A222084 -id:A222084 - cp's OEIS frontend

A222086 a(n) is the least number k for which A000005(k)/A222084(k) = n.

1, 30, 360, 840, 11088, 18018, 1713600, 32760, 327600, 350064, 39437798400, 180180, 8532905472000, 47361600, 720720, 1750320

Offset: 1

Paolo P. Lava, Feb 12 2013

a(17) <= 214555365177753600. a(18) = 3423420, a(20) = 4084080, a(24) = 6126120, a(30) = 46558512, a(32) = 38798760. - Hiroaki Yamanouchi, Oct 03 2014

			For k=18018, tau(k)=48: the 48 divisors of k are 1, 2, 3, 6, 7, 9, 11, 13, 14, 18, 21, 22, 26, 33, 39, 42, 63, 66, 77, 78, 91, 99, 117, 126, 143, 154, 182, 198, 231, 234, 273, 286, 429, 462, 546, 693, 819, 858, 1001, 1287, 1386, 1638, 2002, 2574, 3003, 6006, 9009, 18018.
The least common multiple of the first 8 divisors, (1, 2, 3, 6, 7, 9, 11, 13), is again 18018, but the least common multiple of the first 7 divisors, (1, 2, 3, 6, 7, 9, 11), is less than 18018.
Since tau#(k)=8 (see A222084 for the definition of tau#(n)), tau(k)/tau#(k) = 48/8 = 6, and since 18018 is the minimum number k to have this ratio, a(6)=18018.

Cf. A000005, A000961, A001358, A003418, A005179, A024619, A034444, A077610, A222084, A222085.

with(numtheory);
A222086:=proc(q)
local a,b,c,d,j,n,t,v;
v:=array(1..100); for j from 1 to 100 do v[j]:=0; od; t:=0;
for n from 1 to q do
  a:=ifactors(n)[2]; b:=nops(a); c:=0;
  for j from 1 to b do if a[j][1]^a[j][2]>c then c:=a[j][1]^a[j][2]; fi; od;
  a:=op(sort([op(divisors(n))])); b:=nops(divisors(n));
  for j from 1 to b do if a[j]=c then break; fi; od;
  if type(tau(n)/j,integer)  then if tau(n)/j=t+1
       then t:=t+1; lprint(t,n); while v[t+1]>0 do t:=t+1; lprint(t,v[t]); od;
       else if tau(n)/j>t+1 then if v[tau(n)/j]=0 then v[tau(n)/j]:=n; fi; fi;
fi; fi; od; end:
A222086(1000000000000000);

a(1) corrected and a(11), a(13) and a(14) added by Hiroaki Yamanouchi, Oct 03 2014

A222087 Numbers n for which A222084(n)= A222084(A222085(n)).

Original entry on oeis.org

1, 2, 6, 14, 15, 20, 22, 24, 33, 38, 46, 49, 51, 52, 62, 63, 86, 87, 91, 92, 93, 95, 118, 119, 121, 141, 142, 143, 145, 147, 148, 153, 158, 159, 165, 166, 169, 183, 198, 206, 210, 215, 217, 219, 244, 247, 249, 253, 262, 267, 270, 278, 279, 286, 287, 295, 301

Offset: 1

Paolo P. Lava, Feb 13 2013

nonn

Similar to A037197 but using tau#(n), number of the least divisors of n whose LCM is equal to n, as defined in A222084, and sigma#(n), sum of the least divisors of n whose LCM is equal to n, as defined in A222085: tau#(n)=tau#(sigma#(n)).

If we add also the restriction tau(n)=tau(sigma(n)) we have: 2, 52, 1525, 83667, 116162,…

			n=20; tau#(20)=4, sigma#(20)=12 and tau#(sigma#(20))=tau#(12)=4.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A000005, A000961, A001358, A003418, A005179, A024619, A034444, A077610, A222084, A222085.

with(numtheory);
A222087:=proc(q)
local a,b,c,j,n,t,v;
print(1);
for n from 2 to q do
  a:=ifactors(n)[2]; b:=nops(a); c:=0; v:=0;
  for j from 1 to b do if a[j][1]^a[j][2]>c then c:=a[j][1]^a[j][2]; fi; od;
  a:=op(sort([op(divisors(n))])); b:=nops(divisors(n));
  for j from 1 to b do v:=v+a[j]; if a[j]=c then break; fi; od; t:=j;
  a:=ifactors(v)[2]; b:=nops(a); c:=0;
  for j from 1 to b do if a[j][1]^a[j][2]>c then c:=a[j][1]^a[j][2]; fi; od;
  a:=op(sort([op(divisors(v))])); b:=nops(divisors(v));
  for j from 1 to b do if a[j]=c then break; fi; od; if t=j then print(n);
fi; od; end:
A222087(10000000000);

A222089 Where records occur in A222084.

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 48, 96, 192, 384, 480, 768, 960, 1920, 3840, 5760, 7680, 11520, 19440, 23040, 26880, 46080, 53760, 80640, 107520, 161280, 215040, 322560, 612360, 645120, 967680, 1224720, 1290240, 1935360, 2580480, 3870720, 5160960, 6451200, 7096320

Offset: 1

Paolo P. Lava, Feb 13 2013

nonn

Like A002182 but using tau#(n), as defined in A222084, instead of tau(n).

			tau#(8)=4 and not until n=16 this value is overtaken: tau#(16)=5.

Cf. A000005, A000961, A001358, A003418, A005179, A024619, A034444, A077610, A222084, A222085.

with(numtheory);
A222089:=proc(q)
local a,b,c,j,n,t;
t:=0;
for n from 1 to q do
a:=ifactors(n)[2]; b:=nops(a); c:=0;
for j from 1 to b do if a[j][1]^a[j][2]>c then c:=a[j][1]^a[j][2]; fi; od;
a:=op(sort([op(divisors(n))])); b:=nops(divisors(n));
for j from 1 to b do if a[j]=c then break; fi; od;
if j>t then t:=j; print(n); fi;
od; end:
A222089(10000000000);

f[n_]:= Module[{d = Divisors[n], k = 1}, While [LCM@@d[[1;;k]] != n, k++]; k]; fm=0; s={}; Do[f1=f[n]; If[f1>fm, fm=f1; AppendTo[s,n]], {n, 1, 10000}]; s (* Amiram Eldar, Jan 19 2019 *)

f(n) = {my(d = divisors(n), k = 1); while (lcm(vector(k, j, d[j])) != n, k++); k; } \\ A222084
lista(nn) = {my(m=0, nm); for( n=1, nn, if ((nm=f(n)) > m, print1(n, ", "); m = nm););} \\ Michel Marcus, Jan 19 2019

a(2) inserted and a(32)-a(39) added by Amiram Eldar, Jan 19 2019

A222085 Sum of the least divisors of n whose LCM is equal to n.

Original entry on oeis.org

1, 3, 4, 7, 6, 6, 8, 15, 13, 8, 12, 10, 14, 10, 9, 31, 18, 21, 20, 12, 11, 14, 24, 24, 31, 16, 40, 14, 30, 11, 32, 63, 15, 20, 13, 25, 38, 22, 17, 20, 42, 19, 44, 18, 18, 26, 48, 52, 57, 43, 21, 20, 54, 66, 17, 22, 23, 32, 60, 15, 62, 34, 20, 127, 19, 23, 68

Offset: 1

Paolo P. Lava, Feb 11 2013

nonn

			The divisors of 20 are 1, 2, 4, 5, 10, 20 while the least divisors of 20 whose LCM is equal to 20 are 1, 2, 4, 5. Then a(20) = 1+2+4+5 = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)

Cf. A000005, A000961, A001358, A003418, A005179, A024619, A034444, A077610, A222084.

with(numtheory);
A222085:=proc(q)
local a,b,c,j,n,v; print(1);
for n from 2 to q do a:=ifactors(n)[2]; b:=nops(a); c:=0;
  for j from 1 to b do if a[j][1]^a[j][2]>c then c:=a[j][1]^a[j][2]; fi; od;
  a:=op(sort([op(divisors(n))])); b:=nops(divisors(n)); v:=0;
  for j from 1 to b do v:=v+a[j]; if a[j]=c then break; fi; od; print(v);
od; end:
A222085(100000000);

s[n_] := Module[{sum=0, L=1}, Do[sum+=d; L = LCM[L, d]; If[L == n, Break[]], {d, Divisors[n]}]; sum]; Array[s, 67] (* Amiram Eldar, Nov 05 2019 *)

a(n)=my(s,L=1);fordiv(n,d,s+=d;L=lcm(L,d);if(L==n,return(s))) \\ Charles R Greathouse IV, Feb 14 2013

A222088 Numbers n for which A222085(n)=A222085(n+1).

Original entry on oeis.org

5, 23, 44, 84, 132, 185, 335, 368, 1342, 2226, 3354, 4726, 7359, 7598, 8436, 10234, 12123, 18078, 18744, 19848, 20492, 20922, 21823, 21902, 23218, 24069, 24221, 31678, 36510, 36849, 40235, 45046, 46916, 49356, 49769, 50560, 51780, 52716, 53079, 59942, 60150

Offset: 1

Paolo P. Lava, Feb 13 2013

nonn

Like A002961 but using sigma#(n), sum of the least divisors of n whose LCM is equal to n, as defined in A222085, instead of sigma(n):

sigma#(n)=sigma#(n+1).

			n=44; sigma#(44)=18 and sigma#(45)=18.

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

Cf. A000005, A000961, A001358, A003418, A005179, A024619, A034444, A077610, A222084, A222085.

with(numtheory);
A222088:=proc(q)
local a,b,c,d,j,n,t,v;
d:=1;
for n from 2 to q do
  a:=ifactors(n)[2]; b:=nops(a); c:=0; v:=0;
  for j from 1 to b do if a[j][1]^a[j][2]>c then c:=a[j][1]^a[j][2]; fi; od;
  a:=op(sort([op(divisors(n))])); b:=nops(divisors(n));
  for j from 1 to b do v:=v+a[j]; if a[j]=c then break; fi; od;
  if d=v then print(n-1); fi; d:=v; od; end:
A222088(1000000);

s[n_] := Module[{sum=0, L=1}, Do[sum+=d; L = LCM[L, d]; If[L == n, Break[]], {d, Divisors[n]}]; sum]; s1=1; seq={}; Do[s2=s[n]; If[s1==s2, AppendTo[seq, n-1]]; s1=s2, {n, 2, 10^4}]; seq (* Amiram Eldar, Nov 05 2019 *)

A306312 Number of terms of the set of divisors of n that are not the product of any other two distinct divisors.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 4, 2, 3, 3, 3, 2, 4, 2, 4, 3, 3, 2, 4, 3, 3, 3, 4, 2, 4, 2, 3, 3, 3, 3, 5, 2, 3, 3, 4, 2, 4, 2, 4, 4, 3, 2, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 3, 2, 5, 2, 3, 4, 3, 3, 4, 2, 4, 3, 4, 2, 5, 2, 3, 4, 4, 3, 4, 2, 4, 3, 3, 2, 5, 3, 3, 3, 4

Offset: 1

Paolo P. Lava, Feb 06 2019

nonn

Sets contain 1, primes and powers of primes.
a(n) <= A000005(n), a(n) <= A222084(n) and a(p) = 2 with p prime.
Record values for:
a(1) = 1
a(2) = 2
a(4) = 3
a(12) = 4
a(36) = 5
a(180) = 6
a(900) = 7
a(6300) = 8
a(44100) = 9
a(485100) = 10, ...
Records are obtained for A061742 U A228593. - David A. Corneth, Feb 06 2019

			Divisors of 198 are 1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198. Here the set is 1, 2, 3, 9, 11 because 2*3 = 6, 2*9 = 18, 2*11 = 22, 3*11 = 33, 6*11 = 66, 9*11 = 99, 2*99 = 198. Then a(198) = 5.

Cf. A000005, A061742, A222084, A228593.

with(numtheory): with(combinat): P:=proc(q) local a,b,c,k,n;
for n from 2 to q do if isprime(n) then print(2) else a:=sort([op(divisors(n) minus {1})]); b:=choose(a,2); c:=[];
for k from 1 to nops(b) do c:=[op(c),b[k][1]*b[k][2]]; od;
a:=[1,op({op(a)} minus {op(c)})]; print(nops(a)); fi; od; end: P(10^6);

a(n) = my(f = factor(n)[, 2]); sum(i = 1, #f, min(2, f[i])) \\ David A. Corneth, Feb 06 2019

cp's OEIS Frontend

A222086 a(n) is the least number k for which A000005(k)/A222084(k) = n.

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A222087 Numbers n for which A222084(n)= A222084(A222085(n)).

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A222089 Where records occur in A222084.

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A222085 Sum of the least divisors of n whose LCM is equal to n.

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A222088 Numbers n for which A222085(n)=A222085(n+1).

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A306312 Number of terms of the set of divisors of n that are not the product of any other two distinct divisors.

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