(sigma(n) >= (n+n), i++; n = A003961(n)); (i); };
A336389(n) = for(i=1,oo,if(A336835(i)>=n,return(i)));

For n >= 0, A336835(a(n)) >= n.
For all n >= 1, a(n) <= A337202(n-1) [= 2*A246277(A047802(n-1))].
a(n) = A025487(A337477(n)).
a(n) = A108951(A337478(n)).

a(4) - a(6) from combined work of David A. Corneth and Peter Munn Aug 13-26 2020

A115414 Odd abundant numbers not divisible by 3.

Original entry on oeis.org

5391411025, 26957055125, 28816162375, 33426748355, 34393484125, 37739877175, 40342627325, 48150877775, 50866790975, 53356378075, 55959128225, 59305521275, 60711976325, 61164628525, 63395557225, 64899009175, 67275433225, 68972878975, 70088343325, 74922022175, 75665665075

Offset: 1

Sergio Pimentel, Mar 08 2006

nonn

An odd abundant number (A005231) not divisible by 3 must have at least 7 distinct prime factors (e.g., 5^4*7^2*11^2*13*17*19*23) and be >= 5*29#/3# = 5^2*7*11*13*17*19*23*29 = 5391411025 = A047802(2) = a(1). This is most easily seen by writing the relative abundancy A(N) = sigma(N)/2N = sigma[-1](N) as A(Product p_i^e_i) = (1/2)*Product (p_i-1/p_i^e_i)/(p_i-1) < (1/2)*Product p_i/(p_i-1). See A064001 for odd abundant numbers not divisible by 5. - M. F. Hasler, Jul 27 2016

This is not a subsequence of A248150. For example, 81324229811825 and 37182145^2 = 1382511906801025 are terms, with sigma(.) == 2 (mod 4) and sigma(.) == 3 (mod 4) respectively. - Amiram Eldar, Aug 24 2020

			a(1)=5391411025 because it is the smallest abundant number (sigma(n)/n =~ 2.003) that is not divisible by 2 or 3.

David A. Corneth, Table of n, a(n) for n = 1..42320 (terms < 10^14, terms 1..394 from Donovan Johnson, terms 395..4343 from Giovanni Resta)
Eric Weisstein's World of Mathematics, Abundant Number

Cf. A005101, A005231, A064001, A112640, A112644, A248150, A325311.

is(n)=gcd(n,6)==1 && sigma(n,-1)>2 \\ Charles R Greathouse IV, Jul 28 2016

Added missing term 55959128225 and a(14)-a(16) from Donovan Johnson, Dec 29 2008
a(17)-a(20) from Donovan Johnson, Dec 01 2011
More terms from M. F. Hasler, Jul 28 2016

A064001 Odd abundant numbers not divisible by 5.

Original entry on oeis.org

81081, 153153, 171171, 189189, 207207, 223839, 243243, 261261, 279279, 297297, 351351, 459459, 513513, 567567, 621621, 671517, 729729, 742203, 783783, 793611, 812889, 837837, 891891, 908523, 960687, 999999, 1024947, 1054053, 1072071

Offset: 1

Harvey P. Dale, Sep 17 2001

nonn

Or, odd abundant numbers that do not end in 5.
All terms below 2000000 are divisible by 21 (so by 3). Moreover, except for a few, most are divisible by 231. - Labos Elemer, Sep 15 2005 [The least term that is not divisible by 21 is a(908) = 28683369. - Amiram Eldar, Jan 27 2025]
An odd abundant number (see A005231) not divisible by 3 nor 5 must have at least 15 distinct prime factors (e.g., 61#/5#*7^2*11*13*17, where # is primorial) and be >= 67#/5#*77 = A047802(3) ~ 2.0*10^25. -- The smallest non-primitive abundant number (cf. A006038) in this sequence is 7*a(1) = 567567 = a(14). - M. F. Hasler, Jul 27 2016
There are 26 terms less than 10^6 and a surprising fact is that 18 of them are doublets (cf. A020338). - Omar E. Pol, Jan 17 2025
The numbers of terms that do not exceed 10^k, for k = 5, 6, ..., are 1, 26, 290, 3071, 31600, 320948, 3174762, 31693948, ... . Apparently, the asymptotic density of this sequence equals 0.000031... . Therefore, the least term not divisible by 3 that was mentioned above is a(~6*10^20) = 20169691981106018776756331. - Amiram Eldar, Jan 27 2025

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Rev. ed. 1997, p. 169.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
Carlos Rivera, Puzzle 329. Odd abundant numbers not divided by 2 or 3, The Prime Puzzles and Problems Connection.
Jay L. Schiffman, Odd Abundant Numbers, Mathematical Spectrum, Volume 37, Number 2 (January 2005), pp 73-75.

Intersection of A005231 and A047201.
Cf. A006038, A047802, A110585.
Cf. A020338.

Select[ Range[ 1, 10^6, 2 ], DivisorSigma[ 1, # ] - 2# > 0 && Mod[ #, 5 ] != 0 & ]
ta={{0}};Do[g=n;s=DivisorSigma[1, n]-2*n; If[Greater[s, 0]&&!Equal[Mod[n, 2], 0]&& !Equal[Mod[n, 5], 0], Print[n];ta=Append[ta, n]], {n, 1, 2000000}] ta=Delete[ta, 1] (* Labos Elemer, Sep 15 2005 *)

{ n=0; forstep (m=1, 10^9, 2, if (m%5 && sigma(m) > 2*m, write("b064001.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 05 2009

More terms from Robert G. Wilson v, Sep 28 2001
Further terms from Labos Elemer, Sep 15 2005
Entry revised by N. J. A. Sloane, Mar 28 2006

A112640 Odd abundant numbers divisible neither by 5 nor 7.

Original entry on oeis.org

28683369, 36165987, 38660193, 86050107, 108497961, 115980579, 131339637, 138428433, 140397543, 153393669, 160876287, 258150321, 277272567, 296394813, 315517059, 325493883, 347941737, 353761551, 372883797, 373715199, 392006043

Offset: 1

Labos Elemer, Sep 15 2005

nonn

Subsequence of A064001 which itself is a subsequence of A005231. All 500 terms in b-file are divisible by 99. Cf. also A047802. - Zak Seidov, Mar 30 2011
From Amiram Eldar, Aug 15 2024: (Start)
The least term that is not divisible by 99 is a(1718) = 21097921689.
The least term that is not divisible by 3 is 149#/7# = Product_{k=5..35} prime(k) = 7105630242567996762185122555313528897845637444413640621. (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from Zak Seidov)
Carlos Rivera, Puzzle 329. Odd abundant numbers not divided by 2 or 3, The Prime Puzzles and Problems Connection.

Cf. A005231, A047802, A064001, A110585.

Do[If[Mod[n,5]>0&&Mod[n,7]>0&&DivisorSigma[1, n]>2n, Print[n]], {n, 28683369, 6*10^9,2}] (* Zak Seidov, Mar 30 2011 *)

is(n)=gcd(n,70)==1 && sigma(n,-1)>2 \\ Charles R Greathouse IV, Feb 21 2017

Extended by Ray Chandler, Sep 19 2005

A358412 Least number k coprime to 2 and 3 such that sigma(k)/k >= n.

Original entry on oeis.org

1, 5391411025, 5164037398437051798923642083026622326955987448536772329145127064375

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) = 5^2*7*...*29 and a(3) = 5^4*7^3*11^2*13^2*17*...*157 ~ 5.16404*10^66. a(4) = 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 is too large to display.

			a(2) = A047802(2) = 5391411025 is the smallest abundant number coprime to 2 and 3.
Even if there is a number k coprime to 2 and 3 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=3..m+2} (prime(i)/(prime(i)-1)) => m >= 33, and we have k >= prime(3)^2*...*prime(35)^2 ~ 6.18502*10^112 > A358413(2) ~ 5.16403*10^66. So a(3) = A358413(2).
Even if there is a number k coprime to 2 and 3 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=3..m+2} (prime(i)/(prime(i)-1)) => m >= 140, and we have k >= prime(3)^2*...*prime(140)^2*prime(141)*prime(142) ~ 2.65585*10^669 > A358414(2) ~ 1.83947*10^370. So a(4) = A358414(2).

Jianing Song, Table of n, a(n) for n = 1..4
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), this sequence (p=5), A358418 (p=7), A358419 (p=11).

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

Original entry on oeis.org

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).

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).

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A337202 a(n) = 2*A246277(A047802(n)).

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A005231 Odd abundant numbers (odd numbers m whose sum of divisors exceeds 2m).

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A336835 Number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=n.

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A336389 The least positive integer k for which A336835(k) >= n, where A336835(k) is the number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=k.

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A115414 Odd abundant numbers not divisible by 3.

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A064001 Odd abundant numbers not divisible by 5.

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A112640 Odd abundant numbers divisible neither by 5 nor 7.

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