A377070 -id:A377070 - cp's OEIS frontend

A377072 a(n) = sum of row n of A377070.

1, 2, 3, 4, 5, 19, 7, 8, 9, 39, 11, 65, 13, 67, 49, 16, 17, 65, 19, 203, 79, 147, 23, 211, 25, 199, 27, 477, 29, 410, 31, 32, 163, 327, 109, 211, 37, 403, 217, 1031, 41, 786, 43, 1625, 272, 579, 47, 665, 49, 203, 349, 2595, 53, 211, 201, 3355, 427, 903, 59, 2261

Offset: 1

Michael De Vlieger, Nov 14 2024

Cf. A000203, A024619, A244974, A376248, A377070.

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

A377071 a(n) = binomial(bigomega(n) + omega(n) - 1, omega(n) - 1), where bigomega = A001222 and omega = A001221.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 4, 1, 4, 3, 3, 1, 5, 1, 3, 1, 4, 1, 10, 1, 1, 3, 3, 3, 5, 1, 3, 3, 5, 1, 10, 1, 4, 4, 3, 1, 6, 1, 4, 3, 4, 1, 5, 3, 5, 3, 3, 1, 15, 1, 3, 4, 1, 3, 10, 1, 4, 3, 10, 1, 6, 1, 3, 4, 4, 3, 10, 1, 6, 1, 3, 1, 15, 3

Offset: 1

Michael De Vlieger, Oct 25 2024

Number of permutations of the integer partitions of omega(n) supplemented with zeros such that there are bigomega(n) parts, whose sum equals bigomega(n).

a(n) = cardinality of { m : rad(m) | n, bigomega(m) = bigomega(n) }, where rad = A007947.

			For n = 6, omega(6) = 2, bigomega(6) = 2, we have 3 exponent combinations [2,0], [1,1], [0,2]. Raising prime factors {2, 3} to these exponents yields {4, 6, 9}, i.e., row 6 of A377070.
For n = 10, omega(10) = 2, bigomega(10) = 2, we have 3 exponent combinations [2,0], [1,1], [0,2]. Raising prime factors {2, 5} to these exponents yields {4, 10, 25}, i.e., row 10 of A377070.
For n = 12, omega(12) = 2, bigomega(12) = 3, we have 4 exponent combinations [3,0], [2,1], [1,2], [0,3]. Raising prime factors {2, 3} to these exponents yields {8, 12, 18, 27}, i.e., row 6 of A377070.

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

Cf. A000005, A000961, A001221, A001222, A007947, A024619, A376248, A377070.

Array[Binomial[#2 + #1 - 1, #1 - 1] & @@ {PrimeNu[#], PrimeOmega[#]} &, 120]

a(n) is the length of row n of A377070.
For prime power p^k, k >= 0, a(p^k) = 1.
For n in A024619, a(n) > 1.

A378180 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) < bigomega(n), where rad = A007947 and bigomega = A001222.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 1, 3, 1, 2, 5, 1, 1, 2, 3, 4, 6, 9, 1, 1, 2, 7, 1, 3, 5, 1, 2, 4, 8, 1, 1, 2, 3, 4, 6, 9, 1, 1, 2, 4, 5, 10, 25, 1, 3, 7, 1, 2, 11, 1, 1, 2, 3, 4, 6, 8, 9, 12, 18, 27, 1, 5, 1, 2, 13, 1, 3, 9, 1, 2, 4, 7, 14, 49, 1, 1, 2, 3, 4, 5, 6, 9, 10, 15, 25

Offset: 2

Michael De Vlieger, Nov 19 2024

Row n is a finite set of products of prime power factors p^k (i.e., p^k | n) such that Sum_{p|n} k < bigomega(n).
Row n contains numbers m such that rad(m) | n, where the number of prime factors of m with repetition is less than that of n.
Row 1 of this sequence is {}, hence offset of this sequence is set to 2.
For n = p^k (in A246655), row n contains p^j, j = 0..k-1.
For prime p, row p = {1}.
For n in A024619, row n of this sequence does not match row n of A162306, since the former contains gpf(n)^bigomega(n) = A006530(n)^A001222(n), which is larger than n, and since row n of A162306 contains n itself.

			Select rows n, showing nondivisors k parenthetically (i.e., k not in row n of A027750), and numbers k > n in brackets (i.e., k neither in row n of A162306 nor in row n of A027750):
   n    row n of this sequence:
  -------------------------------------------
   2:   1;
   3:   1;
   4:   1, 2;
   6:   1, 2, 3;
   8:   1, 2, 4;
   9:   1, 3;
  10:   1, 2, 5;
  12:   1, 2, 3,  4,   6,  (9);
  18:   1, 2, 3, (4),  6,   9;
  20:   1, 2, 4,  5,  10, [25];
  24:   1, 2, 3,  4,   6,   8, (9), 12, (18), [27];
  28:   1, 2, 4,  7,  14, [49];
  30:   1, 2, 3, (4),  5,   6, (9), 10,  15,  (25);
  36:   1, 2, 3,  4,   6,   8,  9,  12,  18,  (27);

Michael De Vlieger, Table of n, a(n) for n = 2..10325

Cf. A000961, A001221, A001222, A007947, A010846, A024619, A027750, A162306, A376567, A376248, A377070, A378181, A378183.

Table[Clear[p]; MapIndexed[Set[p[First[#2]], #1] &, FactorInteger[n][[All, 1]]];
 k = PrimeOmega[n]; w = PrimeNu[n];
 Union@ Map[Times @@ MapIndexed[p[First[#2]]^#1 &, #] &,
  Select[Tuples[Range[0, k], w], Total[#] < k &]], {n, 120}]

Row n of this sequence is { m : rad(m) | n, bigomega(m) < bigomega(n) } = S \ T, where S is row n of A376248, and T is row n of A377070.
A378181(n) = binomial(bigomega(n) + omega(n) - 1, omega(n)) = Length of row n, where omega = A001221.
A378183(n) = rad(n)^binomial(omega(n) + bigomega(n) - 1, bigomega(n)-2) = A377073(n)/A377379(n) = product of row n.

A377073 a(n) = rad(n)^binomial(bigomega(n) + omega(n) - 1, omega(n)), where rad = A007947, bigomega = A001222, and omega = A001221.

Original entry on oeis.org

1, 2, 3, 4, 5, 216, 7, 8, 9, 1000, 11, 46656, 13, 2744, 3375, 16, 17, 46656, 19, 1000000, 9261, 10648, 23, 60466176, 25, 17576, 27, 7529536, 29, 590490000000000, 31, 32, 35937, 39304, 42875, 60466176, 37, 54872, 59319, 10000000000, 41, 17080198121677824, 43, 113379904

Offset: 1

Michael De Vlieger, Oct 27 2024

Product of row n of A377070.

Michael De Vlieger, Table of n, a(n) for n = 1..6719

Cf. A000203, A001221, A001222, A007947, A024619, A244974, A376248, A377070.

Table[Apply[Times, FactorInteger[n][[All, 1]]]^Binomial[PrimeOmega[n] + PrimeNu[n] - 1, PrimeNu[n]], {n, 44}]

For prime power p^k, k >= 0, a(p^k) = p^k.

For n in A024619, a(n) > n.

cp's OEIS Frontend

A377072 a(n) = sum of row n of A377070.

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A377071 a(n) = binomial(bigomega(n) + omega(n) - 1, omega(n) - 1), where bigomega = A001222 and omega = A001221.

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A378180 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) < bigomega(n), where rad = A007947 and bigomega = A001222.

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A377073 a(n) = rad(n)^binomial(bigomega(n) + omega(n) - 1, omega(n)), where rad = A007947, bigomega = A001222, and omega = A001221.

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