Ivan N. Ianakiev - cp's OEIS frontend

A384515 Unitary s-Zumkeller numbers.

60, 140, 420, 660, 1224, 1820, 2660, 2820, 4620, 5460, 7140, 7980, 8580, 9660, 11220, 12180, 12540, 13020, 13260, 13580, 13860, 14140, 14420, 14820, 15180, 15540, 16380, 17220, 17940, 18060, 18200, 19140, 19380, 19740, 20020, 20460, 22260, 22620, 23460, 24180, 24420

Offset: 1

Ivan N. Ianakiev, Jun 01 2025

nonn

A positive integer k is called a unitary s-Zumkeller number if the set D of proper positive unitary divisors of k can be partitioned as {A,B}, such that the sum of the squares of the elements of A equals the sum of the squares of the elements of B equals (sigma2U(k) - k^2)/2, where sigma2U(k) is the sum of the squares of the unitary divisors of k.

All terms are even numbers.

			The set of proper unitary divisors of 60 is A = {1,3,4,5,12,15,20}. The set of their squares is B= {1,9,16,25,144,225,400} and the sum of its elements is 820. B = {1,9,400} union {16,25,144,225}, which makes 60 a term of the present sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program.
Jayanta Kalita and Helen K. Saikia, A Study On Generalized Zumkeller Numbers, INTEGERS, Volume 24 (2024), A95, pp. 7-10.

Cf. A034448, A077610, A083207, A290466.

(* Naive code to illustrate the terms found by Kalita and Saikia *)
uDiv[n_]:=Block[{d=Divisors[n]},Select[d,GCD[#,n/#]==1&]];
sigma2UDiv[n_]:=Total[uDiv[n]^2]; propUDiv[n_]:=uDiv[n]//Most;
subsetsPropUDivQ[1]:=False; subsetsPropUDivQ[n_]:=
Select[Subsets[propUDiv[n]],Total[Flatten[#]^2]==(sigma2UDiv[n]-n^2)/2&]!={};
Select[Range[1820],subsetsPropUDivQ[#]&]
(* or *)
q[n_] := Module[{d = Select[Divisors[n], # < n && CoprimeQ[#, n/#] &]^2, sum, x}, sum = Total[d]; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[2, 1000], q] (* Amiram Eldar, Jun 01 2025 *)

PARI
```
\\ See Corneth link
```

a(7)-a(16) from Amiram Eldar, Jun 01 2025

More terms from David A. Corneth, Jun 01 2025

A384493 Composite integers k such that sigma(k) | (k + 1)*tau(k) where tau is number of divisors of k.

Original entry on oeis.org

20, 35, 104, 207, 399, 464, 650, 1519, 1952, 2015, 2774, 2915, 2975, 4454, 11339, 22847, 32318, 63503, 97019, 122499, 130304, 352835, 522752, 924482, 1949375, 7366463, 8382464, 9486399, 15857855, 30222023, 39992975, 49280399, 63483104, 65094623, 69291935, 95309054

Offset: 1

David A. Corneth and Ivan N. Ianakiev, May 31 2025

nonn

			104 is in the sequence as tau(104) = 8, sigma(104) = 210 and sigma(104) = 210 | 840 = (104 + 1) * 8 = (104 + 1) * tau(104).

Composites in A384354.

Select[Range[4, 2^20], And[CompositeQ[#1], Divisible[(#1 + 1)*#2, #3]] & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* Michael De Vlieger, May 31 2025 *)

is(n) = my(f = factor(n), nd = numdiv(f)); nd > 2 && ((n+1)*nd) % sigma(f) == 0

from sympy import divisors, isprime
def ok(n): return n > 3 and not isprime(n) and (n+1)*len(d:=divisors(n))%sum(d) == 0
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, May 31 2025

A384354 Numbers k such that the arithmetic mean of the divisors of k evenly divides k+1.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 20, 23, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 104, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 207, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Ivan N. Ianakiev, May 27 2025

nonn

A term k with a fractional arithmetic mean of divisors is allowed as long as that arithmetic mean evenly divides k+1.
There exist triples such as (19,20,21) and quadruples such as (1,2,3,4) of consecutive numbers where the arithmetic mean of the divisors of every earlier number evenly divides the immediately following number. Are there similar quintuples?
Contains every prime p since (1+p)/2 evenly divides 1+p. - Michael S. Branicky, May 29 2025

			2 is a term since (1+2)/2 = 3/2 and 3/2 evenly divides 3.
19 is a term since (1+19)/2 is 10 and 10 evenly divides 20.
20 is a term since (1+2+4+5+10+20)/6 = 7 and 7 evenly divides 21.

Cf. A000005, A000040 (subsequence), A000203.

fQ[n_]:=Divisible[n+1,Mean[Divisors[n]]]; Select[Range[300],fQ]

isok(k) = my(f=factor(k)); denominator((k+1)/(sigma(f)/numdiv(f))) == 1; \\ Michel Marcus, May 31 2025

from sympy import divisors
def ok(n): return n and (n+1)*len(d:=divisors(n))%sum(d) == 0
print([k for k in range(300) if ok(k)]) # Michael S. Branicky, May 29 2025

A382389 Numbers k such that k, prime(k) and primepi(reverse(prime(k))) are emirps (A006567).

Original entry on oeis.org

7673, 9001, 12491, 17749, 31481, 75041, 93887, 95881, 102061, 104479, 112621, 113557, 118429, 139999, 722713, 743891, 749927, 999133, 1001941, 1086353, 1115071, 1165511, 1233907, 1861913, 1861973, 1881697, 1927903, 1972259

Offset: 1

Ivan N. Ianakiev, Mar 23 2025

			Take for example the two emirps 133963 and 369331. Their indices (in A000040) are 12491 and 31481, which are also emirps. So, those indices are terms of the present sequence.

Cf. A006567 (supersequence).

emirpQ[n_]:=AllTrue[{n,IntegerReverse[n]},PrimeQ]&&!PalindromeQ[n];
Select[Prime[Range[10000]],AllTrue[{#,Prime[#],PrimePi[IntegerReverse[Prime[#]]]},emirpQ]&]

A381874 Numbers whose set of divisors can be partitioned: a) into two disjoint subsets with equal sums and cardinalities, and b) into two disjoint subsets with equal products and cardinalities.

Original entry on oeis.org

24, 30, 42, 54, 60, 66, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 186, 198, 204, 210, 216, 220, 222, 224, 228, 240, 246, 258, 260, 264, 270, 276, 280, 282, 306, 308, 312, 318, 330, 336, 340, 342, 348, 352, 354, 360, 364, 366, 372, 378, 380, 384, 390, 402

Offset: 1

Ivan N. Ianakiev, Mar 09 2025

nonn

If y is a term of this sequence, then so is p*y, where p is a prime that is coprime to y.

It seems that the maximum first difference is 24.

			Let D be the set of the divisors of 24, i.e. {1,2,3,4,6,8,12,24}. On the one hand (sums) D = {1,2,3,24} union {4,6,8,12}. On the other hand (products) D = {1,2,12,24} union {3,4,6,8}. So, 24 is a term.

Cf. A083207 (supersequence), A347063 (supersequence).

sumQ[set_]:=AnyTrue[Subsets[set,{Length[set]/2}],(Plus@@#==Total[Complement[set,#]]&)];
prodQ[set_]:=AnyTrue[Subsets[set,{Length[set]/2}],(Times@@#==Times@@Complement[set,#]&)];
Select[Range[500],sumQ[Divisors[#]]&&prodQ[Divisors[#]]&]//Quiet

A380980 Place 2n distinct positive integers on an n-gon, n on the vertices and n on the sides such that the sums of the three values on all sides are equal. a(n) is the minimal sum of all the integers used.

Original entry on oeis.org

21, 38, 55, 81, 105, 140

Offset: 3

Ivan N. Ianakiev, Feb 10 2025

			For n=7 the first 14 positive integers suffice. Their permutation is 1,11,7,8,4,9,6,10,3,14,2,12,5,13 and the sum on each side of the heptagon is 19.

Cf. A380853.

mod[n_]:=Append[n,First[n]];
plus[n_]:=Table[mod[n][[i]]+mod[n][[i+1]],{i,1,Length[mod[n]]-1}];
goodPermutations[n_]:=Select[Permutations[n],Length[plus[#]]==Length[Union[plus[#]]]&];
min[n_]:=Min[Max/@plus/@goodPermutations[Range[n]]];
bestPermutation[n_]:=Select[goodPermutations[Range[n]],Max[plus[#]]==min[n]&,1];
plusBP[n_]:=plus/@bestPermutation[n]; max[n_]:=Max[Max/@plusBP[n]];
unit[n_]:=max[n]+n+1; sum[n_]:=n*unit[n]-Total@@plusBP[n]+n*(n+1)/2; sum/@Range[3,8]

A380289 Unitary Double Zumkeller numbers: numbers whose set of unitary divisors can be partitioned into two disjoint sets with equal sums and equal cardinalities.

Original entry on oeis.org

30, 42, 66, 78, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822, 834, 858, 870, 894, 906, 910, 930, 942, 966, 978, 990

Offset: 1

Ivan N. Ianakiev, Jan 19 2025

nonn

Based on checking the first 151 terms of this sequence it seems it is a subsequence of A342398. The first number that belongs to A342398, but not to this sequence is 2394. It also seems a subsequence of Zumkeller numbers (A083207). It is not a subsequence of Sphenic numbers (A007304). For example, 150 = 2*3*5*5 does not belong to A007304.
If y is a term, then so is x*y, where x is coprime to y.
It seems that 12k+6 is a term, where k>0 and k == 0 or 2 mod 3. Verified for k <= 191.

			Let D be the set of unitary divisors of 210. D = {1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210} = {1,2,5,6,14,15,35,210}union{3,7,10,21,30,42,70,105}.

Bhabesh Das, On unitary Zumkeller numbers, Notes on Number Theory and Discrete Mathematics, Vol. 30, No. 2 (2024), pp. 436-442.
Eric Weisstein's World of Mathematics, Unitary Divisor Function.
Wikipedia, Unitary divisor.

Cf. A083207, A290466 (supersequence), A342398, A347063.

uDiv[n_]:=Block[{d=Divisors[n]},Select[d,GCD[#,n/#]==1&]];uZNQ[n_]:=Module[{d=uDiv[n],t,ds,x},ds=Plus@@d;If[Mod[ds,2]>0,False,t=CoefficientList[Product[1+x^i,{i,d}],x];t[[1+ds/2]]>0]];dZNQ[n_]:=Block[{div=uDiv[n]},!IntegerQ@Sqrt[n]&&MemberQ[Total/@Subsets[div,{Length@div/2}],Total@div/2]];Select[Range[1000],uZNQ[#]&&dZNQ[#]&]

A376550 Smallest primes that generate a record number of deranged primes (see Comments).

Original entry on oeis.org

2, 13, 197, 1097, 10259, 10273, 10369, 10723, 10739, 10937, 13729, 31729, 38791, 101267, 101273, 102139, 102359, 102367, 102397, 105379, 107839, 108793, 109387, 109537, 109873, 130579, 150379, 709831, 1002739, 1002973, 1003879, 1005937, 1008379, 1012369, 1012379

Offset: 1

Ivan N. Ianakiev, Nov 27 2024

A derangement is a permutation of the elements of a set in which no element appears in its original position. In the present case, elements and set stand for digits and number.
When counting deranged primes, leading zeros are allowed.
The respective number of deranged primes is: 0,1,2,5,6,8,9,10,12,14,15,17,18,...
From David A. Corneth, Nov 28 2024: (Start)
a(105) > 185000000. Likely a(105)..a(110) are 324908761, 364150279, 403157269, 504921367, 539021467, 724908361 and a(111) onwards > 10^9. If there is any term between 10^8 and 10^9 that is not listed in the b-file and not listed in the candidates I gave then any such term t meets A328447(t) > 103567778 and t > 18500000. (End)
The likely terms a(105)..a(110) above are indeed terms. - Michael S. Branicky, Nov 30 2024

			The smallest prime generating one deranged prime is 13, where the deranged prime is 31. The smallest prime generating two deranged primes is 197, where the deranged primes are 719 and 971. So, 13 and 197 are terms. Although 10753 is the smallest prime that generates 7 deranged primes it is not a term, since the smaller prime 10273 generates a record of 8 deranged primes.

Michael S. Branicky, Table of n, a(n) for n = 1..138
David A. Corneth, PARI program
Wikipedia, Derangement

Cf. A000040, A000166, A328447.

(* How to find a(4) among the 4-digit candidates, where z=9 is the number of derangements of a 4-element set *)
Derangements[list_]:=Module[{n=Length[list],perms,isDerangement},
perms=Permutations[list];isDerangement[perm_]:=And@@Table[perm[[i]]!=list[[i]],{i,n}]; Select[perms,isDerangement]];
numberOfPrimeDerangements[n_]:=Length[Select[FromDigits/@Derangements[IntegerDigits[n]],PrimeQ]];
listOf4digitCandidates=Table[{z,Select[Prime/@Range[169,1229],numberOfPrimeDerangements[#]==z&,1]},{z,3,9}]

PARI
```
\\ See Corneth link
```

from itertools import islice
from sympy import isprime, nextprime
from sympy.utilities.iterables import multiset_derangements as md
def f(n): return len(set(tuple(d) for d in md(list(str(n))) if isprime(int("".join(d)))))
def agen(): # generator of terms
    record, p = -1, 2
    while True:
        v = f(p)
        if v > record:
            yield p
            record = v
        p = nextprime(p)
print(list(islice(agen(), 26))) # Michael S. Branicky, Nov 27 2024

a(14) and beyond from Michael S. Branicky, Nov 27 2024

2 prepended by David A. Corneth, Nov 27 2024

A377156 Numbers k such that the set of divisors of k that are Zumkeller numbers can be partitioned into two disjoint subsets with equal sum.

Original entry on oeis.org

120, 168, 240, 336, 432, 480, 528, 624, 660, 672, 780, 864, 924, 960, 1020, 1056, 1092, 1140, 1248, 1260, 1296, 1320, 1344, 1380, 1428, 1560, 1596, 1632, 1728, 1740, 1760, 1800, 1824, 1848, 1920, 1932, 2040, 2080, 2100, 2112, 2184, 2208, 2280, 2436, 2464, 2496, 2520, 2592

Offset: 1

Ivan N. Ianakiev, Oct 18 2024

nonn

If k is a term, then so is 2k.

Below 6000 terms which are not Zumkeller are: 1296, 1800, 2592, 3528, 3600, 4050, 5184.

			The set D of the divisors of 120 that are Zumkeller numbers is {6,12,20,24,30,40,60,120}. D = {6,30,120} union {12,20,24,40,60}, so 120 is a term.

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

Cf. A083207.

zQ[n_]:=Module[{d=Divisors[n],t,ds,x},ds=Plus@@d;If[Mod[ds,2]>0,False,
t=CoefficientList[Product[1+x^i,{i,d}],x];t[[1+ds/2]]>0]];
zDiv[n_]:= Select[Divisors[n],zQ]; myQ[n_]:=Select[Subsets[zDiv[n]], #!={}&&EvenQ[(Plus@@zDiv[n])/2]&&Plus@@#==(Plus@@zDiv[n])/2&,1]!={};
Select[Range[336],myQ] (* zQ by T. D. Noe at A083207 *)
(* second program *)
zumQ[n_] := zumQ[n] = Module[{d = Divisors[n], sum, x}, sum = Plus @@ d; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; q[n_] := Module[{d = Select[Divisors[n], zumQ], sum, x}, sum = Plus @@ d; sum > 0 && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[2600], q] (* Amiram Eldar, Oct 19 2024 *)

More terms from Amiram Eldar, Oct 19 2024

A376862 Unitary Zumkeller numbers whose divisors can be partitioned into two disjoint subsets with equal sums and cardinalities.

Original entry on oeis.org

30, 42, 60, 66, 78, 90, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690

Offset: 1

Ivan N. Ianakiev, Oct 07 2024

A unitary divisor of n is a divisor d such that gcd(d,n/d)=1.
This sequence is an intersection of A290466 and A347063 and seemingly a subsequence of A293188.
From the facts: a) for n>2 every primorial(n), i.e. A002110(n), is a Zumkeller number, b) a(1) = 30 = 2*3*5 is primorial(3), c) if n is squarefree, than sigma(n) = usigma(n), d) the number of unitary divisors of n is 2^k, where k is the number of distinct prime factors of n, and e) p*y belongs to A347063, where p is a prime coprime to y and y belongs to A347063, it follows that the present sequence is infinite, since for m >= 3 primorial(m) is a term.
It seems that for k >= 0 all numbers of the form 30 + 36k are terms.

			The set of divisors of 90 is {1,2,3,5,6,9,10,15,18,30,45,90}, which is a union of the sets {1,2,3,6,15,90} and {5,9,10,18,30,45}, which have equal sums (117) and cardinalities (6). So, 90 is a term.

Eric Weisstein's World of Mathematics, Unitary Divisor Function
Wikipedia, Unitary divisor

Cf. A083207, A290466, A347063.

uzn=Cases[Import["https://oeis.org/A290466/b290466.txt","Table"],{,}][[All,2]];
dzn=Select[Range@700,!IntegerQ@Sqrt@#&&(d=Divisors@#; MemberQ[Total/@Subsets[d,{Length@d/2}],Total@d/2])&]; Intersection[uzn,dzn] (* Thanks to Giorgos Kalogeropoulos at A347063 *)

cp's OEIS Frontend

User: Ivan N. Ianakiev

A384515 Unitary s-Zumkeller numbers.

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A384493 Composite integers k such that sigma(k) | (k + 1)*tau(k) where tau is number of divisors of k.

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A384354 Numbers k such that the arithmetic mean of the divisors of k evenly divides k+1.

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A382389 Numbers k such that k, prime(k) and primepi(reverse(prime(k))) are emirps (A006567).

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A381874 Numbers whose set of divisors can be partitioned: a) into two disjoint subsets with equal sums and cardinalities, and b) into two disjoint subsets with equal products and cardinalities.

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Mathematica

A380980 Place 2n distinct positive integers on an n-gon, n on the vertices and n on the sides such that the sums of the three values on all sides are equal. a(n) is the minimal sum of all the integers used.

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Examples

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Mathematica

A380289 Unitary Double Zumkeller numbers: numbers whose set of unitary divisors can be partitioned into two disjoint sets with equal sums and equal cardinalities.

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Mathematica

A376550 Smallest primes that generate a record number of deranged primes (see Comments).

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Mathematica