A071395 -id:A071395 - cp's OEIS frontend

A307098 The primitive abundant numbers k (A071395) arranged by the decreasing values of their abundancy index sigma(k)/k.

3465, 15015, 4095, 1430, 19635, 16796, 20, 21945, 5355, 692835, 2584, 5985, 23205, 49742, 20332, 22309287, 26565, 188955, 1870, 216315, 838695, 25935, 3128, 22724, 6084351, 7245, 2090, 60214, 2107575, 937365, 1542773001, 25636, 28129101, 33495, 13066965, 3016174

Offset: 1

Amiram Eldar, Mar 25 2019

nonn

Cohen proved that for any given eps > 0 there are only finitely many primitive abundant numbers k with sigma(k)/k >= 2 + eps. Thus the primitive abundant numbers can be arranged by their decreasing value of their abundancy index. In case of more than one primitive abundant number with the same abundancy index, the terms are ordered by their value.

Cohen calculated the first 91 terms of this sequence, all the terms with abundancy index >= 2.05 - see the link for the corresponding values of the abundancy index.

			a(1) = 3465 since it is the primitive abundant number (A071395) with the largest possible abundancy index among the primitive abundant numbers: sigma(3465)/3465 = 832/385 = 2.161003...

Amiram Eldar, Table of n, a(n) for n = 1..91
Graeme L. Cohen, On primitive abundant numbers, Journal of the Australian Mathematical Society, Vol. 34 No. 1 (1983), pp. 123-137.
Amiram Eldar, Table of n, a(n), prime factorization of a(n), sigma(a(n))/a(n) (rounded) for n = 1..91

Cf. A000203, A005100, A005101, A071395.

A335290 Primitive pseudoperfect numbers (A006036) that are not primitive abundant (A071395).

Original entry on oeis.org

6, 28, 350, 490, 496, 770, 910, 1190, 1330, 1610, 2030, 2170, 2590, 2870, 3010, 3290, 3710, 4130, 4270, 4690, 4970, 5110, 5530, 5810, 6230, 6790, 7070, 7210, 7490, 7630, 7910, 8128, 8890, 9170, 9196, 9590, 9730, 15884, 19228, 24244, 25916, 30932, 34276, 35948

Offset: 1

Amiram Eldar, May 30 2020

nonn

Includes all the perfect numbers (A000396). The nonperfect terms have an abundant proper divisor which is not pseudoperfect, i.e., a proper divisor which is a weird number (A006037).
The first term with one weird divisor is a(3) = 350, having the weird divisor 70.
The first term with 2 weird divisors is a(202) = 658312, having the 2 weird divisors 9272 and 10792.
The first term with 3 weird divisors is a(353) = 1574930, having the 3 weird divisors 70, 10430 and 10570.

			350 is a term since it is pseudoperfect: 1 + 5 + 14 + 35 + 50 + 70 + 175 = 350. All of its proper divisors, {1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175} are not pseudoperfect, and it is not primitive abundant, since its divisor 70 is abundant.

Amiram Eldar, Table of n, a(n) for n = 1..2500

Cf. A000396, A006036, A006037, A071395, A306987.

pspQ[n_] := Module[{d = Most @ Divisors[n], x}, Plus @@d >= n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0]; primPspQ[n_] := pspQ[n] && AllTrue[Most @ Divisors[n], !pspQ[#] &]; primAbQ[n_] := DivisorSigma[1, n] > 2*n && AllTrue[Most @ Divisors[n], DivisorSigma[1, #] < 2*# &]; Select[Range[1000], primPspQ[#] && !primAbQ[#] &]

A337469 a(n) is the least k that is a multiple of A071395(n) (the n-th primitive abundant number) for which A003961(k) is abundant.

Original entry on oeis.org

120, 420, 1320, 1560, 4080, 4560, 5520, 6960, 1650, 3432, 3900, 4488, 7524, 1890, 17760, 19680, 20640, 4290, 22560, 3150, 25440, 5610, 28320, 29280, 12012, 6270, 4410, 6630, 7410, 7590, 23256, 8970, 28152, 9570, 9690, 10230, 6930, 52440, 22620, 59160, 24180, 12210, 8190, 63240, 64320

Offset: 1

Antti Karttunen and Peter Munn, Sep 07 2020

nonn

A003961(k) replaces each prime factor of k with the next larger prime. Thus for all terms a(n), A003961(a(n)) is an odd abundant number (some of which are also primitive abundant numbers, starting with n = 1, 2, 9, 10, 12, ...).

			The table below shows a(n), for n less than 16, alongside A071395(n) and its prime factors, and the additional prime factors that are needed to produce a(n).
   n   a(n)               A071395(n)
   1    120 / (2 * 3)  =    20  =  2^2 * 5,
   2    420 / (2 * 3)  =    70  =  2 * 5 * 7,
   3   1320 / (3 * 5)  =    88  =  2^3 * 11,
   4   1560 / (3 * 5)  =   104  =  2^3 * 13,
   5   4080 / (3 * 5)  =   272  =  2^4 * 17,
   6   4560 / (3 * 5)  =   304  =  2^4 * 19,
   7   5520 / (3 * 5)  =   368  =  2^4 * 23,
   8   6960 / (3 * 5)  =   464  =  2^4 * 29,
   9   1650 / (3)      =   550  =  2 * 5^2 * 11,
  10   3432 / (2 * 3)  =   572  =  2^2 * 11 * 13,
  11   3900 / (2 * 3)  =   650  =  2 * 5^2 * 13,
  12   4488 / (2 * 3)  =   748  =  2^2 * 11 * 17,
  13   7524 / (3 * 3)  =   836  =  2^2 * 11 * 19,
  14   1890 / (2)      =   945  =  3^3 * 5 * 7,
  15  17760 / (3 * 5)  =  1184  =  2^5 * 37, ...

See A000203 and A005101 for the definition of abundant.
A003961 and A071395 are used to define the sequence.
Sequences with related definitions: A337386, A337479, A337538.
Cf. A003973.

Map[Block[{k = 1}, While[DivisorSigma[1, #] <= 2 # &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[k #] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]], k++]; # k] &, Select[Range[5*10^3], DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ #] == 1 &]] (* Michael De Vlieger, Oct 05 2020 *)

isA071395(n) = if(sigma(n) <= 2*n, 0, fordiv(n, d, if((d != n)&&(sigma(d) >= 2*d), return(0))); (1)); \\ After code in A071395
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337386(n) = { my(x=A003961(n)); (sigma(x)>=2*x); };
for(n=1,2^13,if(isA071395(n), i=0; for(k=1,oo,if(isA337386(k*n),i++; print1(k*n,", "); break))));

a(n) = A071395(n) * A337538(n).

A362053 Primitive abundant numbers k (A071395) whose abundancy index sigma(k)/k has a record low value.

Original entry on oeis.org

20, 70, 88, 104, 464, 650, 1888, 1952, 4030, 5830, 8925, 17816, 32128, 77744, 91388, 128768, 130304, 442365, 521728, 522752, 1848964, 8353792, 8378368, 8382464, 35021696, 45335936, 120888092, 134193152, 775397948, 1845991216, 2146926592, 2146992128, 3381872252

Offset: 1

Amiram Eldar, Apr 06 2023

nonn

The abundancy index of an integer k is sigma(k)/k, where sigma is the sum-of-divisors function (A000203).

Terms k of A071395 such that sigma(k)/k < sigma(m)/m for all smaller terms m < k of A071395.

			The abundancy indices of the first terms are 21/10 > 72/35 > 45/22 > 105/52 > 465/232 > 651/325 > 945/472 > ... > 2.

Cf. A000203, A071395, A071927.

Other sequences related to records in A071395: A083873, A334419.

f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1);
(* Returns the abundancy index of n if n is primitive abundant, and 0 otherwise: *)
abIndex[n_] := If[(r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f < 2, r, 0]; abIndex[1] = 0;
seq[kmax_] := Module[{s = {}, ab, abm = 3}, Do[If[0 < (ab = abIndex[k]) < abm, abm = ab; AppendTo[s, k]], {k, 1,  kmax}]; s]; seq[10^6]

abindex(n) = {my(f = factor(n), r, p, e); r = sigma(f, -1); if(r <= 2, return(0)); if(vecmax(vector(#f~, i, p = f[i, 1]; e = f[i, 2]; (p^(e + 1) - p)/(p^(e + 1) - 1))) * r < 2, r, 0);} \\ Returns the abundancy index of n if n is primitive abundant, and 0 otherwise.
lista(kmax) = {my(ab, abm = 3); for(k = 1, kmax, ab = abindex(k); if(ab > 0 && ab < abm, abm = ab; print1(k, ", "))); }

A298157 Number of primitive abundant numbers (A071395) with n prime factors, counted with multiplicity.

Original entry on oeis.org

0, 0, 2, 25, 906, 265602, 13232731828

Offset: 1

Gianluca Amato, Feb 15 2018

This uses the first definition of primitive abundant numbers, A071395: having only deficient proper divisors. The second definition (A091191: having no abundant proper divisors) would yield infinite a(3), since all numbers 6*p, p > 3, are in that sequence.
See A287728 for the number of ODD primitive abundant numbers with n prime factors, counted with multiplicity and A295369 for the number of squarefree primitive abundant numbers with n distinct prime factors.
It appears that a(n) is just slightly larger than A295369(n).

			For n=3, the only two primitive abundant numbers (PAN) are 2*2*5 = 20 and 2*5*7 = 70. The latter is also a primitive weird number, see A002975.
For n=4, the 25 PAN range from 2^3*11 = 88 to 2*5*11*53 = 5830.

G. Amato, Primitive Weirds and Abundant Numbers, GitHub.
Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton, Primitive abundant and weird numbers with many prime factors, arXiv:1802.07178 [math.NT], 2018.

Cf. A071395 (primitive abundant numbers), A091191 (alternative definition), A287728 (counts of odd PAN), A295369 (counts of squarefree PAN).

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A306986 Number of primitive abundant numbers (A071395) < 10^n.

Original entry on oeis.org

0, 3, 14, 98, 441, 1734, 8667, 41653, 213087, 1123424

Offset: 1

Amiram Eldar, Mar 18 2019

			There are 3 terms of A071395 below 100 (20, 70, and 88), thus a(2) = 3.

Michael R. Avidon, On the distribution of primitive abundant numbers, Acta Arithmetica, Vol. 77, No. 2 (1996), pp. 195-205.
Paul Erdős, On primitive abundant numbers, J. London Math. Soc., Volume s1-10, Issue 1 (1935), pp. 49-58, alternative link.
Aleksandar Ivić, The distribution of primitive abundant numbers, Studia Sci. Math. Hungar., Vol. 20 (1985), pp. 183-187.

Cf. A071395, A302992.

paQ[n_] := DivisorSigma[1, n] > 2n && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ n] == 1;  c = 0; k = 1;  seq={}; Do[ While[ k < 10^n, If[ paQ[k], c++ ]; k ++]; AppendTo[seq, c], {n, 1, 5}]; seq (* after Michael De Vlieger at A071395 *)

ispab(n) = {my(f = factor(n), r, p, e); r = sigma(f, -1); if(r <= 2, return(0)); if(vecmax(vector(#f~, i, p = f[i, 1]; e = f[i, 2]; (p^(e + 1) - p)/(p^(e + 1) - 1))) * r < 2, 1, 0);}
lista(nmax) = {my(c = 0, r = 10); for(k = 1, 10^nmax, if(ispab(k), c++); if(k+1 == r, print1(c, ", "); r *= 10));} \\ Amiram Eldar, Mar 26 2023

a(10) from Amiram Eldar, Mar 26 2023

A091191 Primitive abundant numbers: abundant numbers (A005101) having no abundant proper divisor.

Original entry on oeis.org

12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114, 138, 174, 186, 196, 222, 246, 258, 272, 282, 304, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 550, 572, 582, 606, 618, 642, 644, 650, 654, 678, 748, 762, 786, 812, 822

Offset: 1

Reinhard Zumkeller, Dec 27 2003

nonn

A080224(a(n)) = 1.
This is a supersequence of the primitive abundant number sequence A071395, since many of these numbers will be positive integer multiples of the perfect numbers (A000396). - Timothy L. Tiffin, Jul 15 2016
If the terms of A071395 are removed from this sequence, then the resulting sequence will be A275082. - Timothy L. Tiffin, Jul 16 2016

			12 is a term since 1, 2, 3, 4, and 6 (the proper divisors of 12) are either deficient or perfect numbers, and thus not abundant. - _Timothy L. Tiffin_, Jul 15 2016

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On the density of the abundant numbers, J. London Math. Soc. 9 (1934), pp. 278-282.
Eric Weisstein's World of Mathematics, Abundant Number

Cf. A006038 (odd terms), A005101 (abundant numbers), A091192.
Cf. A027751, A071395 (subsequence), supersequence of A275082.
Cf. A294930 (characteristic function), A294890.

a091191 n = a091191_list !! (n-1)
a091191_list = filter f [1..] where
   f x = sum pdivs > x && all (<= 0) (map (\d -> a000203 d - 2 * d) pdivs)
         where pdivs = a027751_row x
-- Reinhard Zumkeller, Jan 31 2014

isA005101 := proc(n) is(numtheory[sigma](n) > 2*n ); end proc:
isA091191 := proc(n) local d; if isA005101(n) then for d in numtheory[divisors](n) minus {1,n} do if isA005101(d) then return false; end if; end do: return true; else false; end if; end proc:
for n from 1 to 200 do if isA091191(n) then printf("%d\n",n) ; end if;end do: # R. J. Mathar, Mar 28 2011

t = {}; n = 1; While[Length[t] < 100, n++; If[DivisorSigma[1, n] > 2*n && Intersection[t, Divisors[n]] == {}, AppendTo[t, n]]]; t (* T. D. Noe, Mar 28 2011 *)
Select[Range@ 840, DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] <= 2 # &, Most@ Divisors@ #] == 1 &] (* Michael De Vlieger, Jul 16 2016 *)

is(n)=sumdiv(n,d,sigma(d,-1)>2)==1 \\ Charles R Greathouse IV, Dec 05 2012

Erdős shows that a(n) >> n log^2 n. - Charles R Greathouse IV, Dec 05 2012

A006038 Odd primitive abundant numbers.

Original entry on oeis.org

945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 7425, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 18585, 19215, 19635, 21105, 21945, 22365, 22995, 23205, 24885, 25935, 26145, 26565, 28035, 28215

Offset: 1

N. J. A. Sloane

nonn

Dickson proves that there are only a finite number of odd primitive abundant numbers having n distinct prime factors. Sequence A188342 lists the smallest such numbers. - T. D. Noe, Mar 28 2011
Sequence A188439 sorts the numbers in this sequence by the number of distinct prime factors. Eight numbers have exactly three prime factors; 576 have exactly four prime factors. - T. D. Noe, Apr 04 2011
Any multiple of an abundant number (A005101) is again an abundant number. Primitive abundant numbers (A091191) are those not of this form, i.e., without an abundant proper divisor. We don't know any odd perfect number (A000396), so the (odd) terms here have only deficient proper divisors (A071395), and their prime factors p are less than sigma(n/p)/deficiency(n/p). See A005231 (odd abundant numbers) for an explanation why all terms have at least 3 distinct prime factors, and 5 prime factors when counted with multiplicity (A001222), whence a(1) = 3^3*5*7. All known terms are semiperfect (A005835, and thus in A006036): no odd weird number (A006037) is known, but if it exists, the smallest one is in this sequence. - M. F. Hasler, Jul 28 2016
So far, a(173) = 351351 is the only known term of A122036, i.e., which can't be written as sum of its proper divisors > 1. - M. F. Hasler, Jan 26 2020

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
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.
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
Jacob Liddy, An algorithm to determine all odd primitive abundant numbers with d prime divisors, Honors Research Projects (2018), 728.
Eric Weisstein's World of Mathematics, Primitive Abundant Number

Cf. A005101, A005231. Subsequence of A091191.

Cf. A000203, A027751, A379949 (subsequence of square terms).

a006038 n = a006038_list !! (n-1)
a006038_list = filter f [1, 3 ..] where
   f x = sum pdivs > x && all (<= 0) (map (\d -> a000203 d - 2 * d) pdivs)
         where pdivs = a027751_row x
-- Reinhard Zumkeller, Jan 31 2014

isA005101 := proc(n) is(numtheory[sigma](n) > 2*n ); end proc:
isA005100 := proc(n) is(numtheory[sigma](n) < 2*n ); end proc:
isA006038 := proc(n) local d; if type(n,'odd') and isA005101(n) then for d in numtheory[divisors](n) minus {1,n} do if not isA005100(d) then return false; end if; end do: return true;else false; end if; end proc:
n := 1 ; for a from 1 by 2 do if isA006038(a) then printf("%d %d\n",n,a) ; n := n+1 ; end if; end do: # R. J. Mathar, Mar 28 2011

t = {}; n = 1; While[Length[t] < 50, n = n + 2; If[DivisorSigma[1, n] > 2 n && Intersection[t, Divisors[n]] == {}, AppendTo[t, n]]]; t (* T. D. Noe, Mar 28 2011 *)

is(n)=n%2 && sumdiv(n,d,sigma(d,-1)>2)==1 \\ Charles R Greathouse IV, Jun 10 2013

is_A006038(n)=bittest(n,0) && sigma(n)>2*n && !for(i=1,#f=factor(n)[,1],sigma(n\f[i],-1)>2&&return) \\ More than 5 times faster. - M. F. Hasler, Jul 28 2016

A006039 Primitive nondeficient numbers.

Original entry on oeis.org

6, 20, 28, 70, 88, 104, 272, 304, 368, 464, 496, 550, 572, 650, 748, 836, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4030

Offset: 1

N. J. A. Sloane, R. K. Guy

nonn

A number n is nondeficient (A023196) iff it is abundant or perfect, that is iff A001065(n) is >= n. Since any multiple of a nondeficient number is itself nondeficient, we call a nondeficient number primitive iff all its proper divisors are deficient. - Jeppe Stig Nielsen, Nov 23 2003
Numbers whose proper multiples are all abundant, and whose proper divisors are all deficient. - Peter Munn, Sep 08 2020
As a set, shares with the sets of k-almost primes this property: no member divides another member and each positive integer not in the set is either a divisor of 1 or more members of the set or a multiple of 1 or more members of the set, but not both. See A337814 for proof etc. - Peter Munn, Apr 13 2022

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..8671
L. E. Dickson, Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors, Amer. J. Math., 35 (1913), 413-426.
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
Jared Duker Lichtman, The reciprocal sum of primitive nondeficient numbers, Journal of Number Theory, Vol. 191 (2018), pp. 104-118.
Joshua Zelinsky, The Sum of the Reciprocals of the Prime Divisors of an Odd Perfect or Odd Primitive Non-deficient Number, Integers (2025) Vol. 25, Art. No. A59. See p. 2.
Index entries for sequences where any odd perfect numbers must occur

Cf. A001065 (aliquot function), A023196 (nondeficient), A005101 (abundant), A091191.
Subsequences: A000396 (perfect), A071395 (primitive abundant), A006038 (odd primitive abundant), A333967, A352739.
Positions of 1's in A341620 and in A337690.
Cf. A180332, A337479, A337688, A337689, A337691, A337814, A338133 (sorted by largest prime factor), A338427 (largest prime(n)-smooth), A341619 (characteristic function), A342669.

k = 1; lst = {}; While[k < 4050, If[DivisorSigma[1, k] >= 2 k && Min@Mod[k, lst] > 0, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Mar 09 2017 *)

Union of A000396 (perfect numbers) and A071395 (primitive abundant numbers). - M. F. Hasler, Jul 30 2016

Sum_{n>=1} 1/a(n) is in the interval (0.34842, 0.37937) (Lichtman, 2018). - Amiram Eldar, Jul 15 2020

A337690 a(n) is the number of primitive nondeficient numbers (A006039) dividing 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, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 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 and Peter Munn, Sep 15 2020

nonn

As a simple consequence of the definition of a primitive nondeficient number, a(n) is nonzero if and only if n is nondeficient.

			The least nondeficient number, therefore the least primitive nondeficient number is 6. So a(1) = a(2) = a(3) = a(4) = a(5) = 0 as all primitive nondeficient numbers are larger, and therefore not divisors; and a(6) = 1, as only 1 primitive nondeficient number divides 6, namely 6 itself.
60 has the following 12 divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Of these, only 6 and 20 are in A006039, thus a(60) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n).

A006039 (or equivalently, its characteristic function, A341619) is used to define this sequence.
See A000203 and A023196 for definitions of deficient and nondeficient.
Sequences with similar definitions: A080224, A294927, A337539, A341620.
Cf. A302110, A341604.
Positions of 0's: A005100.
Positions of numbers >= k: A023196 (k=1), A337688 (k=2), A337689 (k=3).
Positions of first appearances are given in A337691.
Differs from its derived sequence A341618 for the first time at n=120, where a(120)=2, while A341618(120)=1.

A341619(n) = if(sigma(n) < (2*n), 0, fordiv(n, d, if((d= 2*d), return(0))); (1)); \\ After code in A071395
A337690(n) = sumdiv(n, d, A341619(d));

a(n) = Sum_{d|n} A341619(d) = Sum_{d|n} [1==A341620(d)]. - Corrected by Antti Karttunen, Feb 21 2021
a(A005100(n)) = 0.
a(A006039(n)) = 1.
a(A023196(n)) >= 1.
a(A337479(n)) = A337539(n).
a(n) <= A341620(n). - Antti Karttunen, Feb 22 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=1} 1/A006039(n) = 0.3... (see A006039 for a better estimate of this constant). - Amiram Eldar, Jan 01 2024

Data section extended to 120 terms by Antti Karttunen, Feb 21 2021

cp's OEIS Frontend

A307098 The primitive abundant numbers k (A071395) arranged by the decreasing values of their abundancy index sigma(k)/k.

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A335290 Primitive pseudoperfect numbers (A006036) that are not primitive abundant (A071395).

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A337469 a(n) is the least k that is a multiple of A071395(n) (the n-th primitive abundant number) for which A003961(k) is abundant.

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PARI

Formula

A362053 Primitive abundant numbers k (A071395) whose abundancy index sigma(k)/k has a record low value.

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Mathematica

PARI

A298157 Number of primitive abundant numbers (A071395) with n prime factors, counted with multiplicity.

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SageMath

A306986 Number of primitive abundant numbers (A071395) < 10^n.

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Mathematica

PARI

Extensions

A091191 Primitive abundant numbers: abundant numbers (A005101) having no abundant proper divisor.

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Haskell

Maple

Mathematica

PARI

Formula

A006038 Odd primitive abundant numbers.

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