A377485 -id:A377485 - cp's OEIS frontend

A377486 a(n) = product of {p^k : p | n, k = 1..floor(log n/log p)}, a(1) = 1.

1, 2, 3, 8, 5, 24, 7, 64, 27, 320, 11, 1728, 13, 448, 135, 1024, 17, 27648, 19, 5120, 189, 11264, 23, 27648, 125, 13312, 729, 7168, 29, 93312000, 31, 32768, 8019, 557056, 875, 23887872, 37, 622592, 9477, 4096000, 41, 167215104, 43, 360448, 91125, 753664, 47, 23887872

Offset: 1

Michael De Vlieger, Oct 29 2024

Compare with A064446, where A064446(n) = Product_{p|n} p^floor(log n / log p).

			Let S(n) = row n of A377485 = { p^k : p | n, p^k <= n, k > 0 }.
a(4) = 8 since S(4) = {2, 4} and the product of these is 8.
a(6) = 24 since S(6) = {2, 3, 4} and the product of these is 24.
a(12) = 1728 since S(12) = {2, 3, 4, 8, 9}, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A064446, A162306, A246655, A377485.

{1}~Join~Table[Times @@ Flatten@ Map[#^Range[Floor@ Log[#, n]] &, FactorInteger[n][[All, 1]]], {n, 2, 120}]

a(p) = p for prime p.
a(p^k) = Product_{j=1..k} p^j = p^(k*(k+1)/2) = p^A000217(k).
a(n) = Product_{p|n} p^(k*(k+1)/2), where k = floor(log n / log p).
Product of row n of A377485.

A377488 Irregular triangle r ead by rows where row n lists powers p^k for primes p | n such that k = floor(log n/log p).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 4, 7, 8, 9, 5, 8, 11, 8, 9, 13, 7, 8, 5, 9, 16, 17, 9, 16, 19, 5, 16, 7, 9, 11, 16, 23, 9, 16, 25, 13, 16, 27, 7, 16, 29, 16, 25, 27, 31, 32, 11, 27, 17, 32, 7, 25, 27, 32, 37, 19, 32, 13, 27, 25, 32, 41, 7, 27, 32, 43, 11, 32, 25, 27, 23, 32

Offset: 1

Michael De Vlieger, Nov 01 2024

Row 1 is {1} by convention, since 1 is the empty product.

			Table of the first 12 rows:
   1:  1;
   2:  2;
   3:  3:
   4:  4;
   5:  5;
   6:  3, 4;
   7:  7;
   8:  8;
   9:  9;
  10:  5, 8;
  11: 11;
  12:  8, 9.

Michael De Vlieger, Table of n, a(n) for n = 1..11739 (rows n = 1..5000, flattened)
Michael De Vlieger, Log log scatterplot of rows n of this sequence, n = 1.2^20 (2996045 terms).

Cf. A001221, A064446, A339378, A377485.

{{1}}~Join~Table[Union[Join @@ Map[#^Range[Floor@ Log[#, n]] &, FactorInteger[n][[All, 1]] ] ], {n, 2, 30}]

Row n is { p^k : p | n, k = floor(log n/log p) }.
Row p^k = { p^k } for prime p and k > 0.
A001221(n) = length of row n for n > 1.
A064446(n) = product of row n.
A339378(n) = sum of row n for n > 1.

A380870 a(n) = A381798(n) - A361373(n) - 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 2, 0, 1, 4, 6, 0, 0, 0, 8, 0, 1, 0, 1, 0, 0, 3, 3, 7, 2, 0, 13, 0, 1, 0, 4, 0, 7, 6, 6, 0, 1, 0, 15, 14, 8, 0, 13, 3, 0, 15, 23, 0, 1, 0, 0, 5, 0, 5, 7, 0, 3, 9, 12, 0, 2, 0, 30, 18, 14, 10, 6, 0, 3, 0, 14, 0

Offset: 1

Michael De Vlieger, Apr 08 2025

nonn

a(n) = cardinality of the intersection of A024619 and row n of A381799.
Let S(n,p) = {p^m : p | n, m = 1..floor(log_p n)}. Therefore S(10,2) = {1,2,4,8} and S(30,3) = {1,3,9,27}. Then U({S(n,p) : p|n}) = row n of A377485.
Let T(n,p) = {p^m (mod n) : p | n} the set of prime divisor power residues r (mod n) == p^m, p | n. Thus T(10,2) = {1,2,4,8,6} and T(30,3) = {1,3,9,27,21}. Then U({T(n,p) : p|n}) = row n of A381799.

			Table of n, a(n), and H(n) = intersection of row n of A381799 with A024619.
 n   facs(n)   a(n)  H(n)
--------------------------------------------
 6   2 * 3       0   -
10   2 * 5       1   {6}
12   2^2 * 3     0   -
14   2 * 7       0   -
15   3 * 5       3   {6, 10, 12}
18   2 * 3^2     2   {10, 14}
20   2^2 * 5     1   {12}
21   3 * 7       4   {6, 12, 15, 18}
22   2 * 11      6   {6, 10, 12, 14, 18, 20}
24   2^3 * 3     0   -
30   2 * 3 * 5   1   {21}
.
a(6) = 0 since Q(6) = R(6) = {1,2,3,4}, i.e., all terms in row 6 of A381799 are powers of primes.
a(10) = 1 since Q(10) = {1,2,4,5,8} but R(10) = {1,2,4,5,6,8}; the latter set contains 1 term (i.e., 6) that is not a member of the former set.
a(14) = 0 since R(14) = {1,2,4,7,8} are all powers of primes.
a(15) = 3 since R(15) = {1,3,5,6,9,10,12} has 3 terms {6,10,12} that are not powers of primes.
a(18) = 2 since R(18) = {1,2,3,4,8,9,10,14,16} has 2 terms {6,10} that are not powers of primes, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16384

Cf. A000961, A024619, A361373, A377485, A381750, A381798, A381799.

f[x_, p_] := Block[{m = 2, r, c},
  Which[
    PrimePowerQ[x],
    Join[{0}, #1^Range[0, #2 - 1]] & @@ FactorInteger[x][[1]],
    PowerMod[p, m, x] == p, {1, p},
    True, c[_] := False;
    c[1] = c[p] = True; {1, p}~Join~
      Reap[While[r = PowerMod[p, m, x]; ! c[r], Sow[r];
        c[r] = True; m++] ][[-1, 1]] ] ]
Table[Count[Union@ Flatten@ Map[f[n, #] &, FactorInteger[n][[All, 1]] ], _?(And[# > 1, ! PrimePowerQ[#]] &)], {n, 120}]

Let Q(n) = {1} joined to row n > 1 of A377485 and let R(n) = row n of A381799.
a(n) = card(U(Q(n) \ R(n))).
a(p^m) = 0 for prime power p^m, m >= 0.
a(n) = 0 for n in A381750.

A384442 Smallest k such that A361373(k) = n.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 18, 40, 36, 30, 60, 102, 84, 132, 150, 264, 210, 540, 330, 420, 660, 630, 840, 1050, 2100, 2340, 2520, 3150, 2310, 2730, 4290, 4620, 6930, 9240, 15960, 16170, 17850, 18480, 20790, 34650, 62370, 68250, 30030, 62790, 60060, 78540, 90090, 117810

Offset: 0

Michael De Vlieger, Jun 12 2025

nonn

For n > 1, a(n) is composite, since A361373(p) = 1 for prime p.

For n = 0..2, a(n) = 2^n. For n > 2, a(n) is in A024619.

			Table of n, a(n) for n = 1..12, showing row a(n) of A377485.
          log n/log p
 n  a(n)  p_1 p_2 p_3  row n of A377485
-------------------------------------------------------------------------
 1:   2   1            {p}
 2:   4   2            {p, p^2}
 3:   6   2   1        {p, q, p^2}
 4:  10   3   1        {p, p^2, q, p^3}
 5:  12   3   2        {p, q, p^2, p^3, q^2}
 6:  18   4   2        {p, q, p^2, p^3, q^2, p^4}
 7:  40   5   2        {p, p^2, q, p^3, p^4, q^2, p^5}
 8:  36   5   3        {p, q, p^2, p^3, q^2, p^4, q^3, p^5}
 9:  30   4   3   2    {p, q, p^2, r, p^3, q^2, p^4, r^2, q^3}
10:  60   5   3   2    {p, q, p^2, r, p^3, q^2, p^4, r^2, q^3, p^5}
11: 102   6   4   1    {p, q, p^2, p^3, q^2, p^4, r, q^3, p^5, p^6, q^4}
12:  84   6   4   2    {p, q, p^2, r, p^3, q^2, p^4, q^3, p^5, r^2, p^6, q^4}

Cf. A361373, A377845.

nn = 30030; t[_] := 0; u = 1; Do[(If[t[#] == 0, t[#] = n]; If[# == u, While[t[u] != 0, u++]]) &[Total@ Map[Floor@ Log[#, n] &, FactorInteger[n][[All, 1]] ] ], {n, 2, nn}]; {1}~Join~Array[t, u - 1]

A384936 a(n) = Sum_{k=1..n} floor( log(A002110(n)) / log(prime(k)) ).

Original entry on oeis.org

0, 1, 3, 9, 16, 28, 42, 57, 76, 97, 121, 148, 177, 208, 242, 279, 316, 359, 401, 446, 493, 545, 596, 651, 708, 767, 829, 893, 958, 1026, 1096, 1170, 1246, 1319, 1400, 1484, 1567, 1657, 1742, 1834, 1923, 2021, 2119, 2218, 2316, 2419, 2526, 2635, 2745, 2857, 2972

Offset: 0

Michael De Vlieger, Jun 12 2025

A384442(a(n)) = A002110(n) for n <= 8; does it hold for n > 5?

			Table of n, a(n) for n = 0..10, listing terms in row n of A287010:
      Terms in row n of A287010 corresponding
      to the primes listed in the header
 n\k   2   3   5   7  11  13  17  19  23  29   a(n)
---------------------------------------------------
 0:    0   .   .   .   .   .   .   .   .   .     0
 1:    1   .   .   .   .   .   .   .   .   .     1
 2:    2   1   .   .   .   .   .   .   .   .     3
 3:    4   3   2   .   .   .   .   .   .   .     9
 4:    7   4   3   2   .   .   .   .   .   .    16
 5:   11   7   4   3   3   .   .   .   .   .    28
 6:   14   9   6   5   4   4   .   .   .   .    42
 7:   18  11   8   6   5   5   4   .   .   .    57
 8:   23  14   9   8   6   6   5   5   .   .    76
 9:   27  17  11   9   8   7   6   6   6   .    97
10:   32  20  14  11   9   8   7   7   7   6   121

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A002110, A287010, A361373, A377485, A384442.

P = 2; s = {2}; {0}~Join~Reap[Do[Sow@ Total@ Map[Floor@ Log[#, P] &, s]; (AppendTo[s, #]; P *= #) &[Prime[k]], {k, 2, 51}] ][[-1, 1]]

a(n) = my(v=primes(n), pp=vecprod(v)); sum(k=1, n, log(pp)\log(v[k])); \\ Michel Marcus, Jun 14 2025

a(n) = A361373(A002110(n)).

Row sums of A287010.

cp's OEIS Frontend

A377486 a(n) = product of {p^k : p | n, k = 1..floor(log n/log p)}, a(1) = 1.

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A377488 Irregular triangle r ead by rows where row n lists powers p^k for primes p | n such that k = floor(log n/log p).

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A380870 a(n) = A381798(n) - A361373(n) - 1.

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A384442 Smallest k such that A361373(k) = n.

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A384936 a(n) = Sum_{k=1..n} floor( log(A002110(n)) / log(prime(k)) ).

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