A112643 -id:A112643 - cp's OEIS frontend

A087248 Squarefree abundant numbers.

30, 42, 66, 70, 78, 102, 114, 138, 174, 186, 210, 222, 246, 258, 282, 318, 330, 354, 366, 390, 402, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 642, 654, 678, 690, 714, 762, 770, 786, 798, 822, 834, 858, 870, 894, 906, 910, 930, 942, 966, 978

Offset: 1

Labos Elemer, Sep 05 2003

nonn

First odd term is 15015 = 3 * 5 * 7 * 11 * 13, with 32 divisors that add up to 32256 = 2*15015 + 2226. See A112643. - Alonso del Arte, Nov 06 2017
The lower asymptotic density of this sequence is larger than 1/(2*Pi^2) = 0.05066... which is the density of its subsequence of squarefree numbers larger than 6 and divisible by 6. The number of terms below 10^k for k=1,2,... is 0, 5, 53, 556, 5505, 55345, 551577, 5521257, 55233676, 552179958, 5521420147, ..., so it seems that this sequence has an asymptotic density which equals to about 0.05521... - Amiram Eldar, Feb 13 2021
The asymptotic density of this sequence is larger than 0.0544 (Wall, 1970). - Amiram Eldar, Apr 18 2024

			Checking that 30 = 2 * 3 * 5 and sigma(30) = 1 + 2 + 3 + 5 + 6 + 10 + 15 + 30 = 72, which is more than twice 30, we verify that 30 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Charles Robert Wall, Topics related to the sum of unitary divisors of an integer, Ph.D. diss., University of Tennessee, 1970.

Cf. A087244, A087245, A087246, A087247, A087248, A008683, A005117, A005101, A112643.

isA005101 := proc(n)
    simplify(numtheory[sigma](n)>2*n);
end proc:
isA087248 := proc(n)
    isA005101(n) and numtheory[issqrfree](n) ;
end proc:
for n from 1 to 500 do
    if isA087248(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

Select[Range[10^3], SquareFreeQ@ # && DivisorSigma[1, #] > 2 # &] (* Michael De Vlieger, Feb 05 2017 *)

isA087248(i) = (sigma(i) > 2*i) && issquarefree(i) \\ Michel Marcus, Mar 09 2013

A005117 INTERSECT A005101.

A129485 Odd unitary abundant numbers.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035

Offset: 1

Ant King, Apr 17 2007

This sequence is different from A112643. The two sequences agree for the first 50 terms but differ thereafter. The exceptions, i.e. those odd unitary abundant numbers that are not squarefree ordinary abundant numbers, are in A129486.
22309287 is the smallest term not divisible by 5. 33426748355 is the smallest term not divisible by 3. - Donovan Johnson, May 15 2013
The numbers of terms not exceeding 10^k, for k = 5, 6, ..., are 34, 137, 1714, 16918, 181744, 1752337, 17290556, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00017... . - Amiram Eldar, Sep 02 2022

			The third odd unitary abundant number is 21945. Hence a(3) = 21945.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A034683, A129486, A034460, A034448, A129487, A002827, A129468, A112643.

# see A034683 for the code of isA034683()
isA129485 := proc(n)
    type(n,'odd') and isA034683(n) ;
end proc:
for n from 1 do
    if isA129485(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

UnitaryDivisors[n_Integer?Positive]:=Select[Divisors[n],GCD[ #,n/# ]==1&];sstar[n_]:=Plus@@UnitaryDivisors[n]-n;Select[Range[1,10^5,2],sstar[ # ]># &]

This sequence contains the odd members of A034683. i.e. odd numbers with a positive unitary abundance (A129468).

A249263 Primitive, odd, squarefree abundant numbers.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035, 116655, 118965

Offset: 1

Derek Orr, Oct 23 2014

nonn

The subsequence of primitive terms (not multiples of smaller terms) of A112643.
The subsequence of squarefree terms of A006038.
The subsequence of odd terms of A249242.
Not the same as A129485. Does not contain, for example, 195195, 255255, 285285, 333795, 345345, 373065, which are in A129485. - R. J. Mathar, Nov 09 2014
Sequences A287590, A188342 and A287581 list the number, smallest* and largest of all squarefree odd primitive abundant numbers with n prime factors. (*At least whenever A188342(n) is squarefree, which appears to be the case for all n >= 5.) - M. F. Hasler, May 29 2017

Giovanni Resta, Table of n, a(n) for n = 1..10000

Intersection of A112643 and A006038.
Cf. A249242, A129485.
Cf. A188342 (least with n factors), A287581 (largest with n factors), A287590 (number of terms with n factors).

# see A112643 and A006038 for the coding of isA112643 and isA006038
isA249263 := proc(n)
    isA112643(n) and isA006038(n) ;
end proc:
for n from 1 do
    if isA249263(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

PrimAbunQ[n_] := Module[{x, y},
   y = Most[Divisors[n]]; x = DivisorSigma[1, y];
   DivisorSigma[1, n] > 2 n  &&  AllTrue[x/y, # <= 2  &]];
Select[Range[1, 120000, 2], PrimAbunQ[#] &&
AllTrue[FactorInteger[#][[All, 2]], # == 1 &]  &] (* Robert Price, Sep 26 2019 *)

v=[]; for(k=1, 10^5, n=2*k+1; if(issquarefree(n) && sigma(n)>2*n, for(i=1, #v, n%v[i] || next(2)); print1(n, ", "); v=concat(v, n))) \\ Improved (from 20 sec to 0.2 sec) by M. F. Hasler, May 27 2017

A285993 Largest odd abundant number (A005231) equal to the product of n consecutive primes.

Original entry on oeis.org

15015, 255255, 4849845, 111546435, 33426748355, 1236789689135, 50708377254535, 2180460221945005, 102481630431415235, 5431526412865007455, 320460058359035439845, 19548063559901161830545, 1309720258513377842646515, 1357656019974967471687377449, 107254825578022430263302818471

Offset: 5

M. F. Hasler, Apr 30 2017

nonn

The smallest term is a(5) = 3*5*7*11*13, there is no odd abundant number (A005231) equal to the product of less than 5 consecutive primes.
The smallest odd abundant number (A005231) equal to the product of n consecutive primes is equal (when it exists, i.e., for n >= 5) to the least odd number with n (distinct) prime divisors, equal to the product of the first n odd primes = A070826(n+1) = A002110(n+1)/2.
See A188342 = (945, 3465, 15015, 692835, 22309287, ...) for the least odd primitive abundant number (A006038) with n distinct prime factors, and A275449 for the least odd primitive abundant number with n prime factors counted with multiplicity.
The terms are in general not primitive abundant numbers (A091191), in particular this cannot be the case when a(n) is a multiple of a(n-1), as is the case for most of the terms, for which a(n) = a(n-1)*A117366(a(n-1)). In the other event, spf(a(n)) = nextprime(spf(a(n-1))), and a(n) is in A007741(2,3,4...). These are exactly the primitive terms in this sequence.

			For n < 5, there is no odd abundant number equal to the product of n distinct primes.
For 5 <= n <= 8, the largest odd abundant number equal to the product of n consecutive primes is 3*...*prime(n+1).
For 9 <= n <= 17, the largest odd abundant number equal to the product of n consecutive primes is 5*...*prime(n+2).
For 18 <= n <= 30, the largest odd abundant number equal to the product of n consecutive primes is 7*...*prime(n+3).
For 31 <= n <= 45, the largest odd abundant number equal to the product of n consecutive primes is 11*...*prime(n+4).
For 46 <= n <= 66, the largest odd abundant number equal to the product of n consecutive primes is 13*...*prime(n+5).

Amiram Eldar, Table of n, a(n) for n = 5..341

Cf. A005231, A006038, A007707, A007708, A007741.

A subsequence of A112643 (odd squarefree abundant numbers); see also A108227 (~ A107705) which give indices of primitive terms = those with smallest prime factor larger than that of earlier terms.

a(r,f=vector(r,i,prime(i+1)),o)={ while(sigma(factorback(f),-1)>2, o=f; f=concat(f[^1],nextprime(f[r]+1)));factorback(o)} \\ Intentionally throws an error when n < 5.

a(n) >= a(n-1)*p where p = A117366(a(n-1)) = A151800(A006530(a(n-1))) = nextprime(gpf(a(n-1))), an odd abundant number equal to the product of n consecutive primes. We have strict inequality for n = 9, 18, 31, 46, 67, ..., in which case a(n) = a(n-1)*p*p'/q, where p' = nextprime(p), q = least prime factor of a(n-1). This is the case if a(n) is in A007741.

A360526 Odd numbers k such that A360522(k) > 2*k.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035, 116655

Offset: 1

Amiram Eldar, Feb 10 2023

nonn

First differs from A112643, A129485, A249263 at n = 46: a(46) = 165165 is not a term of these sequences.

			15015 is a term since A360522(15015) = 32256 > 2*15015.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A360522.
Subsequence of A005101, A005231 and A360525.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A348275, A348605.

f[p_, e_] := p^e + e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := s[n] > 2*n; Select[Range[1, 10^5, 2], q]

isab(n) = { my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,2]) > 2*n;}
is(n) = n%2 && isab(n);

A129486 Odd unitary abundant numbers that are not odd, squarefree, ordinary abundant numbers.

Original entry on oeis.org

195195, 333795, 416955, 1786785, 1996995, 2417415, 2807805, 3138135, 3318315, 3708705, 3798795, 4103715, 4339335, 4489485, 4789785, 4967655, 5120115, 5420415, 5552085, 5660655, 5731635, 6051045, 6111105, 6263565, 6342105, 6695535, 6771765, 6938295, 7000455, 7088235

Offset: 1

Ant King, Apr 17 2007

The first 50 members of A129485 and A112643 are the same. However, the sequences differ thereafter and this sequence contains those integers that are included in A129485 but are not included in A112643.

			The third integer which is an odd unitary abundant number but is not an ordinary, squarefree abundant number is 416955. Hence a(3)=416955.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein, Unitary Divisor.

Cf. A034683, A129485, A034460, A034448, A129487, A002827, A129468, A112643.

UnitaryDivisors[ n_Integer?Positive ] := Select[ Divisors[ n ], GCD[ #, n/# ] == 1 & ]; sstar[ n_ ] := Plus @@ UnitaryDivisors[ n ] - n; UnitaryAbundantNumberQ[ k_ ] := If[ sstar[ k ] > k, True, False ]; data1 = Select[ Range[ 1, 10^7, 2 ], UnitaryAbundantNumberQ[ # ] & ]; data2 = Select[ Range[ 1, 10^7, 2 ], DivisorSigma[ 1, # ] - 2 # > 0 && ! MoebiusMu[ # ] == 0 & ]; Complement[ data1, data2 ]
uaQ[n_] := Module[{f = FactorInteger[n]}, Max[f[[;;,2]]] > 1 && Times@@(1 + Power @@@ f) > 2n]; Select[Range[3, 2*10^6, 2], uaQ] (* Amiram Eldar, May 13 2019 *)

The complement of A129485 and A112643.

More terms from Amiram Eldar, May 13 2019

A339936 Odd coreful abundant numbers: the odd terms of A308053.

Original entry on oeis.org

99225, 165375, 231525, 297675, 496125, 694575, 826875, 893025, 1091475, 1157625, 1225125, 1289925, 1488375, 1620675, 1686825, 1819125, 1885275, 2083725, 2149875, 2282175, 2480625, 2546775, 2679075, 2811375, 2877525, 3009825, 3075975, 3142125, 3274425, 3472875

Offset: 1

Amiram Eldar, Dec 23 2020

nonn

The asymptotic density of this sequence is Sum_{n>=1} f(A356871(n)) = 9.1348...*10^(-6), where f(n) = (4/(Pi^2*n))*Product_{prime p|n}(p/(p+1)) if n is odd and 0 otherwise. - Amiram Eldar, Sep 02 2022

			99225 is a term since it is odd and the sum of its coreful divisors is A057723(99225) = 201600 > 2 * 99225.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A005408 and A308053.
Subsequence of A321147.
Cf. A057723, A356871.
Similar sequences: A005231, A094889, A112643, A127666, A129485, A293186, A321147, A333950.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[1, 10^6, 2], s[#] > 2*# &]

A339938 Odd non-coreful abundant numbers: the odd terms of A308127.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 75075, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 105105

Offset: 1

Amiram Eldar, Dec 23 2020

nonn

First differs from A112643, A129485 and A249263 at n = 28.

			15015 is a term since it is odd and the sum of its non-coreful divisors is A308135(15015) = 17241 > 15015.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A005408 and A308127.
Cf. A308135.
Similar sequences: A005231, A094889, A112643, A127666, A129485, A249263, A293186, A321147, A333950.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; s[1] = 0; s[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]); Select[Range[1, 10^5, 2], s[#] > # &]

A379031 Odd modified exponential abundant numbers: odd numbers k such that A241405(k) > 2*k.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035, 116655

Offset: 1

Amiram Eldar, Dec 14 2024

First differs from its subsequences A112643 and A249263 at n = 51: a(51) = 195195 is not a term of these two sequences.
First differs from its subsequence A129485 at n = 363: a(363) = 2537535 is not a term of A129485.
First differs from A339938 at n = 28: A339938(28) = 75075 is not a term of this sequence.
First differs from A360526 at n = 46: A360526(46) = 165165 is not a term of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A005408 and A379029.
Subsequence of A005231.
Subsequences: A112643, A129485, A249263.
Cf. A241405.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A339938, A348275, A360526.

f[p_, e_] := DivisorSum[e + 1, p^(# - 1) &]; mesigma[1] = 1; mesigma[n_] := Times @@ f @@@ FactorInteger[n]; meAbQ[n_] := mesigma[n] > 2*n; Select[Range[1, 10^5, 2], meAbQ]

is(k) = if(!(k%2), 0, my(f=factor(k)); prod(i=1, #f~, sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1))) > 2*k);

A112644 Odd and squarefree abundant numbers not divisible by 5.

Original entry on oeis.org

22309287, 28129101, 30069039, 34051017, 35888853, 36399363, 38057019, 39768729, 40681641, 41708667, 43444401, 45588543, 45894849, 48141093, 48555507, 50489439, 51294243, 51408357, 53804751, 54777723, 55186131, 56429373, 57228171, 58555497, 59168109

Offset: 1

Labos Elemer, Sep 20 2005

nonn

The least term that is not divisible by 3 is 73#/5# = Product_{k=4..21} prime(k) = 1357656019974967471687377449. - Amiram Eldar, Aug 15 2024

			99906807 = 3*7*11*13*17*19*103 is a term since it is an odd squarefree number that is not divisible by 5, and sigma(99906807) = 201277440 > 2*99906807.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A087248, A046391, A112643, A112642, A112640.

Cf. A005231, A047802.

ta={{0}};Do[g=n;s=DivisorSigma[1, n]-2*n; If[Greater[s, 0]&&Equal[Abs[MoebiusMu[n]], 1]&& !Equal[Mod[n, 2], 0]&&!Equal[Mod[n, 5], 0], Print[n, PrimeFactorList[n], s];ta=Append[ta, n]], {n, 10000000, 100000000}];{ta=Delete[ta, 1], g}

issfab(k) = my(f = factor(k)); issquarefree(f) && sigma(f, -1) > 2;
is(k) = gcd(k, 10) == 1 && issfab(k); \\ Amiram Eldar, Aug 15 2024

cp's OEIS Frontend

A087248 Squarefree abundant numbers.

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A129485 Odd unitary abundant numbers.

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A249263 Primitive, odd, squarefree abundant numbers.

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A285993 Largest odd abundant number (A005231) equal to the product of n consecutive primes.

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A360526 Odd numbers k such that A360522(k) > 2*k.

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A129486 Odd unitary abundant numbers that are not odd, squarefree, ordinary abundant numbers.

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A339936 Odd coreful abundant numbers: the odd terms of A308053.

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