A115414 -id:A115414 - cp's OEIS frontend

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

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

A337386 Numbers k for which A003973(k) >= 2*A003961(k).

Original entry on oeis.org

120, 180, 240, 300, 360, 420, 480, 504, 540, 600, 630, 660, 720, 780, 840, 900, 924, 960, 990, 1008, 1020, 1050, 1080, 1092, 1140, 1170, 1200, 1260, 1320, 1380, 1440, 1470, 1500, 1512, 1560, 1620, 1650, 1680, 1740, 1800, 1848, 1860, 1890, 1920, 1980, 2016, 2040, 2100, 2160, 2184, 2220, 2280, 2310, 2340, 2400, 2460

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Provided that there are no odd perfect numbers, then these are equal to numbers k for which A003961(k) is in A005231, i.e., numbers that become odd abundant numbers when prime-shifted once.

Not all terms are even. The first odd term is a(8313165) = 334639305 = A064989(A115414(1)). (See A337385). For any odd term x present, A064989(x) is also present, for example, A064989(334639305) = 19399380 = a(482324).

Cf. A000203, A003961, A003973, A005231, A115414, A336389, A337468.
Subsequence of A005101, of A337381, and of A246282.
Subsequences: A337385 (odd terms), A337479 (primitive elements).

Select[Range[2500], If[# == 1, 1, DivisorSigma[1, # ]] >= 2# &@ Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Prime[PrimePi@ p + 1]^e] &] (* 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); };
isA337386(n) = (sigma(A003961(n))>=2*A003961(n));

A306497 Abundant numbers that differ from the next abundant number by 5.

Original entry on oeis.org

5391411025, 26957055120, 28816162375, 33426748350, 34393484125, 37739877175, 40342627320, 48150877770, 50866790970, 53356378075, 59305521270, 60711976320, 61164628525, 63395557225, 64899009175, 67275433225, 70088343325, 74922022170, 75665665075, 76781129425

Offset: 1

Sergio Pimentel, Feb 19 2019

nonn

Since all multiples of 6 are abundant, numbers in this sequence have to be abundant numbers of the form 6n or 6n + 1. Most common difference between abundant numbers is 6, followed by 2, 4, 3, 1. 5 is the least common.

			a(1) = 5391411025 is in the sequence since it is abundant and the next abundant number is 5391411030 which is a(1)+5 and all the numbers in between are deficient.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A005101, A096399, A231086, A228382, A316098, A316099, A115414, A125115.

isok(n) = for(k=1, 4, if(sigma(n+k) > 2*(n+k), return(0))); (sigma(n) > 2*n) && (sigma(n+5) > 2*(n+5)); \\ Daniel Suteu, Jul 24 2019

Either a(n) or a(n)+5 are in A115414. - Amiram Eldar, Jul 16 2019

More terms from Amiram Eldar, Jul 16 2019

A343306 Numbers k such that there is only 1 abundant number (A005101) among 6k+1 through 6k+5.

Original entry on oeis.org

3, 6, 9, 11, 13, 14, 16, 17, 18, 23, 26, 29, 32, 33, 34, 36, 37, 43, 45, 46, 50, 51, 53, 56, 60, 61, 63, 65, 66, 69, 73, 74, 76, 77, 79, 81, 83, 86, 88, 90, 91, 93, 95, 96, 101, 102, 103, 106, 107, 108, 113, 116, 117, 121, 122, 123, 124, 126, 128, 130, 133, 135

Offset: 1

Jianing Song, Apr 11 2021

nonn

The smallest k such that 6*k+3 is the only abundant number among 6*k+1 through 6*k+5 is k = 157, with 6*k+3 = 945 = A005231(1).
The smallest k such that 6*k+1 is the only abundant number among 6*k+1 through 6*k+5 is k = 898568504, with 6*k+1 = 5391411025 = A115414(1).
The smallest k such that 6*k+5 is the only abundant number among 6*k+1 through 6*k+5 is k = 4492842520, with 6*k+5 = 26957055125 = A115414(2).

			13 is a term since 80 is the only abundant number among 79, 80, 81, 82 and 83.
962 is not a term since there are 2 abundant numbers (5775 and 5776) among 5773, 5774, 5775, 5776 and 5777.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A005101 (abundant numbers), A005231 (odd abundant numbers), A115414 (5-rough abundant numbers), A343301.

q[n_] := Count[Range[5], ?(DivisorSigma[-1, 6*n + #] > 2 &)] == 1; Select[Range[0, 135], q] (* _Amiram Eldar, Mar 21 2024 *)

isA343306(k) = (sum(i=1, 5, sigma(6*k+i) > 2*(6*k+i)) == 1)

A387165 Nondeficient numbers k for which A324644(k)/A324198(k) = 2.

Original entry on oeis.org

38745, 77805, 78435, 118755, 141075, 157815, 210735, 237195, 241605, 294975, 300105, 323505, 364455, 371925, 390195, 409185, 455715, 475335, 499905, 567945, 607635, 660825, 701415, 733005, 766395, 806085, 809325, 872235, 885465, 891135, 937755, 964845, 978705, 1101555, 1150065, 1201095, 1229445, 1265355, 1293705

Offset: 1

Antti Karttunen, Aug 28 2025

First three nonmultiples of 5 occur at a(138), a(276), a(356) = 4446981, 8909901, 11234223. (Cf. A005231, A064001).

Intersection of A023196 and A364286.
Cf. A000203, A276086.
Cf. also A371082, A386422, A387163 and A047802, A064001, A115414.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A387165(n) = if(sigma(n)<2*n, 0, my(u=A276086(n)); (gcd(sigma(n),u)==2*gcd(n,u)));

{k | sigma(k) >= 2*k, A324644(k) = 2*A324198(k)}.

A133849 Least odd primitive abundant numbers with no factor 3 and with 5^n but not 5^(n+1) as a factor.

Original entry on oeis.org

20169691981106018776756331, 33426748355, 5391411025, 26957055125, 134785275625, 673926378125, 3369631890625, 16848159453125, 84240797265625, 421203986328125, 2106019931640625, 10530099658203125, 52650498291015625, 263252491455078125, 1316262457275390625, 6581312286376953125

Offset: 0

Pierre CAMI, Jan 06 2008

nonn

A subsequence of A115414, odd abundant numbers (A005231) not divisible by 3. The smallest of these equals a(2). All subsequent terms are a(n) = 5*a(n-1). - M. F. Hasler, Jul 28 2016

			a(0) = 20169691981106018776756331 = 5^0*7^2*11^2*13*17*19*23*29*31*37*41*43*47*53*59*61*67 = A047802(3), the least odd abundant number with no factor 3 or 5.
a(1) = 33426748355 = 5^1*7*11*13*17*19*23*29*31.
a(2) = 5391411025 = 5^2*7*11*13*17*19*23*29 = A115414(1) = A047802(2), the least odd abundant number with no factor 3.

Cf. A115414, A005231, A005101.

A133849(n)=215656441*if(n>1,5^n,[3016998806898461,5][n+1]*31) \\ M. F. Hasler, Jul 28 2016

For all n >= 2, a(n) = 5^n*7*11*13*17*19*23*29. This can be seen from sigma[-1](5^n) = (5-1/5^n)/4 and sigma[-1](29#/5#) = 1.615... > 2/sigma[-1](5^n) for all n >= 2 (but not for n = 1), while sigma[-1](23#/5#) = 1.56... < 2*4/5 (and idem for any other factor omitted) is never large enough. - M. F. Hasler, Jul 28 2016

Edited, a(3) corrected, and more terms added by M. F. Hasler, Jul 28 2016

A295042 Numbers k such that both k and (k+1) are abundant, and neither is divisible by 3.

Original entry on oeis.org

55959128224, 68972878975, 91653987424, 171967420624, 350441716624, 372944997424, 386136575824, 711480344575, 769856312224, 789255692224, 818564922175, 997039218175, 1071710665024, 1216042052224, 1340586071824, 1925671372624, 1954925637664, 2045947528624

Offset: 1

Amiram Eldar, Nov 13 2017

nonn

Subsequence of A096399.

All terms are of the form 3j+1, with j = 18653042741, 22990959658, 30551329141, 57322473541, 116813905541, 124314999141, 128712191941, 237160114858, 256618770741, 263085230741, 272854974058, 332346406058, ...

			k = 55959128224 is in the sequence as sigma(k) > 2*k and sigma(k + 1) > 2*(k + 1). - _David A. Corneth_, Apr 11 2021

David A. Corneth, Table of n, a(n) for n = 1..988 (first 87 terms from Giovanni Resta)

Cf. A005101, A096399, A115414.

abQ[n_] := Mod[n, 3] > 0 && DivisorSigma[1, n] > 2 n; abQ1[n_] := abQ[n - 1]; abQ2[n_] := abQ[n + 1]; s = Import["b115414.txt", "Data"][[All, -1]]; s1 = Select[s, abQ1] - 1; s2 = Select[s, abQ2]; seq = Union[s1, s2] (* using the b-File from A115414 *)

isoka(n) = (n%3) && (sigma(n) > 2*n);
isok(n) = isoka(n) && isoka(n+1); \\ Michel Marcus, Nov 13 2017

a(13)-a(18) from Giovanni Resta, Aug 22 2018

A337385 Odd numbers k for which A003973(k) >= 2*A003961(k).

Original entry on oeis.org

334639305, 1003917915, 1265809545, 1353106755, 1673196525, 2109682575, 2255177925, 2342475135, 2553826275, 2691663975, 2729952225, 2953555605, 2982654675, 3011753745, 3128150025, 3157249095, 3234846615, 3258330075, 3419140725, 3442113675, 3681032355, 3797428635, 3855626775, 4059320265, 4292112825, 4350310965

Offset: 1

Antti Karttunen, Aug 31 2020

nonn

Provided that there are no odd perfect numbers, then applying A003961 to each term and sorting into ascending order gives A115414.

Apparently, all terms are divisible by 255255 = 3*5*7*11*13*17. - Hugo Pfoertner, Sep 24 2020

Subsequence of A337386.

Cf. A000203, A003961, A003973, A115414.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337385(n) = if(!(n%2),0,my(x=A003961(n)); (sigma(x)>=2*x));

A343357 7-rough abundant numbers.

Original entry on oeis.org

20169691981106018776756331, 21373852696395930345517903, 21975933054040886129898689, 23476198863254546445077041, 23782174126975753483041047, 23836908704943476736166573, 24137500239684251978741183, 24272002214551310731350839, 24955720586792192723783257, 24986334842265665051802619

Offset: 1

David A. Corneth, Apr 12 2021

nonn

Each term has at least A001276(4) = 15 distinct prime factors and A108227(4) = 18 prime factors counted with multiplicity. - Jianing Song, Apr 13 2021

The smallest term with exactly 15 distinct prime factors is a(830) = 465709156638373299218537971 = 7^3 * 11^2 * 13^2 * 17^2 * 19 * 23 * ... * 61. - Jianing Song, Apr 14 2021

			k = 20169691981106018776756331 is in the sequence as its smallest prime factor is at least 7 and it is abundant as sigma(k) > 2*k.

David A. Corneth, Table of n, a(n) for n = 1..2187

Cf. A005101, A007775, A047802, A115414.

is(n) = gcd(n, 30) == 1 && sigma(n) > 2*n

A133688 Least odd primitive abundant number with 3^n as a divisor, but not 3^(n+1).

Original entry on oeis.org

5391411025, 5775, 1575, 945, 81081, 78975, 1468935, 6375105, 436444281, 5356826865, 21873816315, 371922783705, 2241870572475, 158639164165575, 297836412308955, 1429674513582825, 13431279259253115, 100139192108634825

Offset: 0

Pierre CAMI, Jan 04 2008

			5391411025=3^0 * 5^2 * 7 * 11 * 13 * 17 * 19 * 23 * 29 least odd abundant number with no factor 3.
5775 = 3^1 * 5^2 * 7 * 11.
1575 = 3^2 * 5^2 * 13.
945 = 3^3 * 5 * 7.
81081 = 3^4 * 7 * 11 * 13.
78975 = 3^5 * 5^2 * 13.
1468935 = 3^6 * 5 * 13 * 31.
6375105 = 3^7 * 5 * 11 * 53.
436444281 = 3^8 * 7 * 13 * 17 * 43.

Cf. A006038 (odd primitive abundant numbers).

Cf. A115414 (odd abundant numbers not divisible by 3).

isprab(v) = {my(sig = sigma(v)); if (sig < 2*v, return (0)); if (sig == 2*v, return (1)); fordiv (v, d, if ((d != v) && (sigma(d)>=2*d), return (0));); return (1);}
a(n) = {my(p = 3^n, k = 1); while (1, if (k % 3 != 0, v = p * k; if (isprab(v), return (v));); k += 2;);}
\\ Michel Marcus, Mar 07 2013

Some terms corrected and a(9)-a(13) from Michel Marcus, Mar 07 2013

a(14)-a(17) from David A. Corneth, Oct 26 2024

cp's OEIS Frontend

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

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A337386 Numbers k for which A003973(k) >= 2*A003961(k).

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A306497 Abundant numbers that differ from the next abundant number by 5.

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A343306 Numbers k such that there is only 1 abundant number (A005101) among 6*k+1 through 6*k+5.

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A387165 Nondeficient numbers k for which A324644(k)/A324198(k) = 2.

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A133849 Least odd primitive abundant numbers with no factor 3 and with 5^n but not 5^(n+1) as a factor.

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A295042 Numbers k such that both k and (k+1) are abundant, and neither is divisible by 3.

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A343306 Numbers k such that there is only 1 abundant number (A005101) among 6k+1 through 6k+5.