A287483 -id:A287483 - cp's OEIS frontend

A336568 Numbers that are not a product of two numbers each having distinct prime multiplicities.

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 330, 345, 354, 357, 366, 370, 374, 385, 390, 399, 402, 406, 410, 418, 420, 426, 429

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

First differs from A007304 and A093599 in having 210.
First differs from A287483 in having 222.
First differs from A350352 in having 420.
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			Selected terms together with their prime indices:
   660: {1,1,2,3,5}
   798: {1,2,4,8}
   840: {1,1,1,2,3,4}
  3120: {1,1,1,1,2,3,6}
  9900: {1,1,2,2,3,3,5}

A336500 has zeros at these positions.
A007425 counts divisors of divisors.
A056924 counts divisors greater than their quotient.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A336424 counts factorizations using A130091.
A336422 counts divisible pairs of divisors, both in A130091.
A327498 is the maximum divisor with distinct prime multiplicities.
A336423 counts chains in A130091, with maximal version A336569.
A336571 counts divisor sets using A130091, with maximal version A336570.
Cf. A001055, A002033, A098859, A124010, A167865, A253249, A336420.

strsig[n_]:=UnsameQ@@Last/@FactorInteger[n]
Select[Range[100],Function[n,Select[Divisors[n],strsig[#]&&strsig[n/#]&]=={}]]

A287691 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of squarefree numbers A002110(n) <= m <= (A002110(n+1)-1) such that A001221(m) = n and m is divisible by A002110(k).

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 3, 7, 8, 1, 5, 12, 23, 17, 1, 6, 16, 44, 56, 29, 1, 9, 24, 78, 130, 139, 41, 1, 9, 30, 107, 214, 351, 224, 59, 1, 11, 39, 154, 332, 707, 650, 389, 76, 1, 17, 64, 261, 598, 1475, 1637, 1489, 640, 112, 1, 21, 82, 378, 902, 2496, 3155, 3782

Offset: 0

Michael De Vlieger, May 29 2017

Let p_n# = A002110(n).
T(n,n) = 1 since p_n# is the only primorial divisible by p_n#.
Maxima for the first rows are {1, 2, 4, 8, 23, 56, 139, 351, 707, 1637, 3782, 8843, 18442, 38103, 77355, 177358, 387470, ...} at positions {1, 1, 2, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7, 9, 10, 10, 10, ...}.
A287484(n) = sum of row n. - Michael De Vlieger, Jun 07 2017

			The triangle starts:
   n |   0    1    2    3     4     5     6     7    8    9   10
   -------------------------------------------------------------
   0 |   1
   1 |   2    1
   2 |   2    4    1
   3 |   3    7    8    1
   4 |   5   12   23   17     1
   5 |   6   16   44   56    29     1
   6 |   9   24   78  130   139    41     1
   7 |   9   30  107  214   351   224    59     1
   8 |  11   39  154  332   707   650   389    76    1
   9 |  17   64  261  598  1475  1637  1489   640  112    1
  10 |  21   82  378  902  2496  3155  3782  2505 1041  144    1
      ...
Let p_n# = A002110(n).
There are A287484(2) = 7 squarefree numbers m between p_2# = 6 and p_3# - 1 = 29: {6, 10, 14, 15, 21, 22, 26}. Of these, {15, 21} are divisible by p_0# = 1, {10, 14, 22, 26} are divisible by p_1# = 2, and {6} is divisible by p_2# = 6. Thus, T(2,k) = {2, 4, 1}.
Note that the terms {15, 21}, {10, 14, 22, 26}, and {6} pertaining to the above example appear in row n of A287483 sorted as {6, 10, 14, 15, 21, 22, 26}. - _Michael De Vlieger_, Jun 07 2017

Michael De Vlieger, Table of n, a(n) for n = 0..230 (rows 0 <= n <= 20).

Cf. A001221, A002110, A287483, A287484.

Table[Length /@ Split@ Sort@ Map[Block[{k = 1}, While[Divisible[#, Prime@ k], k++]; k] &, Select[Range[#, Prime[n + 1] #], And[SquareFreeQ@ #, PrimeOmega@ # == n] &] &@ Product[Prime@ i, {i, n}]], {n, 0, 6}] // Flatten (* Michael De Vlieger, May 29 2017 *)

A376294 The product of n's prime powers, with the base and exponent concatenated.

Original entry on oeis.org

1, 21, 31, 22, 51, 651, 71, 23, 32, 1071, 111, 682, 131, 1491, 1581, 24, 171, 672, 191, 1122, 2201, 2331, 231, 713, 52, 2751, 33, 1562, 291, 33201, 311, 25, 3441, 3591, 3621, 704, 371, 4011, 4061, 1173, 411, 46221, 431, 2442, 1632, 4851, 471, 744, 72, 1092, 5301

Offset: 1

Haines Hoag, Sep 19 2024

Multiplicative with a(p^e) = the concatenation of p and e.
A prime which occurs just once in the factorization of n is taken as exponent 1 so that for instance n = 7 = 7^1 becomes 71.
It is unknown whether there are any fixed points a(n) = n beyond n=1.

			For n=5, a(5) = 51 since 5=5^1.
For n=20, a(20) = 22*51 = 1122 since 20=2^2*5^1.

Haines Hoag, Table of n, a(n) for n = 1..10000
Combo Class, The Mysterious Pattern I Found Within Prime Factorizations, Sep 12 2024.

Cf. A376254 (indices of a(n) < n), A287483 (indices of local maxima).

a:= n-> mul(parse(cat(i[])), i=ifactors(n)[2]):
seq(a(n), n=1..51);  # Alois P. Heinz, Sep 19 2024

f[p_, e_] := 10^IntegerLength[e] * p + e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 26 2024 *)

a(n)={my(f=factor(n)); prod(i=1, #f~, my([b,e]=f[i,]); b*10^(1+logint(e,10))+e)} \\ Andrew Howroyd, Sep 19 2024

from math import prod
from sympy import factorint
def a(n): return prod(int(str(p)+str(e)) for p, e in factorint(n).items())
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Sep 27 2024

For primes p, a(p) = 10p + 1.

A287692 Triangle read by rows: T(n,k) is the greatest difference between prime factors among squarefree numbers A002110(n) <= m <= (A002110(n+1)-1) such that A001221(m) = n and m is divisible by A002110(k).

Original entry on oeis.org

3, 2, 5, 2, 3, 9, 2, 3, 5, 18, 2, 2, 4, 7, 30, 2, 2, 3, 5, 10, 42, 2, 2, 3, 4, 6, 13, 60, 2, 2, 3, 4, 5, 8, 17, 77, 2, 2, 3, 3, 4, 6, 10, 22, 113, 2, 2, 2, 3, 4, 5, 8, 12, 25, 145, 2, 2, 2, 3, 4, 5, 6, 9, 15, 32, 179, 2, 2, 2, 3, 4, 4, 6, 7, 11, 19, 36, 229

Offset: 1

Michael De Vlieger, Jun 15 2017

Let p_n# = A002110(n).
T(n,1) is the greatest index of the smallest prime divisor p of terms m in row n.
T(n,n) = A120941(n).
Consider the use of A287352 as a method for formulating squarefree numbers with n distinct prime factors. The values in row n serve as a limit beyond which we need not search further for terms p_n# <= m <= (p_(n+1)# - 1). A287352 defines a squarefree number using a sequence of nonzero positive terms, beginning with the index of the smallest prime factor, then listing differences between indexes of subsequent prime factors in order of their magnitude. We can direct increment to the largest prime index as long as the number m < p_(n+1), then increment the index before it, etc. to produce the entire tree of factors that code numbers m.

			Triangle begins:
  n\k|  1   2   3   4   5   6   7   8    9   10   11   12
---------------------------------------------------------
   1 |  3
   2 |  2   5
   3 |  2   3   9
   4 |  2   3   5  18
   5 |  2   2   4   7  30
   6 |  2   2   3   5  10  42
   7 |  2   2   3   4   6  13  60
   8 |  2   2   3   4   5   8  17  77
   9 |  2   2   3   3   4   6  10  22  113
  10 |  2   2   2   3   4   5   8  12   25  145
  11 |  2   2   2   3   4   5   6   9   15   32  179
  12 |  2   2   2   3   4   4   6   7   11   19   36  229
  ...
Let p_n# = A002110(n). For n = 2, there are A287484(2) = 7 squarefree numbers p_2# <= m <= (p_3# - 1) such that omega(m) = n. These are {6, 10, 14, 22, 26, 15, 21}. These numbers m have A287352(m) = {{1,1}, {1,2}, {1,3}, {1,4}, {1,5}, {2,1}, {2,2}} respectively; the largest values in both positions are {2,5}, thus row n = 2 of a(n) is {2,5}.

Cf. A001221, A002110, A120941, A287483, A287484, A287691.

f[n_] := If[n == 0, {{1}}, Block[{P = Product[Prime@ i, {i, n}], lim, k = 1, c, w = ConstantArray[1, n]}, lim = Prime[n + 1] P; {w}~Join~Reap[Do[w = If[k == 1, MapAt[# + 1 &, w, -k], Join[Drop[MapAt[# + 1 &, w, -k], -k + 1], ConstantArray[1, k - 1]]]; c = Times @@ Map[If[# == 0, 1, Prime@ #] &, Accumulate@ w]; If[c < lim, Sow[w]; k = 1, If[k == n, Break[], k++]], {i, Infinity}] ][[-1, 1]] ] ]; Table[Max /@ Transpose@ f@ n, {n, 14}] // Flatten (* Michael De Vlieger, Jun 15 2017 *)

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A336568 Numbers that are not a product of two numbers each having distinct prime multiplicities.

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A287691 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of squarefree numbers A002110(n) <= m <= (A002110(n+1)-1) such that A001221(m) = n and m is divisible by A002110(k).

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A376294 The product of n's prime powers, with the base and exponent concatenated.

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A287692 Triangle read by rows: T(n,k) is the greatest difference between prime factors among squarefree numbers A002110(n) <= m <= (A002110(n+1)-1) such that A001221(m) = n and m is divisible by A002110(k).

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