A376248 -id:A376248 - cp's OEIS frontend

A377378 a(n) = sum of row n of A376248.

1, 3, 4, 7, 6, 25, 8, 15, 13, 47, 12, 90, 14, 77, 58, 31, 18, 90, 20, 250, 90, 161, 24, 301, 31, 215, 40, 554, 30, 490, 32, 63, 178, 347, 122, 301, 38, 425, 234, 1281, 42, 902, 44, 1786, 330, 605, 48, 966, 57, 250, 370, 2810, 54, 301, 218, 3909, 450, 935, 60, 2751

Offset: 1

Michael De Vlieger, Nov 14 2024

For prime p, a(p) = A244974(p) = A000203(p) = p+1.
For prime power p^k, a(p^k) = A244974(p^k) = A000203(p^k).
For n in A024619, a(n) != A244974(n).

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

Cf. A000203, A024619, A244974, A376248.

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}]]

A376567 a(n) = binomial(bigomega(n) + omega(n), omega(n)), where bigomega = A001222 and omega = A001221.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 4, 3, 6, 2, 10, 2, 6, 6, 5, 2, 10, 2, 10, 6, 6, 2, 15, 3, 6, 4, 10, 2, 20, 2, 6, 6, 6, 6, 15, 2, 6, 6, 15, 2, 20, 2, 10, 10, 6, 2, 21, 3, 10, 6, 10, 2, 15, 6, 15, 6, 6, 2, 35, 2, 6, 10, 7, 6, 20, 2, 10, 6, 20, 2, 21, 2, 6, 10, 10, 6, 20, 2

Offset: 1

Michael De Vlieger, Oct 09 2024

For prime power p^k, a(p^k) = A010846(p^k) = A000005(p^k) = k+1. Therefore, for prime p, a(p) = A010846(p) = A000005(p) = 2.

For n in A024619, a(n) != A010846(n) and A010846(n) > A000005(n).

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

Cf. A000005, A001221, A001222, A007947, A010846, A024619, A068795, A376248, A376846, A376847.

with(NumberTheory):
a := n -> binomial(Omega(n) + Omega(n, distinct), Omega(n, distinct)):
seq(a(n), n = 1..79);  # Peter Luschny, Oct 25 2024

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

a(n) = length of row n of A376248.

a(n) = A010846(n) - A376846(n) + A376847(n).

A377070 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, 2, 3, 4, 5, 4, 6, 9, 7, 8, 9, 4, 10, 25, 11, 8, 12, 18, 27, 13, 4, 14, 49, 9, 15, 25, 16, 17, 8, 12, 18, 27, 19, 8, 20, 50, 125, 9, 21, 49, 4, 22, 121, 23, 16, 24, 36, 54, 81, 25, 4, 26, 169, 27, 8, 28, 98, 343, 29, 8, 12, 18, 20, 27, 30, 45, 50, 75, 125, 31

Offset: 1

Michael De Vlieger, Oct 25 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), that is, numbers m such that rad(m) | n and m has the same number of prime factors with repetition than does n.

			Triangle begins:
    n    row n of this sequence:
   -------------------------------------------
    1:   {1}
    2:   {2}
    3:   {3}
    4:   {4}
    5:   {5}
    6:   {4, 6, 9}
    7:   {7}
    8:   {8}
    9:   {9}
   10:   {4, 10, 25}
   ...                       (Select rows appear below)
   12:   {8, 12, 18, 27}
   14:   {4, 14, 49}
   15:   {9, 15, 25}
   18:   {8, 12, 18, 27}
   20:   {8, 20, 50, 125}
   24:   {16, 24, 36, 54, 81}
   30:   {8, 12, 18, 20, 27, 30, 45, 50, 75, 125}
   42:   {8, 12, 18, 27, 28, 42, 63, 98, 147, 343}
   60:   {16, 24, 36, 40, 54, 60, 81, 90, 100, 135, 150, 225, 250, 375, 625}.
.
Diagrams of the rank omega(n)-1 simplexes created by row n of this sequence for select n, ordering k in row n by prime decomposition. The number k = n appears in brackets:
Rank 3:
   n = 30:                    n = 42:
             8                         8
           /  \                      /  \
         12 -- 20                  12 -- 28
        /  \  /  \                /  \  /  \
      18 --[30]-- 50            18 --[42]-- 98
     /  \  /  \  /  \          /  \  /  \  /  \
   27 -- 45 -- 75 -- 125     27 -- 63 --147 -- 343
.
   n = 60:     16
              /  \
            24 -- 40
           /  \  /  \
         36 --[60]-- 50
        /  \  /  \  /  \
      54 -- 90 -- 75 --125
     /  \  /  \  /  \  /  \
   81 --150 --135 --375 --625
.
Rank 4:
   n = 210:
   16
        40
   24   56
             100
        60   140
   36   84   196
                   250
             150   350
        90  [210]  490
   54  126   294   686
                            625
                     375    875
              225    525   1225
        135   315    735   1715
   81   189   441   1029   2401

Michael De Vlieger, Table of n, a(n) for n = 1..12021, (rows n = 1..1500, flattened)
Michael De Vlieger, Diagrams of select a(n) illustrating rank omega(n)-1 simplexes formed by the arrangement of terms in row n by prime power decomposition.
Michael De Vlieger, Log log scatterplot of a(n), rows n = 1..65536 (1278755 terms).

Cf. A001221, A001222, A007947, A024619, A376248, A377071.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Table[k = PrimeOmega[n]; Select[Range[n^PrimeNu[n]], Divisible[n, rad[#]] && PrimeOmega[#] == k &], {n, 30}]

Row n of this sequence is { m : rad(m) | n, bigomega(m) = bigomega(n) }.
For prime p, row p of this sequence is {p}, generally for prime power p^k, row p^k of this sequence is {p^k}.
For n in A024619, row n of this sequence has more than 1 term.
A377071(n) = length of row n of this sequence.

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.

A377379 a(n) = rad(n)^binomial(bigomega(n) + omega(n), omega(n) + 1), where rad = A007947, bigomega = A001222, and omega = A001221.

Original entry on oeis.org

1, 2, 3, 8, 5, 1296, 7, 64, 27, 10000, 11, 60466176, 13, 38416, 50625, 1024, 17, 60466176, 19, 10000000000, 194481, 234256, 23, 3656158440062976, 125, 456976, 729, 289254654976, 29, 14348907000000000000000, 31, 32768, 1185921, 1336336, 1500625, 3656158440062976

Offset: 1

Michael De Vlieger, Oct 27 2024

Product of row n of A376248.

Michael De Vlieger, Table of n, a(n) for n = 1..1679

Cf. A001221, A001222, A007947, A007955, A010846, A024619, A243103, A376248.

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

For prime p, a(p) = A243103(p) = A007955(p) = p.
For prime power p^k, a(p^k) = A243103(p^k) = A007955(p^k) = p^(k+1).
For n in A024619, a(n) != A010846(n) and A010846(n) > A000005(n).

A376847 Number of m > n such that rad(m) | n and Omega(m) <= Omega(n), where rad = A007947 and Omega = A001222.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 3, 1, 1, 0, 4, 0, 1, 0, 3, 0, 4, 0, 0, 1, 1, 1, 2, 0, 1, 1, 5, 0, 5, 0, 3, 2, 1, 0, 6, 0, 1, 1, 3, 0, 1, 1, 5, 1, 1, 0, 11, 0, 1, 2, 0, 1, 5, 0, 3, 1, 5, 0, 4, 0, 1, 1, 3, 1, 5, 0, 8, 0, 1, 0, 11, 1, 1, 1

Offset: 1

Michael De Vlieger, Oct 13 2024

nonn

			Table of select n such that a(n) > 0:
   n  a(n)  List of m in A376248 such that Omega(m) <= Omega(n)
  -------------------------------------------------------------
   6    1   {9}
  10    1   {25}
  12    2   {18, 27}
  14    1   {49}
  15    1   {25}
  18    1   {27}
  20    3   {25, 50, 125}
  24    4   {27, 36, 54, 81}
  28    3   {49, 98, 343}
  30    4   {45, 50, 75, 125}
  40    5   {50, 100, 125, 250, 625}
  48    6   {54, 72, 81, 108, 162, 243}
  60   11   {75, 81, 90, 100, 125, 135, 150, 225, 250, 375, 625}

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Hasse diagrams of m in select R(n), where R(n) is the union of rows n of A162306 and A376248, indicating in blue those m > n such that Omega(m) <= Omega(n).

Cf. A000961, A001221, A001222, A007947, A162306, A376567, A376846, A068795.

with(NumberTheory):
cond := (m, n) -> irem(n, Radical(m)) = 0 and Omega(m) <= Omega(n):
a := n -> nops(select(m -> cond(m, n), [seq(n+1..A068795(n))])):
seq(a(n), n = 1..87);  # Peter Luschny, Oct 25 2024

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Table[k = PrimeOmega[n]; w = PrimeNu[n]; Binomial[k + w, w] - Count[Range[n], _?(And[Divisible[n, rad[#]], PrimeOmega[#] > k] &)], {n, 120}]

a(n) = card({m > n : rad(m) | n, Omega(m) <= Omega(n) }).
a(n) = 0 for prime power n (in A000961).
a(n) = card(A376248 \ A162306).
a(n) = A376567(n) - A010846(n) + A376546(n) = binomial(A001222(n) + A001221(n), A001221(n)) - A010846(n) + A376546(n).

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

Original entry on oeis.org

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}] ]

cp's OEIS Frontend

A377378 a(n) = sum of row n of A376248.

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A376567 a(n) = binomial(bigomega(n) + omega(n), omega(n)), where bigomega = A001222 and omega = A001221.

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A377070 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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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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A377379 a(n) = rad(n)^binomial(bigomega(n) + omega(n), omega(n) + 1), where rad = A007947, bigomega = A001222, and omega = A001221.

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A376847 Number of m > n such that rad(m) | n and Omega(m) <= Omega(n), where rad = A007947 and Omega = A001222.

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