A067255 -id:A067255 - cp's OEIS frontend

A363537 Rewrite A087980(n) = Product_{i=1..m} p(i)^e(i) instead as Sum_{i=1..m} 2^(i-1), where m = omega(A087980(n)) = A001221(A087980(n)).

0, 1, 2, 4, 3, 8, 5, 16, 9, 32, 6, 17, 64, 10, 33, 128, 18, 7, 65, 12, 256, 34, 11, 129, 20, 512, 66, 19, 257, 36, 1024, 13, 130, 24, 35, 513, 68, 2048, 21, 258, 40, 67, 1025, 132, 4096, 37, 514, 72, 14, 131, 2049, 25, 260, 48, 8192, 69, 1026, 136, 22, 259, 4097, 41, 516, 80, 16384, 133, 2050, 264, 38

Offset: 1

Michael De Vlieger, Jun 09 2023

nonn

Permutation of nonnegative numbers.

Rewriting nonnegative numbers n = Sum_{i=1..A000120(n)} 2^i instead as Product_{i=1..A000120(n)} p(i)^(e(i)+1) gives A362227.

			Table relating this sequence to A087980, where b(n) = A087980(n), f(n) = A067255(n), g(n) = A272011(n), and a(n)_2 the binary expansion of a(n):
   n   b(n)  f(b(n))  a(n)  g(a(n))   a(n)_2
   1     1   0         0
   2     2   1         1    0             1
   3     4   2         2    1            1.
   4     8   3         4    2           1..
   5    12   2,1       3    1,0          11
   6    16   4         8    3          1...
   7    24   3,1       5    2,0         1.1
   8    32   5        16    4         1....
   9    48   4,1       9    3,0        1..1
  10    64   6        32    5        1.....
  11    72   3,2       6    2,1         11.
  12    96   5,1      17    4,0       1...1
  13   128   7        64    6       1......
  14   144   4,2      10    3,1        1.1.
  15   192   6,1      33    5,0      1....1
  16   256   8       128    7      1.......
  17   288   5,2      18    4,1       1..1.
  18   360   3,2,1     7    2,1,0       111
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11157 (a(11157) = 2^64.)
Michael De Vlieger, Log log scatterplot of a(n), n = 1..1203278.
Michael De Vlieger, Plot S(n,k) in row a(n) of A272011 at (x,y) = (n,k), n = 1..2048.

Cf. A000079, A001221, A006939, A087980, A362227.

m = 15; f[n_] := Times @@ MapIndexed[Prime[First[#2]]^(#1 + 1) &, Length[#] - Position[#, 1][[All, 1]]] &[IntegerDigits[n, 2]]; SortBy[Select[Array[{#, f[#]} &, 2^(m + 1)], Last[#] <= 2^m &], Last][[All, 1]]

a(2^k) = 2^(k-1) for k > 0.

a(A006939(k)) = 2^k-1 for k > 0.

A384003 Irregular triangle T(n,k), n >= 0, 0 <= k < 2^(n-1), where T(n,k) = Product_{j=0..n-1} prime(j+1)^((n-j)*d_j), where d_j is the bit with digit weight 2^j in the binary expansion of 2^(n-1)+k.

Original entry on oeis.org

1, 2, 3, 12, 5, 40, 45, 360, 7, 112, 189, 3024, 175, 2800, 4725, 75600, 11, 352, 891, 28512, 1375, 44000, 111375, 3564000, 539, 17248, 43659, 1397088, 67375, 2156000, 5457375, 174636000, 13, 832, 3159, 202176, 8125, 520000, 1974375, 126360000, 4459, 285376, 1083537

Offset: 0

Michael De Vlieger and Peter Munn, May 28 2025

This sequence can be seen as a structured ordering of numbers m that are not divisible by the square of their greatest prime factor and where every prime in the canonical factorization of m has the same sum of prime index and exponent. For example, prime(1)^3 * prime(3)^1 = 2^3 * 5 = 40. The ordering is lexicographic according to prime divisors listed in decreasing order, as used for A019565. Row n has the numbers whose greatest prime factor is prime(n).

			Table begins:
n\k  0    1    2     3    4     5     6       7
-----------------------------------------------
0:   1;
1:   2;
2:   3,  12;
3:   5,  40,  45,  360;
4:   7, 112, 189, 3024, 175, 2800, 4725, 75600;
     ...
Table showing prime power decomposition of a(n), where A067255(a(n)) represents prime(i)^j | a(n), with j in the i-th position, replacing 0 with "." for visibility:
 n     a(n)  A067255(a(n))
--------------------------
 0       1   .
 1       2   1
 2       3   .1
 3      12   21
 4       5   ..1
 5      40   3.1
 6      45   .21
 7     360   321
 8       7   ...1
 9     112   4..1
10     189   .3.1
11    3024   43.1
12     175   ..21
13    2800   4.21
14    4725   .321
15   75600   4321

Michael De Vlieger, Table of n, a(n) for n = 0..16384 (rows n = 0..14, flattened)
Michael De Vlieger, Log log scatterplot of a(n), n = 0..16384.
Michael De Vlieger, Plot prime(i)^j at (x,y) = (n,i), n = 0..2047, 16X vertical exaggeration, with a color function representing j = 1 in black, j = 2 in red, j = 3 in orange, ..., j = 14 in magenta.

Cf. A000040, A003961, A006939, A007947, A019565, A060175, A061395, A071178, A251720.

f[x_] := If[x == 1, {0}, Function[g, ReplacePart[Table[0, {PrimePi[f[[-1, 1]] ]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, g]]@ FactorInteger@ x]; Table[f[Reverse@ Range[Length[#]]*#] &@ Reverse@ IntegerDigits[n, 2], {n, 0, 120}]

T(0,0) = 1; T(1,0) = 2.
Otherwise, T(n,2k) = A003961(T(n-1,k)).
T(n,2k+1) = T(n,2k)*2^n.
T(n,0) = prime(n).
T(n,2^(n-1)-1) = A006939(n).
T(n,2^(n-2)) = A251720(n).
Using a(m) to denote a term of the linear sequence with offset 0: (Start)
A019565(m) = A007947(a(m)).
a(m) = T(n,k) = gcd(A019565(m)^n, A006939(n)).
Equivalently, for p = A000040(i), the i-th prime, p|a(m) iff p|A019565(m), in which case A060175(m,i) = j - i + 1, where j = PrimePi(gpf(A019565(m))) = A061395(A019565(m)).
(End)
For n > 0, A071178(T(n,k)) = 1.

Name edited by Peter Munn, Aug 30 2025

A307458 Composites c where an integer b with 1 < b < c exists such that when the k digits in the base-b expansion of c are considered as exponents in an ordered list of primes prime(1), prime(2), ..., prime(k), then Product_{i=1..k} prime(i)^d[i] = c, where d[h] gives the h-th most significant digit in the expansion.

Original entry on oeis.org

6, 10, 18, 36, 54, 96, 100, 162, 200, 216, 256, 324, 486, 1296, 1458, 2916, 4374, 5832, 13122, 26244, 39366, 46656, 47250, 49000, 65536, 82944, 104976, 118098, 157464, 181500, 236196, 354294, 746496, 1062882, 1492992, 1679616, 1990656, 2125764, 3188646, 3538944

Offset: 1

Felix Fröhlich, Apr 09 2019

In other words, integers k where an integer b with 1 < b < k exists such that row k of A067255 gives the digits of the base-b expansion of k.

Clearly, all terms are even, since all expansions start with a nonzero digit and thus the factorization of each term contains the prime 2.

			The base-4 expansion of 200 is 3020. 2^3 * 3^0 * 5^2 * 7^0 = 200, so 200 is a term of the sequence.

Cf. A067255.

base[n_] := Block[{e, t=0, m, b=0, s=False, p, x, pp}, pp = PrimePi@ FactorInteger[n][[-1, 1]]; If[2^(pp - 1) > n, 0, e = IntegerExponent[n, Prime@ Range@ pp]; m = Max[e] + 1; p = Total[Reverse[e] x^Range[0, Length[e] - 1]]; While[((p x^t) /. x -> m ) <= n, s = Reduce[p x^t == n && m <= x < n, x, Integers]; If[s === False, t++, b = x /. List[ToRules@ s][[1]]; Break[], t++]]; b]]; Select[Range[4, 10^5, 2], base[#] > 0 &] (* Giovanni Resta, Apr 10 2019 *)

is(n) = for(b=2, n-1, my(d=digits(n, b), k=#d, x=1); while(k > 0, x=x*prime(k)^d[k]; k--); if(x==n, return(1))); 0
for(t=1, oo, if(is(2*t), print1(2*t, ", ")))

a(21)-a(40) from Giovanni Resta, Apr 10 2019

A331938 Indices of A002110(n) in A055932.

Original entry on oeis.org

1, 2, 4, 10, 28, 83, 227, 626, 1644, 4290, 11322, 28965, 74469, 189436, 471910, 1166247, 2884920, 7130085, 17349489, 42180190, 101820577, 242907065, 579402163, 1375056009, 3262651085, 7768448187, 18411720785

Offset: 0

Michael De Vlieger, Feb 02 2020

A055932 lists numbers m whose prime divisors p are consecutive primes starting with 2, admitting multiplicity, while A002110 lists numbers m that are products of distinct consecutive primes starting with 2. Therefore, A002110 is a subset of A055932.
Offset is 0 since A002110(0) = 1.
Let 0 <= i <= k, integers. We can write an efficient algorithm to construct a complete list of all terms m of A055932 up to A002110(k) using A067255(m). Every term m in the list has omega(m) = A001221(m) <= k. Starting with A002110(i), we use A067255 to encode m, i.e., the list of multiplicities e pertaining to the 1st..i-th prime p_i, allowing position of the multiplicity e in the list to convey p_i. Thus, the first "recipe" for m = A002110(i) = {1, 1, ..., 1}, a list of i ones. If this does not exceed the limit A002110(k), then we accept it as a value, then increment the last multiplicity. When we have an invalid recipe, we increment the penultimate multiplicity and reset the last to 1, etc., until we have generated all m <= A002110(k). As a measure of efficiency, this algorithm generates 1 <= m <= A002110(12), 74469 terms, in about 2 seconds including sorting, on a 64-bit Intel Xeon E-2286M (2.40 GHz) processor. This is the same amount of time it takes to test numbers 1..400000 to yield the 575 smallest terms of the same sequence.

Cf. A001221, A002110, A055932, A067255.

With[{s = Import["https://oeis.org/A002110/b002110.txt", "Data"][[1 ;; 8, -1]], t = TakeWhile[Import["https://oeis.org/A055932/b055932.txt", "Data"], Length@ # > 0 &][[All, -1]]}, TakeWhile[Map[FirstPosition[t, #][[1]] &, s], IntegerQ] ]

a(21)-a(26) from Giovanni Resta, Feb 03 2020

A387465 Irregular triangle T(n,k), n >= 0, 0 <= k < 2^(n-1), where T(n,k) = Product_{j=0..n-1} prime(n-j)^((j+1)*d_j), where d_j is the bit with digit weight 2^j in the binary expansion of 2^(n-1)+k.

Original entry on oeis.org

1, 2, 4, 12, 8, 40, 72, 360, 16, 112, 400, 2800, 432, 3024, 10800, 75600, 32, 352, 1568, 17248, 4000, 44000, 196000, 2156000, 2592, 28512, 127008, 1397088, 324000, 3564000, 15876000, 174636000, 64, 832, 7744, 100672, 21952, 285376, 2656192, 34530496, 40000, 520000

Offset: 0

Michael De Vlieger and Peter Munn, Aug 29 2025

For n >= 1, row n consists of the even unitary divisors of A006939(n).
The range of properties is best understood when also viewed as a linear sequence with offset 0, so that a(floor(2^(n-1))+k) = T(n,k).
For even n > 0, a(n) is powerful.
For odd n > 1, a(n) is in A332785.
a(1) = 2 is the only prime term.
a(0) = 1 and a(1) = 2 are the only squarefree terms.
a(2^k) = 2^(k+1).
Perfect powers in this sequence include A000079, but also numbers like 400 = prime(1)^4 * prime(3)^2.

			Table begins:
n\k  0    1    2     3    4     5      6       7
------------------------------------------------
0:   1;
1:   2;
2:   4,  12;
3:   8,  40,  72,  360;
4:  16, 112, 400, 2800, 432, 3024, 10800, 75600;
     ...
Table showing prime power decomposition of a(n), where A067255(a(n)) represents prime(i)^j | a(n), with j in the i-th position, replacing 0 with "." for visibility:
 n     a(n)  A067255(a(n))
--------------------------
 0       1   .
 1       2   1
 2       4   2
 3      12   21
 4       8   3
 5      40   3.1
 6      72   32
 7     360   321
 8      16   4
 9     112   4..1
10     400   4.2
11    2800   4.21
12     432   43
13    3024   43.1
14   10800   432
15   75600   4321

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Log log scatterplot of a(n) n = 0..2^14.
Michael De Vlieger, Plot prime(i)^j at (x,y) = (n,i), n = 0..2047, 16X vertical exaggeration, with a color function representing j = 1 in black, j = 2 in red, j = 3 in orange, ..., j = 15 in magenta.

Cf. A001597, A001694, A110765 (squarefree kernel), A286708, A362227, A363250, A384003.
All terms are in A304686.
See the comments for the relationships with A000079, A006939, A332785.
See the formula section for the relationships with A000120, A001221, A001222, A007947, A019565, A029837, A029931, A036044, A064549, A093141, A167747, A242378, A265127.
See the examples for the relationship with A067255.

f[x_] := If[x == 1, {0},
 Function[g,ReplacePart[Table[0, {PrimePi[f[[-1, 1]] ]}], #] &@
     Map[PrimePi@ First@ # -> Last@ # &, g] ]@ FactorInteger@ x];
Table[f[Reverse@ Range[Length[#]]*#] &@ Reverse@ IntegerDigits[n, 2], {n, 0, 120}]

Let row_index(n) = A029837(n+1) = ceiling(log_2(n+1)), giving the row of the triangle that contains term a(n) of the linear sequence.
Let lin_index(n,k) = floor(2^(n-1))+k, giving the index in the linear sequence corresponding to term T(n,k) of the triangle.
a(0) = 1; otherwise:
  a(2n) = A064549(a(n)) = a(n) * rad(a(n));
  a(2n+1) = a(2n) * prime(row_index(n)).
T(n,0) = 2^n.
T(n,k) = T(n,0) * A242378(n - row_index(k), a(k)), where A242378(i,j) "adds i to the indices of the prime factors of j".
T(n,ceiling(2^(n-1))-1) = A006939(n).
For n > 1, T(n,1) = A265127(n) = 2^n * prime(n).
For n > 2, T(n,2^(n-1)) = A167747(n) = phi(6^n) = 2^n * 3^(n-1).
For n > 2, T(n,2^(n-2)) = A093141(n-1) = 4 * 10^(n-1). = 2^n * 5^(n-2).
T(n,k) = max({j >= 1 : j|A006939(n) and gcd(j, A019565(A036044(lin_index(n,k)))) = 1}).
A110765(n) = A007947(a(n)).
A001221(a(n)) = A000120(n).
A001222(a(n)) = A029931(n).

cp's OEIS Frontend

A363537 Rewrite A087980(n) = Product_{i=1..m} p(i)^e(i) instead as Sum_{i=1..m} 2^(i-1), where m = omega(A087980(n)) = A001221(A087980(n)).

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A384003 Irregular triangle T(n,k), n >= 0, 0 <= k < 2^(n-1), where T(n,k) = Product_{j=0..n-1} prime(j+1)^((n-j)*d_j), where d_j is the bit with digit weight 2^j in the binary expansion of 2^(n-1)+k.

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A331938 Indices of A002110(n) in A055932.

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A387465 Irregular triangle T(n,k), n >= 0, 0 <= k < 2^(n-1), where T(n,k) = Product_{j=0..n-1} prime(n-j)^((j+1)*d_j), where d_j is the bit with digit weight 2^j in the binary expansion of 2^(n-1)+k.

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