A376514 -id:A376514 - cp's OEIS frontend

A376936 Powerful numbers divisible by cubes of 2 distinct primes.

216, 432, 648, 864, 1000, 1296, 1728, 1944, 2000, 2592, 2744, 3375, 3456, 3888, 4000, 5000, 5184, 5400, 5488, 5832, 6912, 7776, 8000, 9000, 9261, 10000, 10125, 10368, 10584, 10648, 10800, 10976, 11664, 13500, 13824, 15552, 16000, 16200, 16875, 17496, 17576, 18000

Offset: 1

Michael De Vlieger, Oct 16 2024

Numbers m with coreful divisors d, m/d such that neither d | m/d nor m/d | d, i.e., numbers m such that there exists a divisor pair (d, m/d) such that rad(d) = rad(m/d) but gcd(d, m/d) > 1 is neither d nor m/d, where rad = A007947. Divisors in each pair must be dissimilar and each in A126706.
Proper subset of A320966.
Contains A372695, A177493, and A162142. Does not contain A085986.

			216 is in the sequence since rad(12) | rad(18), but 12 does not divide 18 and 18 does not divide 12.
432 is a term since rad(18) | rad(24), but 18 does not divide 24 and 24 does not divide 18.
Table of coreful divisors d, a(n)/d such that neither d | a(n)/d nor a(n)/d | d for select a(n)
   n |   a(n)   divisor pairs d X a(n)/d
  ---------------------------------------------------------------------------
   1 |   216:   12 X 18;
   2 |   432:   18 X 24;
   3 |   648:   12 X 54;
   4 |   864:   24 X 36, 18 X 48;
   5 |  1000:   20 X 50;
   6 |  1296:   24 X 54;
   7 |  1728:   18 X 96, 36 X 48;
   8 |  1944:   12 X 162, 36 X 54;
   9 |  2000:   40 X 50;
  10 |  2592:   24 X 108, 48 X 54;
  11 |  2744:   28 X 98;
  12 |  3375:   45 X 75;
  13 |  3456:   18 X 192, 36 X 96, 48 X 72;
  22 |  7776:   24 X 324, 48 X 162, 54 X 144, 72 X 108;
  58 | 31104:   48 X 648, 54 X 576, 96 X 324, 108 X 288, 144 X 216, 162 X 192

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Notes on this sequence

Cf. A007947, A030078, A085986, A126706, A162142, A177493, A286708, A307958, A320966, A361430, A370329, A372695, A376514.

Cf. A082020, A082695.

Union@ Select[
  Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[20000],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (15/Pi^2) * (1 + Sum_{prime} 1/((p-1)*(p^2+1))) = 0.021194288968234037106579437374641326044... . - Amiram Eldar, Nov 08 2024

A376271 Numbers k such that there exists at least one proper divisor that is neither squarefree nor a prime power, i.e., m is in A126706.

Original entry on oeis.org

24, 36, 40, 48, 54, 56, 60, 72, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208, 216, 220, 224, 225, 228, 232, 234, 240, 248, 250, 252, 260, 264, 270, 272

Offset: 1

Michael De Vlieger, Sep 28 2024

Numbers k such that A376514(k) > 1. A376514(k) >= 1 for all k in A126706.
Numbers k such that the cardinality of the intersection of row n of A027750 and A126706 exceeds 1.
a(n) is not in A366825, since for k in A366825, there is only one divisor that is in A126706, and that is k itself.

			4 is not in the sequence since 4 is a prime power, and all divisors d | k of prime power k = p^e are also prime powers.
6 is not in the sequence since 6 is squarefree, and all divisors d | k of squarefree k are also squarefree.
12 is not in the sequence since 12 is in A366825, and there is only 1 divisor in A126706, which is 12 itself.
24 is in the sequence since the intersection of A126706 and row 24 of A027750, indicated by bracketed numbers, is {1, 2, 3, 4, 6, [12, 24]}, etc.
Table listing the intersection of A126706 and row a(n) of A027750 for n <= 12:
  24: {12, 24}
  36: {12, 18, 36}
  40: {20, 40}
  48: {12, 24, 48}
  54: {18, 54}
  56: {28, 56}
  60: {12, 20, 60}
  72: {12, 18, 24, 36, 72}
  80: {20, 40, 80}
  84: {12, 28, 84}
  88: {44, 88}
  90: {18, 45, 90}

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A027750, A033987, A126706, A366825, A376514.

Select[Range[300], Function[k, DivisorSum[k, 1 &, Nor[PrimePowerQ[#], SquareFreeQ[#]] &] > 1]]
(* Second program *)
Select[Range[300], And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &] (* Michael De Vlieger, Dec 24 2024 *)

list(lim)=my(v=List()); forfactored(k=24,lim\1, my(e=k[2][,2]); if(#e>1 && vecmax(e)>1 && (#e>2 || vecsum(e)>3), listput(v,k[1]))); Vec(v) \\ Charles R Greathouse IV, Oct 01 2024

Intersection of A033987 and A126706, i.e., { k : bigomega(k) > omega(k) > 1, bigomega(k) > 3 }, where bigomega = A001222 and omega(k) = A001221. - Michael De Vlieger, Dec 24 2024

cp's OEIS Frontend

A376936 Powerful numbers divisible by cubes of 2 distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula