A034683 -id:A034683 - 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] &]

A336671 Unitary barely 3-abundant: numbers m such that 3 < usigma(m)/m < usigma(k)/k for all numbers k < m, where usigma is the sum of unitary divisors function (A034448).

Original entry on oeis.org

30030, 39270, 43890, 46410, 51870, 62790, 67830, 79170, 82110, 91770, 103530, 161070, 166530, 709170, 718410, 723030, 732270, 764610, 778470, 801570, 806190, 815430, 829290, 833910, 847770, 861630, 875490, 884730, 155934030, 264670770, 1234205070, 1448478570

Offset: 1

Amiram Eldar, Jul 29 2020

nonn

The corresponding values of usigma(m)/m are 3.222..., 3.168...., 3.149..., 3.127..., 3.109..., ...

Amiram Eldar, Table of n, a(n) for n = 1..36

The unitary version of A259312.
Subsequence of A285615.
Cf. A034448, A034683, A302570, A336672.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); s = {}; rm = 4; Do[r = usigma[n]/n; If[r > 3 && r < rm, rm = r; AppendTo[s, n]], {n, 1, 10^5}]; s

A360525 Numbers k such that A360522(k) > 2*k.

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 90, 102, 114, 120, 126, 132, 138, 140, 150, 156, 168, 174, 180, 186, 204, 210, 222, 228, 246, 252, 258, 276, 282, 294, 300, 318, 330, 348, 354, 360, 366, 372, 390, 402, 420, 426, 438, 444, 462, 474, 492, 498, 510, 516, 534, 546, 564

Offset: 1

Amiram Eldar, Feb 10 2023

nonn

First differs from A308127 at n = 15.
Analogous to abundant numbers (A005101) with A360522 instead of A000203.
Subsequence of A005101 because A360522(n) <= A000203(n) for all n.
The least odd term is a(1698) = A360526(1) = 15015.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 8, 95, 1135, 10890, 110867, 1104596, 11048123, 110534517, 1105167384, 11051009278, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1105...

			30 is a term since A360522(30) = 72 > 2*30.

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

Cf. A000203, A308127, A360522, A360524, A360526.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

f[p_, e_] := p^e + e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := s[n] > 2*n; Select[Range[1000], q]

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

A374785 Numbers whose unitary divisors have a mean unitary abundancy index that is larger than 2.

Original entry on oeis.org

223092870, 281291010, 300690390, 6469693230, 6915878970, 8254436190, 8720021310, 9146807670, 9592993410, 10407767370, 10485364890, 10555815270, 11125544430, 11532931410, 11797675890, 11823922110, 12095513430, 12328305990, 12598876290, 12929686770, 13162479330

Offset: 1

Amiram Eldar, Jul 20 2024

nonn

Numbers k such that A374783(k)/A374784(k) > 2.
The least odd term is A070826(43) = 5.154... * 10^74, and the least term that is coprime to 6 is Product_{k=3..219} prime(k) = 1.0459... * 10^571.
The least nonsquarefree (A013929) term is a(613) = 148802944290 = 2 * 3 * 5 * 7 * 11 * 13 * 17 *19 * 23^2 * 29.
All the terms are nonpowerful numbers (A052485). For powerful numbers (A001694) k, A374783/(k)/A374784(k) < Product_{p prime} (1 + 1/(2*p)) = 1.242534... (A366586).

			223092870 is a term since A374783(223092870)/A374784(223092870) = 666225/330752 = 2.014... > 2.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Subsequence of A052485.
Cf. A001221, A001694, A013929, A034683, A070826, A366586, A374783, A374784.
Similar sequences: A245214, A374788.

f[p_, e_] := 1 + 1/(2*p^e); r[1] = 1; r[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[4*10^8], s[#] > 2 &]

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

A001221(a(n)) >= 9.

A063875 Numbers k such that sigma(k) - usigma(k) > 3k.

Original entry on oeis.org

831600, 1058400, 1587600, 1663200, 1814400, 1940400, 1965600, 2116800, 2328480, 2494800, 2570400, 2646000, 2721600, 2872800, 2910600, 2948400, 3024000, 3175200, 3326400, 3528000, 3603600, 3628800, 3704400, 3880800, 3931200

Offset: 1

Jason Earls, Aug 27 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..400

Cf. A034448, A048146, A034683, A063846.

u(n) = sumdiv(n,d, if(gcd(d,n/d)==1,d));
for(n=1,10^9, if(sigma(n)-u(n)>3*n,print(n)))

u(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d))
{ n=0; for (m=1, 10^9, if(sigma(m) - u(m) > 3*m, write("b063875.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 01 2009

a(20)-a(25) from Harry J. Smith, Sep 01 2009

A081400 a(n) = d(n) - bigomega(n) - A005361(n).

Original entry on oeis.org

0, 0, 0, -1, 0, 1, 0, -2, -1, 1, 0, 1, 0, 1, 1, -3, 0, 1, 0, 1, 1, 1, 0, 1, -1, 1, -2, 1, 0, 4, 0, -4, 1, 1, 1, 1, 0, 1, 1, 1, 0, 4, 0, 1, 1, 1, 0, 1, -1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 6, 0, 1, 1, -5, 1, 4, 0, 1, 1, 4, 0, 1, 0, 1, 1, 1, 1, 4, 0, 1, -3, 1, 0, 6, 1, 1, 1, 1, 0, 6, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 4, 0, 1, 4, 1, 0, 1, 0, 4, 1, 1, 0, 4, 1, 1, 1, 1, 1, 8, -1, 1

Offset: 1

Labos Elemer, Mar 28 2003

sign

			Negative for true prime powers; zero for 1 and primes; see also A030231, A007304, A034683, A075819 etc. to judge about positivity or magnitude.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A001222, A005361, A030231, A030230, A007304, A034683, A075819.

a(n) = my(f=factor(n)); numdiv(n) - bigomega(n) - prod(k=1, #f~, f[k,2]); \\ Michel Marcus, May 25 2017

from sympy import primefactors, factorint, divisor_count
from operator import mul
def bigomega(n): return 0 if n==1 else bigomega(n/primefactors(n)[0]) + 1
def a005361(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [f[i] for i in f])
def a(n): return divisor_count(n) - bigomega(n) - a005361(n) # Indranil Ghosh, May 25 2017

a(n) = A000005(n) - A001222(n) - A005361(n).

A306720 Even numbers that are not the sum of two unitary abundant numbers (not necessarily distinct).

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 62, 64, 66, 68, 70, 74, 76, 78, 80, 82, 86, 88, 90, 92, 94, 98, 102, 104, 106, 110, 114, 116, 118, 122, 124, 126, 128, 130, 134, 138, 142, 146, 150

Offset: 1

Amiram Eldar, Mar 06 2019

The unitary version of A048242.

a(6066) = 530086 is the last term. te Riele proved that every even number larger than 530086 is the sum of two unitary abundant numbers (not necessarily distinct). The corresponding sequence of odd numbers is also finite, but he did not calculate the last term, and only showed that it is below 2004452254833.

			Since the unitary abundant numbers begin with 30, 42, 66, 70, ... the first integers which are missing from this sequence are 60 = 30 + 30, 72 = 30 +42, 84 = 42 + 42, 96 = 30 + 66, 100 = 30 + 70, ...

Amiram Eldar, Table of n, a(n) for n = 1..6066
Herman J. J. te Riele, On the representation of the positive integers as the sum of two unitary abundant numbers, Stichting Mathematisch Centrum, Numerieke Wiskunde NW 19/75 (1975).

Cf. A034683, A048242.

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

A342399 Unitary pseudoperfect numbers k such that no subset of the nontrivial unitary divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} adds up to k.

Original entry on oeis.org

3510, 3770, 5670, 5810, 6790, 7630, 7910, 9590, 9730, 544310, 740870, 2070970, 4017310, 4095190, 5368510, 5569690, 5762330, 5838770, 5855290, 5856130, 5887630, 5902470, 5985770, 6006070, 6039530, 6075370, 6083630, 6181210, 6259610, 6471290, 7038710, 7065730, 7285390

Offset: 1

Amiram Eldar, Mar 10 2021

nonn

Numbers that are the sum of a proper subset of their aliquot unitary divisors but are not the sum of any subset of their nontrivial unitary divisors.
The unitary perfect numbers (A002827) which are a subset of the unitary pseudoperfect numbers (A293188) are excluded from this sequence since otherwise they would all be trivial terms: if k is a unitary perfect number then the sum of the divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} is k-1, so any subset of them has a sum smaller than k.
The unitary pseudoperfect numbers are thus a disjoint union of the unitary perfect numbers, this sequence and A342398.
The unitary abundant numbers (A034683) are a disjoint union of the unitary weird numbers (A064114), this sequence and A342398.

			3510 is a term since it is a unitary pseudoperfect number, 3510 = 1 + 2 + 5 + 13 + 27 + 54 + 65 + 130 + 135 + 270 + 351 + 702 + 1755, and the set of nontrivial unitary divisors of 3510, {d|3510 : 1 < d < 3510, gcd(d, 3510/d) = 1} = {2, 5, 10, 13, 26, 27, 54, 65, 130, 135, 270, 351, 702, 1755}, has no subset that adds up to 3510.

Amiram Eldar, Table of n, a(n) for n = 1..206

The unitary version of A339343.
Subsequence of A034683 and A293188.
Cf. A002827, A064114, A342398.

q[n_] := Module[{d = Most @ Select[Divisors[n], CoprimeQ[#, n/#] &], x}, Plus @@ d > n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0 && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, 2, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[10^4], q]

A361935 Numbers k such that k and k+1 are both primitive unitary abundant numbers (definition 2, A302574).

Original entry on oeis.org

2457405145194, 2601523139214, 3320774552094, 3490250769005, 3733421997305, 3934651766045, 3954730124345, 4514767592334, 4553585751714, 4563327473705, 5226433847634

Offset: 1

Amiram Eldar, Mar 31 2023

There are no more terms below 10^13.

There are no numbers k such that k and k+1 are both unitary abundant numbers with definition 1 (A302573) below 10^13.

Subsequence of A034683, A302574 and A331412.
Cf. A302573.
Similar sequences: A283418, A330872.

f1[p_, e_] := 1 + 1/p^e; f2[p_, e_] := p^e/(p^e + 1);
puabQ[n_] := (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f <= 2;
Select[Import["https://oeis.org/A331412/b331412.txt", "Table"][[;; , 2]], puabQ[#] && puabQ[# + 1] &]

cp's OEIS Frontend

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

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A336671 Unitary barely 3-abundant: numbers m such that 3 < usigma(m)/m < usigma(k)/k for all numbers k < m, where usigma is the sum of unitary divisors function (A034448).

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A360525 Numbers k such that A360522(k) > 2*k.

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A374785 Numbers whose unitary divisors have a mean unitary abundancy index that is larger than 2.

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A063875 Numbers k such that sigma(k) - usigma(k) > 3k.

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A081400 a(n) = d(n) - bigomega(n) - A005361(n).

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A306720 Even numbers that are not the sum of two unitary abundant numbers (not necessarily distinct).

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

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