A080257 -id:A080257 - cp's OEIS frontend

A084190 Least common multiple of {d-1: d > 1 and d divides n}.

1, 1, 2, 3, 4, 10, 6, 21, 8, 36, 10, 330, 12, 78, 28, 105, 16, 680, 18, 684, 60, 210, 22, 53130, 24, 300, 104, 702, 28, 36540, 30, 3255, 160, 528, 204, 157080, 36, 666, 228, 62244, 40, 31980, 42, 9030, 616, 990, 46, 2497110, 48, 3528, 400, 5100, 52, 468520

Offset: 1

Reinhard Zumkeller, May 18 2003

nonn

Considering the set of divisors > 1 of n reduced by 1, a(n) is the smallest number whose divisors contain this set;
a(n) < n iff n=p^k, p prime and 1 <= k <= 2: a(A001248(n)) < A001248(n), a(A000430(n)) < A000430(n), a(A080257(n))> A080257(n);
a(n) is odd iff n=2^k.

			n=35: divisors > 1 of 35 = {5,7,35}, a(35) = lcm(4,6,34) = 204;
n=37: divisors > 1 of 37 = {37}, a(37) = lcm(36) = 36.

Carl R. White, Table of n, a(n) for n = 1..10000

Cf. A084191(n) = a(a(n)), A007955.
Cf. A027750.
Cf. A258409.

a084190 1 = 1
a084190 n = foldl1 lcm $ map (subtract 1) $ tail $ a027750_row' n
-- Reinhard Zumkeller, May 08 2012

Join[{1}, Table[LCM @@ (Rest[Divisors[n]] - 1), {n, 2, 100}]] (* T. D. Noe, Apr 25 2012 *)

a(n)=if(n>2,lcm(apply(k->k-1,vecextract(divisors(n),"2.."))),1) \\ Charles R Greathouse IV, Apr 25 2012

from math import lcm
from sympy import divisors
def A084190(n): return lcm(*(d-1 for d in divisors(n,generator=True) if d > 1)) # Chai Wah Wu, Jun 25 2022

a(45) was erroneously split into 61 and 6; repaired by Carl R. White, Apr 25 2012

A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A080257 in lacking 60.

			Strict factorizations of selected terms:
  (6)    (12)   (24)     (48)     (216)
  (2*3)  (2*6)  (3*8)    (6*8)    (3*72)
         (3*4)  (4*6)    (2*24)   (4*54)
                (2*12)   (3*16)   (6*36)
                (2*3*4)  (4*12)   (8*27)
                         (2*3*8)  (9*24)
                         (2*4*6)  (12*18)
                                  (2*108)
                                  (3*8*9)
                                  (4*6*9)
                                  (2*3*36)
                                  (2*4*27)
                                  (2*6*18)
                                  (2*9*12)
                                  (3*4*18)
                                  (3*6*12)
                                  (2*3*4*9)

The version for strict integer partitions is A035359.
The version for integer partitions is A046063.
The version for set partitions is A051130.
The non-strict version is A330991.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
Numbers whose number of strict factorizations is odd are A331230.
Numbers whose number of strict factorizations is even are A331231.
The least number with n strict factorizations is A330974(n).
Cf. A001318, A045780, A318286, A328966, A330992, A330993, A330997, A331023/A331024, A331050, A331051, A331200, A331232.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],PrimeQ[Length[strfacs[#]]]&]

A088383 Numbers greater than the 4th power of their smallest prime factor.

Original entry on oeis.org

18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117, 118, 120, 122, 123, 124, 126

Offset: 1

Reinhard Zumkeller, Sep 28 2003

nonn

a(n) > A020639(a(n))^4 = A088379(a(n)); complement of A088382.

a(n) > A088382(k) for n <= 67, a(n) < A088382(k) for n > 67.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A088381, A080257, A020639, A088379, A088382.

Positions of numbers greater than 4 in A307908.

a088383 n = a088383_list !! (n-1)
a088383_list = [x | x <- [1..], x  a020639 x ^ 4]
-- Reinhard Zumkeller, Feb 06 2015

Select[Range[200],#>(FactorInteger[#][[1,1]])^4&] (* Harvey P. Dale, Aug 15 2015 *)

A371734 Maximal length of a factorization of n into factors > 1 all having different sums of prime indices.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 3, 3

Offset: 1

Gus Wiseman, Apr 13 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Sum of prime indices is given by A056239.

Factorizations into factors > 1 all having different sums of prime indices are counted by A321469.

			The factorizations of 90 of this type are (2*3*15), (2*5*9), (2*45), (3*30), (5*18), (6*15), (90), so a(90) = 3.

For set partitions of binary indices we have A000120, same sums A371735.
Positions of 1's are A000430.
Positions of terms > 1 are A080257.
Factorizations of this type are counted by A321469, same sums A321455.
For same instead of different sums we have A371733.
A001055 counts factorizations.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
Cf. A035470, A279787, A305551, A321142, A322794, A326515, A326518, A326534, A336137, A371783, A371789, A371791.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Table[Max[Length/@Select[facs[n],UnsameQ@@hwt/@#&]],{n,100}]

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
all_have_different_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == #facs));
A371734(n, m=n, facs=List([])) = if(1==n, if(all_have_different_sum_of_pis(facs),#facs,0), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s = max(s,A371734(n/d, d, newfacs)))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A293575 Difference between the number of proper divisors of n and the number of squares dividing n.

Original entry on oeis.org

-1, 0, 0, 0, 0, 2, 0, 1, 0, 2, 0, 3, 0, 2, 2, 1, 0, 3, 0, 3, 2, 2, 0, 5, 0, 2, 1, 3, 0, 6, 0, 2, 2, 2, 2, 4, 0, 2, 2, 5, 0, 6, 0, 3, 3, 2, 0, 6, 0, 3, 2, 3, 0, 5, 2, 5, 2, 2, 0, 9, 0, 2, 3, 2, 2, 6, 0, 3, 2, 6, 0, 7, 0, 2, 3, 3, 2, 6, 0, 6, 1, 2, 0, 9, 2, 2, 2, 5, 0, 9, 2, 3, 2, 2, 2, 8, 0, 3, 3, 4, 0, 6, 0, 5, 6

Offset: 1

Juri-Stepan Gerasimov, Oct 14 2017

sign

The difference between the number of ways of writing n = m + k and the number of ways of writing n = r*s, where m|k and r|s.

First occurrence of k beginning with k=-1: 1, 2, 8, 6, 12, 36, 24, 30, 72, 96, 60, 2097152, 216, 576, 120, 210, 1152, 240, 864, etc. - Robert G. Wilson v, Nov 28 2017

			a(6) = 2 because 2 is difference of number of ways of writing n = 1 + 5 = 2 + 4 = 3 + 3 where 1|5, 2|4, 3|3 and number of ways of writing n = 1*6 where 1|6.

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

One less than A056595.
Cf. A000430, A032741, A046951, A166684, A080257.
Cf. A001620, A013661.

f[n_] := Block[{d = Divisors@ n}, Length@ d - Length[ Select[ d, IntegerQ@ Sqrt@# &]] - 1];; Array[f, 105] (* Robert G. Wilson v, Nov 28 2017 *)

a(n) = A032741(n) - A046951(n).
a(n) = A056595(n) - 1. - Antti Karttunen, Oct 30 2017
a(n) = 0 iff n is a prime or a square of a prime, A000430. - Robert G. Wilson v, Nov 28 2017
Sum_{k=1..n} a(k) ~ n*log(n) + (2*gamma - zeta(2) - 2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 01 2023

A348718 Numbers whose divisors can be partitioned into two disjoint sets without singletons whose arithmetic means are both integers.

Original entry on oeis.org

6, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96

Offset: 1

Amiram Eldar, Oct 31 2021

nonn

First differs from A343311 at n = 29.

Differs from A080257 which contains for example 8 and 128. - R. J. Mathar, Nov 03 2021

			6 is a term since its set of divisors, {1, 2, 3, 6}, can be partitioned into the two disjoint sets {1, 3} and {2, 6} whose arithmetic means, 2 and 4 respectively, are both integers.

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

Cf. A003601, A027750, A057020, A057021, A083207, A343311, A348715.

amQ[d_] := IntegerQ @ Mean[d]; q[n_] := Module[{d = Divisors[n], nd, s, subs, ans = False}, nd = Length[d]; subs = Subsets[d]; Do[s = subs[[k]]; If[Length[s] > 1 && Length[s] <= nd/2 && amQ[s] && amQ[Complement[d, s]], ans = True; Break[]], {k, 1, Length[subs]}]; ans]; Select[Range[100], q]

A087719 Least number m such that the number of numbers k <= m with k > spf(k)^n exceeds the number of numbers with k <= spf(k)^n.

Original entry on oeis.org

15, 27, 57, 135, 345, 927, 2577, 7335, 21225, 62127, 183297, 543735, 1618905, 4832127, 14447217, 43243335, 129533385, 388206927, 1163834337, 3489930135, 10466644665, 31393642527, 94168344657, 282479868135

Offset: 1

Reinhard Zumkeller, Sep 29 2003

nonn

mspf(k)^n & 1<=k<=m} <= m/2;

m>=a(n): #{k: k>spf(k)^n & 1<=k<=m} > m/2.

Cf. A088382, A088382, A088380, A088381, A000430, A080257.

Numbers so far satisfy a(n) = 3^n + 3*2^n + 6. - Ralf Stephan, May 10 2004

Empirical G.f.: 3*x*(5-21*x+20*x^2)/(1-x)/(1-2*x)/(1-3*x). - Colin Barker, Feb 22 2012

a(14)-a(24) from Giovanni Resta, May 23 2013

A089748 Numbers k that divide (sum of proper divisors of k + product of proper divisors of k).

Original entry on oeis.org

2, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240

Offset: 1

Joseph L. Pe, Jan 08 2004

All perfect numbers belong to this sequence.
Every term of A007691 is in this sequence. - T. D. Noe, Sep 29 2005
There are two sets of candidates of k: (i) k|A001065(k) and k|A007956(k) individually, or (ii) neither k|A001065(k) nor k|A007956(k) but the remainders of A001065(k)/k and A007956(k)/k sum up to k. If k has at least 4 divisors, the product of the second and penultimate divisor (in the sorted divisors list) is k, so k|A007956(k). This means for all k in A080257 we have k|A007956(k), and the k that do not divide A007956(k) are in A000430, which means k=p or k=p^2 for some prime p. If k=p, A001065(k)+A007956(k) = 1+1 =2, and the requirement here reduces to k|2 and only k=2 is left. If k=p^2, A001065(k) +A007956(k) = 1+p+p = 1+2*p, and the requirement here reduces to p^2 | (1+2*p), which has no solutions. This means case (ii) does not generate any solutions besides k=2. And this means all other solutions are from case (i), and therefore elements A007691 > 1 are the only remaining candidates. - R. J. Mathar, Oct 15 2021

Cf. A001065, A007956, A007691, A080257 (k which divide A007691(k)).

Cf. A219544.

isA087948 := proc(n)
    if modp( A001065(n)+A007956(n),n) = 0 then
        true;
    else
        false;
    end if;
end proc:
for n from 2 do
    if isA087948(n) then
        printf("%d\n",n) ;
    end if;
end do: # R. J. Mathar, Oct 15 2021

l = {}; Do[d = Drop[Divisors[n], -1]; p = Apply[Plus, d]; t = Apply[Times, d]; m = Mod[p + t, n]; If[m == 0, l = Append[l, n]], {n, 2, 10^6}]; l
Select[Range[2,22*10^5],Mod[Total[Most[Divisors[#]]]+Times@@Most[Divisors[#]],#]==0&] (* The program generates the first 11 terms of the sequence. *) (* Harvey P. Dale, Jun 05 2024 *)

from math import prod
from sympy import divisors
def ok(n): d = divisors(n)[:-1]; return n > 1 and (sum(d) + prod(d))%n == 0
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Oct 15 2021

a(11)-a(16) from Michael S. Branicky, Oct 16 2021

A325269 Number of integer partitions of n with 2 distinct parts or at least 3 parts.

Original entry on oeis.org

0, 0, 0, 2, 3, 6, 9, 14, 20, 29, 40, 55, 75, 100, 133, 175, 229, 296, 383, 489, 625, 791, 1000, 1254, 1573, 1957, 2434, 3009, 3716, 4564, 5602, 6841, 8347, 10142, 12308, 14882, 17975, 21636, 26013, 31184, 37336, 44582, 53172, 63260, 75173, 89133, 105556

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The Heinz numbers of these partitions are given by A080257.

Partitions with 2 distinct parts are in A002133(n). Partitions with at least 3 parts are in A004250(n). Some partitions are in both subsets, so A002133(n)+A004250(n) >= a(n). - R. J. Mathar, Dec 13 2022

			The a(1) = 1 through a(8) = 20 partitions:
  (21)   (31)    (32)     (42)      (43)       (53)
  (111)  (211)   (41)     (51)      (52)       (62)
         (1111)  (221)    (222)     (61)       (71)
                 (311)    (321)     (322)      (332)
                 (2111)   (411)     (331)      (422)
                 (11111)  (2211)    (421)      (431)
                          (3111)    (511)      (521)
                          (21111)   (2221)     (611)
                          (111111)  (3211)     (2222)
                                    (4111)     (3221)
                                    (22111)    (3311)
                                    (31111)    (4211)
                                    (211111)   (5111)
                                    (1111111)  (22211)
                                               (32111)
                                               (41111)
                                               (221111)
                                               (311111)
                                               (2111111)
                                               (11111111)

Cf. A001221, A001222, A001358, A001399, A007774, A008284, A060687, A080257, A090858, A116608, A325244.

Cf. A000041, A002133, A004250.

A325269 := proc(n)
    local a,p,s ;
    a := 0 ;
    for p in combinat[partition](n) do
        s := convert(p,set) ;
        if nops(p) >= 3 or nops(s) = 2 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A325269(n),n=0..40) ; # R. J. Mathar, Dec 13 2022

Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==2||Length[#]>2&]],{n,0,30}]

conjecture: a(n) = A000041(n) - A000034(n-1), n>0. - R. J. Mathar, Dec 13 2022

A338112 Least number that is both the sum and product of n distinct positive integers.

Original entry on oeis.org

1, 3, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 1

Rick L. Shepherd, Oct 10 2020

Each a(n) = n! except that a(2) = 1+2 = 3. For n > 0, only each integer >= A000217(n) is the sum of n distinct positive integers. For the integers that are products of these types, see below.

			a(1) = 1 because we define sums and products as sum(m) := prod(m) := m for all integers m in this case where these normally-binary operations only have one operand.
a(3) = 6 because 6 = 1+2+3 = 1*2*3 (with all the distinct positive integers the same in the sum and the product only for this term and a(1)).
a(5) = 120 because 120 = 1+2+3+4+110 (= ... = 22+23+24+25+26) = 1*2*3*4*5.

Cf. A000142, A000217.

Cf. Products of k distinct positive integers: A000027 (k=1), A020725 (k=2), A080257 (k=3), A122181 (k=4).

Array[If[# <= 2, (#^2 - #)/2 &[# + 1], #!] &, 22] (* Michael De Vlieger, Oct 15 2020 *)
With[{nn=30},Rest[CoefficientList[Series[x (2+x-x^2)/(2(1-x)),{x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Aug 10 2021 *)

PARI
```
a(n) = if(n<1, , if(n==2, 3, n!))
```

a(n) = A000142(n) for n = 1 and n > 2; a(2) = 3.
a(n) = max(A000142(n), A000217(n)).
E.g.f.: x*(2 + x - x^2)/(2*(1 - x)). - Stefano Spezia, Oct 11 2020

cp's OEIS Frontend

A084190 Least common multiple of {d-1: d > 1 and d divides n}.

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A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.

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A088383 Numbers greater than the 4th power of their smallest prime factor.

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A371734 Maximal length of a factorization of n into factors > 1 all having different sums of prime indices.

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A293575 Difference between the number of proper divisors of n and the number of squares dividing n.

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A348718 Numbers whose divisors can be partitioned into two disjoint sets without singletons whose arithmetic means are both integers.

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A087719 Least number m such that the number of numbers k <= m with k > spf(k)^n exceeds the number of numbers with k <= spf(k)^n.

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A089748 Numbers k that divide (sum of proper divisors of k + product of proper divisors of k).

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