A171641 -id:A171641 - cp's OEIS frontend

A376881 Numbers that have exactly one Zumkeller divisor.

6, 18, 20, 28, 70, 88, 100, 104, 196, 272, 304, 368, 464, 496, 550, 572, 650, 836, 945, 968, 1184, 1312, 1352, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944

Offset: 1

Peter Luschny, Oct 19 2024

nonn

d is a Zumkeller divisor of n if and only if d is a divisor of n and is Zumkeller (A083207).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Subsequence of A376880.

Cf. A083207, A376877, A376882, A023196, A171641.

# The function 'isZumkeller' is defined in A376880.
zdiv := n -> select(isZumkeller, NumberTheory:-Divisors(n)):
select(n -> nops(zdiv(n)) = 1, [seq(1..4000)]);

If d is the only Zumkeller divisor of n and n is Zumkeller then d = n.

A376882 a(n) is the product of the Zumkeller divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 72, 1, 1, 1, 1, 1, 6, 1, 20, 1, 1, 1, 1728, 1, 1, 1, 28, 1, 180, 1, 1, 1, 1, 1, 72, 1, 1, 1, 800, 1, 252, 1, 1, 1, 1, 1, 82944, 1, 1, 1, 1, 1, 324, 1, 1568, 1, 1, 1, 2592000, 1, 1, 1, 1, 1, 396, 1, 1, 1, 70, 1, 1728, 1, 1, 1, 1, 1, 468, 1, 64000

Offset: 1

Peter Luschny, Oct 19 2024

nonn

d is a Zumkeller divisor of n if and only if d is a divisor of n and is Zumkeller (A083207).

			The Zumkeller divisors of 80 are {20, 40, 80}, so a(80) = 64000.
The divisors of 81 are {1, 3, 9, 27, 81}, none of which is Zumkeller, so a(81) = 1.

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

Cf. A083207, A023196, A171641, A376880 (positions of terms > 1).

# The function 'isZumkeller' is defined in A376880.
with(NumberTheory):
zdiv := n -> select(isZumkeller, Divisors(n));
a := n -> mul(k, k in zdiv(n));
seq(a(n), n = 1..80);

A376882(n) = { my(m=1); fordiv(n,d,if(A083206(d)>0, m *= d)); (m); }; \\ Antti Karttunen, Dec 02 2024

A171642 Non-deficient numbers with odd sigma such that the sum of the even divisors is twice the sum of the odd divisors.

Original entry on oeis.org

18, 162, 450, 882, 1458, 2178, 2450, 3042, 4050, 5202, 6050, 6498, 7938, 8450, 9522, 11250, 13122, 15138, 17298, 19602, 22050, 24642, 27378, 30258, 33282, 36450, 39762, 43218, 46818, 50562, 54450, 58482, 61250, 62658, 66978, 71442, 76050, 80802, 85698

Offset: 1

Peter Luschny, Dec 14 2009

nonn

Numbers which are non-deficient (2n <= sigma(n)) [A023196] such that sigma(n) [A000203] is odd and the sum of the even divisors [A074400] is twice the sum of the odd divisors [A000593].

The sequence of terms which are not of the form 72*k^2 + 72*k + 18 starts: 2450, 6050, 8450, 61250, 120050, 151250, 211250, 296450.

			Divisors(18) = {1, 2, 3, 6, 9, 18}, sigma(18) = 39, and 2 + 6 + 18 = 2*(1 + 3 + 9).

Donovan Johnson, Table of n, a(n) for n = 1..1000
Peter Luschny, Zumkeller Numbers.

Cf. A000203, A023196, A074400, A000593.

Cf. A171641, A083207, A023196, A077591, A137933.

with(numtheory): A171642 := proc(n) local k,s,a;
s := sigma(n); a := add(k,k=select(x->type(x,odd),divisors(n)));
if 3*a = s and 2*n <= s and type(s,odd) then n else NULL fi end:

from sympy import divisors
A171642 = []
for n in range(1, 10**5):
    d = divisors(n)
    s = sum(d)
    if s % 2 and 2*n <= s and s == 3*sum([x for x in d if x % 2]):
        A171642.append(n)
# Chai Wah Wu, Aug 20 2014

A306476 Numbers k, with sigma(k) >= 3k and sigma(k) divisible by 3, that are not in A204830.

Original entry on oeis.org

10556208, 10578672, 10589904, 10612368, 10657296, 10690992, 10702224, 10747152, 10825776, 10859472, 10870704, 10881936, 10938096, 10949328, 10971792, 10983024, 11005488, 11039184, 11050416, 11095344, 11117808, 11196432, 11207664, 11252592, 11286288, 11319984

Offset: 1

Giovanni Resta, Feb 18 2019

nonn

From an idea of Amiram Eldar. Analogous sequence to A171641. The divisors of the listed terms k cannot be arranged in three disjoint sets each of them adding to sigma(k)/3.

Cf. A023197, A083207, A171641, A204830, A319186.

A337740 Weird numbers (A006037) with an even sum of divisors that are not Zumkeller numbers (A083207).

Original entry on oeis.org

73616, 682592, 2081824, 3963968, 4960448, 5440192, 6621632, 8000704, 8134208, 12979264, 31297472, 33736064, 43955584, 55691392, 58433152, 58904704, 160074368, 254533504, 263654656, 266828032, 267369728, 272240768, 352668416, 353383168, 357542656, 431462656, 530110208

Offset: 1

Amiram Eldar, Sep 17 2020

nonn

Non-deficient numbers (A023196) with an even sum of divisors (A000203) that are neither pseudoperfect numbers (A005835) nor Zumkeller numbers (A083207).

Equivalently, numbers k such that sigma(k) >= 2*k and sigma(k) == 0 (mod 2), such that no subset of the aliquot divisors of k sums to k or to sigma(k)/2.

			73616 is a term since sigma(73616) = 147312 is even and larger than 2 * 73616 = 147232. No subset of the aliquot divisors of 73616 sums to 73616 or to sigma(73616)/2 = 73656.

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

Intersection of A006037 and A171641.

Cf. A000203, A083207.

seqQ[n_] := Module[{d = Divisors[n], sum, c, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, c = CoefficientList[Product[1 + x^i, {i, d}], x]; c[[1 + 2*n]] == 0 && c[[1 + sum/2]] == 0]]; Select[Range[10^6], seqQ]

A339983 Coreful abundant numbers (A308053) with an even sum of coreful divisors (A057723) that are not coreful Zumkeller numbers (A339979).

Original entry on oeis.org

108, 216, 432, 540, 756, 864, 972, 1000, 1080, 1188, 1404, 1512, 1728, 1836, 1944, 2000, 2052, 2160, 2376, 2484, 2744, 2808, 3000, 3024, 3132, 3348, 3456, 3672, 3780, 3888, 3996, 4000, 4104, 4320, 4428, 4644, 4752, 4860, 4968, 5076, 5488, 5616, 5724, 5940, 6000

Offset: 1

Amiram Eldar, Dec 25 2020

nonn

			108 is a term since its set of coreful divisors, {6, 12, 18, 36, 54, 108}, has an even sum, 234 > 2*108, and it cannot be partitioned into two disjoint sets of equal sum.

Subsequence of A308053.
Cf. A057723, A339979, A339982.
Similar sequences: A171641, A323341, A323342, A323343, A323344.

q[n_] := Module[{r = Times @@ FactorInteger[n][[;; , 1]], d, sum, x}, d = r*Divisors[n/r]; (sum = Plus @@ d) >= 2*n && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] == 0]; Select[Range[10^4], q]

cp's OEIS Frontend

A376881 Numbers that have exactly one Zumkeller divisor.

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A376882 a(n) is the product of the Zumkeller divisors of n.

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A171642 Non-deficient numbers with odd sigma such that the sum of the even divisors is twice the sum of the odd divisors.

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A306476 Numbers k, with sigma(k) >= 3k and sigma(k) divisible by 3, that are not in A204830.

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A337740 Weird numbers (A006037) with an even sum of divisors that are not Zumkeller numbers (A083207).

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Mathematica

A339983 Coreful abundant numbers (A308053) with an even sum of coreful divisors (A057723) that are not coreful Zumkeller numbers (A339979).

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Mathematica