A272011 -id:A272011 - cp's OEIS frontend

A330350 Table of strictly decreasing sequences with terms in {0, ..., 9}, sorted by length, then lexicographically.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 2, 0, 2, 1, 3, 0, 3, 1, 3, 2, 4, 0, 4, 1, 4, 2, 4, 3, 5, 0, 5, 1, 5, 2, 5, 3, 5, 4, 6, 0, 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 7, 0, 7, 1, 7, 2, 7, 3, 7, 4, 7, 5, 7, 6, 8, 0, 8, 1, 8, 2, 8, 3, 8, 4, 8, 5, 8, 6, 8, 7, 9, 0, 9, 1, 9, 2, 9, 3, 9, 4, 9, 5, 9, 6, 9, 7, 9, 8

Offset: 1

M. F. Hasler, Dec 11 2019

Row n lists the digits of A009995(n), just as row n < 1024 of A272011 lists the digits of A262557(n).

			The first rows start
   n | row n
   1 | 0,
   2 | 1,
    ...
  10 | 9,
  11 | 1, 0,
  12 | 2, 0,
  13 | 2, 1,
  14 | 3, 0,
  15 | 3, 1,
  16 | 3, 2,
  17 | 4, 0,
    ...
The Sury paper lists the first rows of length 3, row 56 = (2, 1, 0), row 57 = (3, 1, 0), row 58 = (3, 2, 0), row 59 = (3, 2, 1), row 60 = (4, 1, 0), ...

M. F. Hasler, Table of n, a(n) for n = 1..5120 (rows 1 .. 1023, flattened).
B. Sury, Macaulay Expansion, Amer. Math. Monthly 121 (2014), no. 4, 359--360. MR3183022.

Cf. A009995, A272011, A262557.

concat(0,[digits(n)|n<-[1..99],is_A009995(n)])

A361376 Rewrite A129912(n), a product of distinct primorials P(i) = A002110(i) instead as a sum of powers 2^(i-1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 16, 11, 17, 12, 13, 18, 19, 32, 14, 33, 20, 15, 21, 34, 35, 22, 24, 64, 23, 36, 25, 65, 37, 26, 66, 38, 27, 67, 40, 128, 39, 41, 28, 68, 129, 29, 69, 42, 130, 48, 43, 30, 70, 72, 131, 49, 31, 71, 44, 73, 256, 132, 45, 50, 257, 133, 74, 51, 46, 80, 75, 258, 134, 136

Offset: 1

Michael De Vlieger, Jun 08 2023

nonn

Permutation of nonnegative numbers.

			a(1) = 0 by convention.
a(8) = 8 comes before a(9) = 7, since we interpret 8 = 2^3 instead as P(4) = 210, while for a(9), 7 = 2^2 + 2^1 + 2^0 becomes P(3)*P(2)*P(1) = 30*6*2 = 360. Because 210 < 360, 8 appears before 7 in this sequence.
Table relating a(n), n=1..19 with the set S(n) of indices of distinct primorial factors of A129912(n):
   n A129912(n)  S(n)   a(n)  A272011(a(n))
  -----------------------------------------
   1         1            0
   2         2   1        1   0
   3         6   2        2   1
   4        12   2,1      3   1,0
   5        30   3        4   2
   6        60   3,1      5   2,0
   7       180   3,2      6   2,1
   8       210   4        8   3
   9       360   3,2,1    7   2,1,0
  10       420   4,1      9   3,0
  11      1260   4,2     10   3,1
  12      2310   5       16   4
  13      2520   4,2,1   11   3,1,0
  14      4620   5,1     17   4,0
  15      6300   4,3     12   3,2
  16     12600   4,3,1   13   3,2,0
  17     13860   5,2     18   4,1
  18     27720   5,2,1   19   4,1,0
  19     30030   6       32   5
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..15303 (a(15303) = 2^29.)
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^6.
Michael De Vlieger, Plot terms S(n) = A272011(a(n)) at (x,y) = (n,S(n,k)) for n = 1..2^11.

Cf. A002110, A129912, A272011, A283477.

a6939[n_] := Product[Prime[n + 1 - i]^i, {i, n}];
g[m_] := Block[{f, j = 1},
  f[n_, i_, e_] :=
   If[n < m, Block[{p = Prime[i + 1]}, If[e == 1, Sow@ n];
     f[n p^e, i + 1, e];
     If[e > 1, f[n p^(e - 1), i + 1, e - 1]]]];
  Sort@ Reap[While[a6939[j] < m, f[2^j, 1, j]; j++]][[-1, 1]] ];
Map[Total@
     Map[2^(# - 1) &,
      Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} ] &[
FactorInteger[#][[All, -1]]] &, g[2^31]] (* Michael De Vlieger, Jun 08 2023, after Giovanni Resta at A129929 *)

Let S(n) be the set of indices of primorials P(i), reverse sorted, such that A129912(n) = Product_{k=1..m} S(n,k), where m = | S(n) |. Then a(n) = Sum_{k=1..m} 2^(S(n,k)-1).

A363235 a(0) = 1; let e be the largest multiplicity such that p^e | a(n); for n>0, a(n) = Sum_{j=1..k} 2^(e(j)-1) where k is the index of the greatest power factor p(k)^e(k) such that p(k-1)^e(k-1) > p(k)^(e(k)+1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 144, 145, 146, 147, 148, 149, 150, 151, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267

Offset: 0

Michael De Vlieger, Jun 09 2023

nonn

A binary compactification of A363250, this sequence rewrites A363250(n) = Product_{i=1..omega(a(n))} p(i)^e(i) instead as Sum_{i=1..omega(a(n))} e(i)-1.

Not a permutation of nonnegative integers.

			a(1) = 1 since 2^1 is a product of the smallest primes p(i) whose prime power factors decrease as i increases; Hence a(1) = 2^(e(i)-1) = 1.
a(2) = 2 since we can find no power 3^e with e>=1 that is smaller than 2^1, we increment the exponent of 2 and have 2^2, hence a(2) = 2^(e(i)-1) = 2.
a(3) = 3 since indeed we may multiply 2^2 by 3^1; 2^2 > 3^1, hence Sum_{i=1..2} 2^(e(i)-1) = 2^1 + 2^0 = 2+1 = 3.
Table relating this sequence to A363250.
b(n) = A363250(n), f(n) = A067255(n), g(n) = A272011(n), with the latter two
   n      b(n)  f(b(n))  a(n)  g(a(n))
  ------------------------------------
   1        1   0          0   -
   2        2   1          1   0
   3        4   2          2   1
   4       12   2,1        3   1,0
   5        8   3          4   2
   6       24   3,1        5   2,0
   7       16   4          8   3
   8       48   4,1        9   3,0
   9      144   4,2       10   3,1
  10      720   4,2,1     11   3,1,0
  11       32   5         16   4
  12       96   5,1       17   4,0
  13      288   5,2       18   4,1
  14     1440   5,2,1     19   4,1,0
  15      864   5,3       20   4,2
  16     4320   5,3,1     21   4,2,0
  17    21600   5,3,2     22   4,2,1
  18   151200   5,3,2,1   23   4,2,1,0
  19       64   6         32   5
  ...
Therefore, a(18) = 23 = 2^4 + 2^2 + 2^1 + 2^0 since b(18) = 151200 = 2^5 * 3^3 * 5^2 * 7^1.
The sequence is a series of intervals, organized so as to begin with 2^k, that begin as follows:
     0
     1
     2..3
     4..5
     8..11
    16..23
    32..39
    64..75
   128..139     144..151
   256..267     272..279
   512..523     528..535     544..559
  1024..1035   1040..1047   1056..1071
  2048..2059   2064..2071   2080..2095   2112..2127
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11210 (a(11210) = 2^28.)
Michael De Vlieger, Binary tree indicating natural numbers k in red that appear in this sequence for k = 1..16383.

Cf. A000079, A067255, A272011, A347284, A363063, A363250.

Select[Range[0, 300], AllTrue[Differences@ MapIndexed[Prime[First[#2]]^#1 &, Length[#] - Position[#, 1][[All, 1]] &@ IntegerDigits[#, 2] + 1], # < 0 &] &]

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)).

Original entry on oeis.org

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.

A372325 Numbers whose binary expansion has an even number of 1's among positions listed in this sequence.

Original entry on oeis.org

0, 2, 5, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 26, 29, 31, 33, 35, 36, 38, 41, 43, 44, 46, 49, 51, 52, 54, 57, 59, 60, 62, 64, 66, 69, 71, 72, 74, 77, 79, 80, 82, 85, 87, 88, 90, 93, 95, 97, 99, 100, 102, 105, 107, 108, 110, 113, 115, 116, 118, 121, 123, 124

Offset: 1

David A. Madore, Apr 27 2024

			118 is in the sequence because 118 = 2^6 + 2^5 + 2^4 + 2^2 + 2^1, and an even number of the exponents 6,5,4,2,1 (namely 2,5) are in the sequence.
8192 is not in the sequence because 8192 = 2^13, and 13 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A133457, A272011.

R:= 0: RL:= [1]: nextp:= 2: m:= 1: count:= 0:
for i from 1 while count < 100 do
  L:= convert(i,base,2);
  if i = nextp then
    nextp:= 2*nextp;
    if R[1+nops(RL)] = m then RL:= [op(RL),m+1] fi;
    m:= m+1;
  fi;
  if convert(L[RL],`+`)::even
  then R:= R,i; count:= count+1
  fi
od:
R; # Robert Israel, May 28 2024

from itertools import count, islice
def agen():  # generator of terms
    aset = 0 # stored as a bitmask
    for k in count(0):
        if (k&aset).bit_count()%2 == 0:
            yield k
            aset += (1<Michael S. Branicky, Apr 28 2024

cp's OEIS Frontend

A330350 Table of strictly decreasing sequences with terms in {0, ..., 9}, sorted by length, then lexicographically.

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A361376 Rewrite A129912(n), a product of distinct primorials P(i) = A002110(i) instead as a sum of powers 2^(i-1).

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A363235 a(0) = 1; let e be the largest multiplicity such that p^e | a(n); for n>0, a(n) = Sum_{j=1..k} 2^(e(j)-1) where k is the index of the greatest power factor p(k)^e(k) such that p(k-1)^e(k-1) > p(k)^(e(k)+1).

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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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A372325 Numbers whose binary expansion has an even number of 1's among positions listed in this sequence.

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