A376567 -id:A376567 - cp's OEIS frontend

A378630 Numbers that set records in A376567.

1, 2, 4, 6, 12, 24, 30, 48, 60, 120, 210, 240, 420, 840, 1680, 3360, 6720, 9240, 13440, 18480, 26880, 36960, 73920, 147840, 240240, 295680, 480480, 591360, 960960, 1921920, 3843840, 7687680, 15375360, 30750720, 32672640, 61501440, 65345280, 123002880, 130690560

Offset: 1

Michael De Vlieger, Dec 02 2024

Numbers n that set records for binomial(bigomega(n)+omega(n), omega(n)), where bigomega = A001222 and omega = A001221.
a(n) is of the form 2^k * P(i), k >= 0, where primorial P = A002110.
Proper subset of A070175.

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

Cf. A000079, A001221, A001222, A002110, A007947, A070175, A376567.

f[x_] := Block[{i, k, m, nn, p}, nn = Product[Prime[j], {j, x}]; Set[{k, i, p}, Range[0, 2]]; {1}~Join~Union@ Reap[Until[i > x, While[Set[m, 2^k*p] <= nn, Sow[m]; k++]; k = 0; i++; p *= Prime[i] ] ][[-1, 1]] ] (* generate A070175 *);
r = 0; Reap[Do[If[# > r, r = #; Sow[n]] &@ Binomial[#2 + #1, #1] & @@ {PrimeNu[n], PrimeOmega[n]}, {n, f[10]}] ][[-1, 1]]

A376248 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, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 4, 6, 9, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 4, 5, 10, 25, 1, 11, 1, 2, 3, 4, 6, 8, 9, 12, 18, 27, 1, 13, 1, 2, 4, 7, 14, 49, 1, 3, 5, 9, 15, 25, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 4, 6, 8, 9, 12, 18, 27, 1, 19, 1, 2, 4, 5, 8, 10, 20, 25, 50, 125

Offset: 1

Michael De Vlieger, Oct 09 2024

Analogous to A162306 regarding m such that rad(m) | n, but instead of taking m <= n, we take m such that bigomega(m) <= bigomega(n).
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).
For prime power n = p^k, k >= 0 (i.e., n in A000961), row p^k of this sequence is the same as row p^k of A027750 and A162306. Therefore, for prime p, row p of this sequence is the same as row p of A027750 and A162306: {1, p}.
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.

			Triangle begins:
   n    row n of this sequence:
  -------------------------------------------
   1:   1;
   2:   1,  2;
   3:   1,  3;
   4:   1,  2   4;
   5:   1,  5;
   6:   1,  2,  3,  4,  6,  9;
   7:   1,  7;
   8:   1,  2,  4,  8;
   9:   1,  3,  9;
  10:   1,  2,  4,  5, 10, 25;
  11:   1, 11;
  12:   1,  2,  3,  4,  6,  8, 9, 12, 18, 27;
        ...
Row n = 10 of this sequence, presented according to 2^k, k = 0..bigomega(n) by columns, 5^i, i = 0..bigomega(n) by rows, showing terms m > n with an asterisk. The remaining m and the parenthetic 8 are in row 10 of A162306:
   1   2   4  (8)
   5  10
  25*
Row n = 12 of this sequence, presented according to 2^k, k = 0..bigomega(n) by columns, 3^i, i = 0..bigomega(n) by rows, showing terms m > n with an asterisk. The remaining m are in row 12 of A162306:
   1   2   4   8
   3   6  12
   9  18*
  27*

Michael De Vlieger, Table of n, a(n) for n = 1..17475 (rows n = 1..1000, flattened)
Michael De Vlieger, Log log scatterplot of rows n = 1..2^16 of this sequence, flattened, (3153752 terms).

Cf. A000961, A001221, A001222, A006530, A007947, A010846, A024619, A027750, A162306, A376567.

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) }.

A376567(n) = binomial(bigomega(n) + omega(n)) = Length of row n, where omega = A001221.

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.

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).

A378182 Sum of row n of A378180.

Original entry on oeis.org

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).

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A378630 Numbers that set records in A376567.

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A376248 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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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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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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A378182 Sum of row n of A378180.

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