A129485 -id:A129485 - cp's OEIS frontend

A293186 Odd bi-unitary abundant numbers: odd numbers k such that bsigma(k) > 2*k, where bsigma is the sum of the bi-unitary divisors function (A188999).

945, 8505, 10395, 12285, 15015, 16065, 17955, 19305, 19635, 21735, 21945, 23205, 23625, 25245, 25515, 25935, 26565, 27405, 28215, 28875, 29295, 29835, 31185, 31395, 33345, 33495, 33915, 34125, 34155, 34965, 35805, 36855, 37125, 38745, 39585, 40635, 41055

Offset: 1

Amiram Eldar, Oct 01 2017

nonn

Analogous to odd abundant numbers (A005231) with bi-unitary sigma (A188999) instead of sigma (A000203).

The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are 1, 2, 82, 559, 6493, 61831, 642468, 6339347, 63112602, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00063... . - Amiram Eldar, Sep 02 2022

			945 is in the sequence since bsigma(945) = 1920 > 2*945.

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

Cf. A005231, A129485, A188999, A292982.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] :=
DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; bOddAbundantQ[n_] := OddQ[n] && bsigma[n] > 2 n; Select[Range[1000], bOddAbundantQ] (* after Michael De Vlieger at A188999 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
biusig(n) = vecsum(biudivs(n));
isok(n) = (n % 2) && (biusig(n) > 2*n); \\ Michel Marcus, Dec 15 2017

A321147 Odd exponential abundant numbers: odd numbers k whose sum of exponential divisors A051377(k) > 2*k.

Original entry on oeis.org

225450225, 385533225, 481583025, 538472025, 672624225, 705699225, 985646025, 1121915025, 1150227225, 1281998025, 1566972225, 1685513025, 1790559225, 1826280225, 2105433225, 2242496025, 2466612225, 2550755025, 2679615225, 2930852925, 2946861225, 3132081225

Offset: 1

Amiram Eldar, Oct 28 2018

nonn

From Amiram Eldar, Jun 08 2020: (Start)
Exponential abundant numbers that are odd are relatively rare: there are 235290 even exponential abundant number smaller than the first odd term, i.e., a(1) = A129575(235291).
Odd exponential abundant numbers k such that k-1 or k+1 is also exponential abundant number exist (e.g. (73#/5#)^2-1 and (73#/5#)^2 are both exponential abundant numbers, where prime(k)# = A002110(k)). Which pair is the least?
The least exponential abundant number that is coprime to 6 is (31#/3#)^2 = 1117347505588495206025. In general, the least exponential abundant number that is coprime to A002110(k) is (A007708(k+1)#/A002110(k))^2. (End)
The asymptotic density of this sequence is Sum_{n>=1} f(A328136(n)) = 5.29...*10^(-9), 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

			225450225 is in the sequence since it is odd and A051377(225450225) = 484323840 > 2 * 225450225.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, e-Divisor.
Eric Weisstein's World of Mathematics, e-Perfect Number.

The exponential version of A005231.
The odd subsequence of A129575.
Cf. A002110, A007708, A051377, A126164, A129485, A293186, A328136.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s={};Do[If[esigma[n]>2n,AppendTo[s,n]],{n,1,10^10,2}]; s

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

A348275 Odd noninfinitary abundant numbers: the odd terms of A348274.

Original entry on oeis.org

99225, 1091475, 1289925, 1334025, 1576575, 1686825, 1715175, 1863225, 1885275, 2027025, 2061675, 2282175, 2304225, 2395575, 2401245, 2436525, 2480625, 2650725, 2723175, 2789325, 2877525, 2962575, 3031875, 3075975, 3132675, 3185325, 3186225, 3296475, 3353805, 3501225

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

The number of terms below 10^k, for k = 5, 6, ..., are 1, 113, 630, 7771, 73685, ... Apparently this sequence has an asymptotic density 0.000007...

			99225 is a term since A348271(99225) = 107207 > 99225.

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

Cf. A348271.
Subsequence of A005231 and A348274.
Similar sequences: A094889, A127666, A129485, A293186, A321147.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := DivisorSigma[1,n] - isigma[n]; Select[Range[1, 2*10^6, 2], s[#] > # &]

A333950 Odd recursive abundant numbers: odd numbers k such that A333926(k) > 2*k.

Original entry on oeis.org

1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 8085, 8415, 8925, 9135, 9555, 9765, 11025, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 17325, 18585, 19215, 19635, 20475, 21105, 21945, 22365, 22995, 23205, 24255, 24885, 25935, 26145, 26565, 26775

Offset: 1

Amiram Eldar, Apr 11 2020

nonn

			1575 is a term since it is odd and A333926(1575) = 3224 > 2 * 1575.

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

Intersection of A005408 and A333928.
Cf. A333926.
Analogous sequences: A005231, A094889 (nonunitary), A129485 (unitary), A127666 (infinitary), A293186 (bi-unitary), A321147 (exponential).

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); Select[2*Range[15000] + 1, recDivSum[#] > 2*# &]

A329188 Odd unitary admirable numbers: the odd terms of A328328.

Original entry on oeis.org

80535, 354585, 403095, 430815, 437745, 442365, 5388495, 6126645, 9338595, 36340395, 130689195, 747242265, 1335049485, 2224695165, 4085490255, 9665740455, 10394173335, 11534750535, 13837748925, 33378237165, 73088757105, 94849396005, 109544822205, 216654032595

Offset: 1

Amiram Eldar, Nov 07 2019

nonn

Of the first 10^4 unitary admirable numbers only 6 are odd.

a(21) > 6*10^10.

Giovanni Resta, Table of n, a(n) for n = 1..43 (terms < 10^13)

The unitary version of A109729.
Intersection of A005408 and A328328.
Subsequence of A129485.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); uadmQ[n_] := (ab = usigma[n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && CoprimeQ[2*n/ab, ab/2]; s = {}; Do[If[uadmQ[n], AppendTo[s, n]], {n, 1, 10^6, 2}]; s

Data corrected by Amiram Eldar, May 12 2020

Terms a(21) and beyond from Giovanni Resta, May 12 2020

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

A335052 Odd unitary abundant numbers whose unitary abundancy is closer to 2 than that of any smaller odd unitary abundant number.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 31395, 33915, 39585, 41055, 45885, 51765, 80535, 83265, 354585, 359205, 361515, 366135, 382305, 389235, 400785, 403095, 407715, 414645, 416955, 423885, 430815, 437745, 442365, 77967015, 132335385, 617102535, 724239285, 1756753845

Offset: 1

Amiram Eldar, May 21 2020

nonn

The unitary abundancy of a number k is usigma(k)/k, where usigma(k) is the sum of unitary divisors of k (A034448).

			The unitary abundancies of the first terms are 2.148..., 2.112..., 2.099..., 2.085..., 2.072..., ...

Cf. A034448, A129485, A188263, A302570, A335053, A335055.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); seq = {}; r = 3; Do[s = usigma[n]/n; If[s > 2 && s < r, AppendTo[seq, n]; r = s], {n, 1, 10^6, 2}]; seq

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*# &]

cp's OEIS Frontend

A293186 Odd bi-unitary abundant numbers: odd numbers k such that bsigma(k) > 2*k, where bsigma is the sum of the bi-unitary divisors function (A188999).

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A321147 Odd exponential abundant numbers: odd numbers k whose sum of exponential divisors A051377(k) > 2*k.

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A249263 Primitive, odd, squarefree abundant numbers.

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A348275 Odd noninfinitary abundant numbers: the odd terms of A348274.

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A333950 Odd recursive abundant numbers: odd numbers k such that A333926(k) > 2*k.

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A329188 Odd unitary admirable numbers: the odd terms of A328328.

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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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