A378180 -id:A378180 - cp's OEIS frontend

A378182 Sum of row n of A378180.

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 25, 1, 10, 9, 15, 1, 25, 1, 47, 11, 14, 1, 90, 6, 16, 13, 77, 1, 80, 1, 31, 15, 20, 13, 90, 1, 22, 17, 250, 1, 116, 1, 161, 58, 26, 1, 301, 8, 47, 21, 215, 1, 90, 17, 554, 23, 32, 1, 490, 1, 34, 90, 63, 19, 212, 1, 347, 27, 152

Offset: 1

Michael De Vlieger, Nov 20 2024

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

Cf. A000203, A007947, A024619, A244974, A376567, A377071, A378180.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Block[{k}, Table[k = PrimeOmega[n];
  Total@ Select[Range[n^PrimeNu[n]],
    Divisible[n, rad[#]] && PrimeOmega[#] < k &], {n, 60}]]

a(n) = A376567(n) - A377071(n).
For prime p, a(p) = 1.
For prime power p^k, a(p^k) = A244974(p^k)-p^k = A000203(p^k)-p^k.
a(2^k) = 2^k - 1.
For n in A024619, a(n) != A244974(n).

A378181 a(1) = 0, a(n) = binomial(bigomega(n) + omega(n) - 1, omega(n)), where bigomega = A001222 and omega = A001221.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 10, 2, 3, 3, 6, 1, 10, 1, 5, 3, 3, 3, 10, 1, 3, 3, 10, 1, 10, 1, 6, 6, 3, 1, 15, 2, 6, 3, 6, 1, 10, 3, 10, 3, 3, 1, 20, 1, 3, 6, 6, 3, 10, 1, 6, 3, 10, 1, 15, 1, 3, 6, 6, 3, 10, 1, 15, 4, 3, 1

Offset: 1

Michael De Vlieger, Nov 19 2024

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

Cf. A000005, A001221, A001222, A007947, A010846, A024619, A378180.

{0}~Join~Table[Binomial[PrimeOmega[n] + # - 1, #] &@ PrimeNu[n], {n, 120}]

a(n) = cardinality of { m : rad(m) | n, bigomega(m) < bigomega(n) }, i.e., row n of A378180.
For prime p, a(p) = A010846(p)-1 = A000005(p)-1 = 1.
For prime power p^k, a(p^k) = A010846(p^k)-1 = A000005(p^k)-1 = k.
For n in A024619, a(n) != A010846(n).

A378183 a(n) = rad(n)^binomial(omega(n) + bigomega(n) - 1, bigomega(n) - 2), where rad = A007947, bigomega = A001222, and omega = A001221.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 8, 3, 10, 1, 1296, 1, 14, 15, 64, 1, 1296, 1, 10000, 21, 22, 1, 60466176, 5, 26, 27, 38416, 1, 24300000, 1, 1024, 33, 34, 35, 60466176, 1, 38, 39, 10000000000, 1, 130691232, 1, 234256, 50625, 46, 1, 3656158440062976, 7, 10000, 51, 456976, 1

Offset: 1

Michael De Vlieger, Nov 19 2024

Michael De Vlieger, Table of n, a(n) for n = 1..3359

Cf. A001221, A001222, A006881, A007947, A007955, A024619, A243103, A377379, A377073, A378180.

Table[Apply[Times, #1]^Binomial[Length[#1] + Total[#2] - 1, Total[#2] - 2] & @@
  Transpose@ FactorInteger[n], {n, 120}]

a(n) = A377379(n)/A377073(n).
For prime p, a(p) = 1.
For prime power p^k, a(p^k) = A243103(p^k)/p^k = A007955(p^k)/p^k = p^(k*(k-1)).
For n in A024619, a(n) != A243103(n).
For squarefree semiprime n (in A006881), a(n) = n.

cp's OEIS Frontend

A378182 Sum of row n of A378180.

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A378181 a(1) = 0, a(n) = binomial(bigomega(n) + omega(n) - 1, omega(n)), where bigomega = A001222 and omega = A001221.

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A378183 a(n) = rad(n)^binomial(omega(n) + bigomega(n) - 1, bigomega(n) - 2), where rad = A007947, bigomega = A001222, and omega = A001221.

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