A129468 -id:A129468 - cp's OEIS frontend

A335252 Numbers k such that k and k+2 have the same unitary abundance (A129468).

12, 63, 117, 323, 442, 1073, 1323, 1517, 3869, 5427, 6497, 12317, 18419, 35657, 69647, 79919, 126869, 133787, 151979, 154007, 163332, 181427, 184619, 333797, 404471, 439097, 485237, 581129, 621497, 825497, 1410119, 2696807, 3077909, 3751619, 5145341, 6220607

Offset: 1

Amiram Eldar, May 28 2020

nonn

Are 12, 442 and 163332 the only even terms?
Are there any unitary abundant numbers (A034683) in this sequence?
No further even terms up to 10^13. - Giovanni Resta, May 30 2020

			12 is a term since 12 and 14 have the same unitary abundance: A129468(12) = usigma(12) - 2*12 = 20 - 24 = -4, and A129468(14) = usigma(14) - 2*14 = 24 - 28 = -4.

Amiram Eldar, Table of n, a(n) for n = 1..300

The unitary version of A330901.

Cf. A034448, A034683, A129468, A129487, A335251.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); udef[n_] := 2*n - usigma[n]; Select[Range[10^5], udef[#] == udef[# + 2] &]

A335251 Numbers k such that k and k+1 have the same unitary abundance (A129468).

Original entry on oeis.org

1, 20, 35, 143, 208, 2623, 5183, 27796, 11177983, 69677008, 920158207, 1099508482048

Offset: 1

Amiram Eldar, May 28 2020

Are there any unitary abundant numbers (A034683) in this sequence?
a(12) > 10^11.
a(13) > 8*10^12. Also terms: 2^36 * 68719644673, 2^48 * 281474901625261, 2^64 * 18446632096776339457. - Giovanni Resta, May 29 2020

			1 is a term since 1 and 2 have the same unitary abundance: A129468(1) = usigma(1) - 2*1 = 1 - 2 = -1, and A129468(2) = usigma(2) - 2*2 = 3 - 4 = -1.

Cf. A034448, A034683, A129468, A129487, A331412, A335252.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); udef[n_] := 2*n - usigma[n]; Select[Range[30000], udef[#] == udef[# + 1] &]

a(12) from Giovanni Resta, May 29 2020

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).

A129487 Unitary deficient numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79

Offset: 1

Ant King, Apr 20 2007

The unitary deficient numbers account for almost 93% of all integers (including all primes (A000040) and prime powers (A000961)) and asymptotically satisfy a(n)~1.0753n. This provides an excellent fit as n grows larger. For example, the one millionth unitary deficient number is 1075293 and the asserted approximation returns 1075300, giving an error of only 0.00065%.

			The sixth integer that exceeds the sum of its proper unitary divisors is 7. Hence a(6)=7.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A034460, A034448, A129468, A034683, A000040, A000961.

a := proc(n) numtheory[divisors](n); select(d -> igcd(d,n/d)=1,%); `if`(add(i,i=%) < 2*n,n,NULL) end: # Peter Luschny, May 03 2009

UnitaryDivisors[n_Integer?Positive]:=Select[Divisors[n],GCD[ #,n/# ]==1&];Select[Range[100],Plus@@UnitaryDivisors[ # ]-2#<0 &]

Integers for which A034460(n) < n, or equivalently for which A034448(n) < 2n.

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

A129499 Records for unitary abundant numbers, i.e., those integers which set a record for having a greater unitary abundance than any of their predecessors.

Original entry on oeis.org

30, 210, 330, 390, 510, 570, 690, 870, 930, 1110, 1230, 1290, 1410, 1470, 1590, 1770, 1830, 2010, 2130, 2190, 2310, 2730, 3570, 3990, 4830, 5610, 6090, 6510, 7590, 7770, 8610, 9030, 9870, 11130, 12390, 12810, 14070, 14910, 15330, 16590, 17430, 18690

Offset: 1

Ant King, Apr 20 2007

			A129498 begins 12, 12, 12, 4, 12, 12, 12, 12, 12, 12, 12, 156, 12, 12. The second record value is 156, which occurs at position 12. As A034683(12)=210, it follows that a(2)=210.

Amiram Eldar, Table of n, a(n) for n = 1..750
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A034448, A034460, A034683, A129468, A129498.

UnitaryDivisors[n_Integer?Positive]:=Select[Divisors[n],GCD[ #,n/# ]==1&];sstar[n_]:=Plus@@UnitaryDivisors[n]-n;RunningMaxima[l_]:=Rest[FoldList[Max,-Infinity,l]] HighWaterMarks[l_]:=Module[{s=Split[RunningMaxima[l]]}, {First/@s,Most[FoldList[Plus,1,Length/@s]]} ];data1=Select[Range[20000],sstar[ # ]-#>0 &];data2=sstar[ # ]-# &/@data1;pos=Last[HighWaterMarks[data2]];champs=data1[[ # ]] &/@pos

Values of A034683 corresponding to those positions in A129498 at which records occur.

A129498 Unitary abundancy of n-th unitary abundant number: usigma(k)-2k if this is > 0.

Original entry on oeis.org

12, 12, 12, 4, 12, 12, 12, 12, 12, 12, 12, 156, 12, 12, 12, 12, 12, 12, 204, 12, 12, 228, 12, 120, 12, 12, 228, 12, 12, 276, 12, 252, 300, 12, 12, 12, 180, 12, 12, 120, 12, 348, 300, 12, 12, 12, 188, 120, 12, 324, 12, 12, 48, 300, 420, 12, 12, 196, 72, 444, 12, 372

Offset: 1

Ant King, Apr 20 2007

The transforms of this sequence are discussed in A129499.

			The fourth unitary abundant number is 70. As the unitary divisors of 70 are 1, 2, 5, 7, 10, 14, 35 and 70, we have a(4) = 1+2+5+7+10+14+35+70-2 * 70 = 4.

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

Cf. A034448, A034460, A129468, A034683, A129499.

uab[1]=-1; uab[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - 2n; seq={}; Do[u=uab[n]; If[u>0, AppendTo[seq, u]], {n, 1, 1000}]; seq (* Amiram Eldar, Jun 18 2019 *)

A034448(k)-2k = A034460(k)-k, whenever these are positive.

a(n) = A129468(A034683(n)). - Amiram Eldar, Jun 18 2019

cp's OEIS Frontend

A335252 Numbers k such that k and k+2 have the same unitary abundance (A129468).

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A335251 Numbers k such that k and k+1 have the same unitary abundance (A129468).

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A129485 Odd unitary abundant numbers.

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A129487 Unitary deficient numbers.

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

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A129499 Records for unitary abundant numbers, i.e., those integers which set a record for having a greater unitary abundance than any of their predecessors.

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A129498 Unitary abundancy of n-th unitary abundant number: usigma(k)-2k if this is > 0.

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