A089633 -id:A089633 - cp's OEIS frontend

A359497 Greatest positive integer whose weakly increasing prime indices have weighted sum (A304818) equal to n.

1, 2, 3, 5, 7, 11, 13, 17, 19, 25, 29, 35, 49, 55, 77, 121, 91, 143, 169, 187, 221, 289, 247, 323, 361, 391, 437, 539, 605, 847, 1331, 715, 1001, 1573, 1183, 1859, 2197, 1547, 2431, 2873, 3179, 3757, 4913, 3553, 4199, 5491, 4693, 6137, 6859, 9317, 14641

Offset: 0

Gus Wiseman, Jan 15 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   25: {3,3}
   29: {10}
   35: {3,4}
   49: {4,4}
   55: {3,5}
   77: {4,5}
The 5 numbers with weighted sum of prime indices 12, together with their prime indices:
  20: {1,1,3}
  27: {2,2,2}
  33: {2,5}
  37: {12}
  49: {4,4}
Hence a(12) = 49.

Andrew Howroyd, Table of n, a(n) for n = 0..500

First position of n in A304818, reverse A318283.
The least instead of greatest is given by A359682, reverse A359679.
The reverse version is A359683.
A112798 lists prime indices, length A001222, sum A056239.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A001248, A029931, A055932, A089633, A243055, A358194, A359043, A359676, A359681, A359755.

nn=10;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ots[y_]:=Sum[i*y[[i]],{i,Length[y]}];
seq=Table[ots[primeMS[n]],{n,1,2^nn}];
Table[Position[seq,k][[-1,1]],{k,0,nn}]

a(n)={ my(recurse(r, k, m) = if(k==1, if(m>=r, prime(r)),
  my(z=0); for(j=1, min(m, (r-k*(k-1)/2)\k), z=max(z, self()(r-k*j, k-1, j)*prime(j))); z));
  if(n==0, 1, vecmax(vector((sqrtint(8*n+1)-1)\2, k, recurse(n, k, n))));
} \\ Andrew Howroyd, Jan 21 2023

Terms a(21) and beyond from Andrew Howroyd, Jan 21 2023

A359679 Least number with weighted sum of reversed (weakly decreasing) prime indices (A318283) equal to n.

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 8, 12, 19, 18, 16, 24, 27, 36, 43, 32, 48, 59, 61, 67, 71, 64, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 0

Gus Wiseman, Jan 14 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i.

			12 has reversed prime indices (2,1,1), with weighted sum 7, and no number < 12 has the same weighted sum of reversed prime indices, so a(7) = 12.

The version for standard compositions is A089633, zero-based A359756.
First position of n in A318283, unreversed A304818.
The unreversed zero-based version is A359676.
The sorted zero-based version is A359680, unreversed A359675.
The zero-based version is A359681.
The unreversed version is A359682.
The greatest instead of least is A359683, unreversed A359497.
The sorted version is A359754, unreversed A359755.
A112798 lists prime indices, length A001222, sum A056239.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A001248, A029931, A053632, A055932, A231204, A243055, A359043, A358194, A359360, A359677.

nn=20;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ots[y_]:=Sum[i*y[[i]],{i,Length[y]}];
seq=Table[ots[Reverse[primeMS[n]]],{n,1,Prime[nn]^2}];
Table[Position[seq,k][[1,1]],{k,0,nn}]

A359675 Positions of first appearances in the sequence of zero-based weighted sums of prime indices (A359674).

Original entry on oeis.org

1, 4, 6, 8, 12, 14, 16, 20, 24, 30, 32, 36, 40, 48, 52, 56, 72, 80, 92, 96, 100, 104, 112, 124, 136, 148, 152, 172, 176, 184, 188, 212, 214, 236, 244, 248, 262, 268, 272, 284, 292, 304, 316, 328, 332, 346, 356, 376, 386, 388, 398, 404, 412, 428, 436, 452, 458

Offset: 1

Gus Wiseman, Jan 13 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.

			The terms together with their prime indices begin:
   1: {}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  12: {1,1,2}
  14: {1,4}
  16: {1,1,1,1}
  20: {1,1,3}
  24: {1,1,1,2}
  30: {1,2,3}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}

Positions of first appearances in A359674.
The unsorted version A359676.
The reverse version is A359680, unsorted A359681.
The reverse one-based version is A359754, unsorted A359679.
The one-based version is A359755, unsorted A359682.
The version for standard compositions is A359756, one-based A089633.
A053632 counts compositions by zero-based weighted sum.
A112798 lists prime indices, length A001222, sum A056239.
A124757 gives zero-based weighted sum of standard compositions, rev A231204.
A304818 gives weighted sum of prime indices, reverse A318283.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A001248, A029931, A055932, A243055, A359043, A358194, A359360, A359497, A359677, A359683.

nn=100;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
seq=Table[wts[primeMS[n]],{n,1,nn}];
Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]

A359680 Positions of first appearances in the sequence of zero-based weighted sums of reversed prime indices (A359677).

Original entry on oeis.org

1, 4, 8, 9, 16, 18, 32, 36, 50, 54, 64, 72, 81, 100, 108, 128, 144, 216, 243, 256, 288, 300, 400, 432, 486, 512, 576, 600, 648, 729, 800, 864, 1024, 1152, 1296, 1350, 1728, 1944, 2048, 2187, 2304, 2400, 2916, 3375, 3456, 3600, 4096, 4374, 4608, 4800, 5184

Offset: 1

Gus Wiseman, Jan 15 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    16: {1,1,1,1}
    18: {1,2,2}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    50: {1,3,3}
    54: {1,2,2,2}
    64: {1,1,1,1,1,1}
    72: {1,1,1,2,2}
    81: {2,2,2,2}
   100: {1,1,3,3}
   108: {1,1,2,2,2}
   128: {1,1,1,1,1,1,1}

The unreversed version is A359675, unsorted A359676.
Positions of first appearances in A359677, unreversed A359674.
This is the sorted version of A359681.
The one-based version is A359754, unsorted A359679.
The unreversed one-based version is A359755, unsorted A359682.
The version for standard compositions is A359756, one-based A089633.
A053632 counts compositions by zero-based weighted sum.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124757 gives zero-based weighted sums of standard compositions, rev A231204.
A304818 gives weighted sum of prime indices, reverse A318283.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A029931, A055932, A243055, A358194, A359043, A359683.

nn=1000;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
seq=Table[wts[Reverse[primeMS[n]]],{n,1,nn}];
Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]

A359683 Greatest positive integer whose reversed (weakly decreasing) prime indices have weighted sum (A318283) equal to n.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 14, 22, 26, 34, 44, 55, 68, 85, 110, 130, 170, 190, 242, 290, 374, 418, 506, 638, 748, 836, 1012, 1276, 1364, 1628, 1914, 2090, 2552, 3190, 3410, 4070, 4510, 5060, 6380, 7018, 8140, 9020, 9922, 11396, 14036, 15004, 17908, 19844, 21692, 23452

Offset: 0

Gus Wiseman, Jan 15 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i.

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      3: {2}
      5: {3}
      7: {4}
     11: {5}
     14: {1,4}
     22: {1,5}
     26: {1,6}
     34: {1,7}
     44: {1,1,5}
     55: {3,5}
     68: {1,1,7}
     85: {3,7}
    110: {1,3,5}
    130: {1,3,6}
    170: {1,3,7}
    190: {1,3,8}
    242: {1,5,5}
    290: {1,3,10}
The 6 numbers with weighted sum of reversed prime indices 9, together with their prime indices:
  18: {1,2,2}
  23: {9}
  25: {3,3}
  28: {1,1,4}
  33: {2,5}
  34: {1,7}
Hence a(9) = 34.

First position of n in A318283, unreversed A304818.
The unreversed version is A359497.
The least instead of greatest is A359679, unreversed A359682.
A112798 lists prime indices, length A001222, sum A056239.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A001248, A029931, A055932, A089633, A243055, A358194, A359043, A359676, A359681, A359755.

nn=10;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ots[y_]:=Sum[i*y[[i]],{i,Length[y]}];
seq=Table[ots[Reverse[primeMS[n]]],{n,1,2^nn}];
Table[Position[seq,k][[-1,1]],{k,0,nn}]

More terms from Jinyuan Wang, Jan 26 2023

A359754 Positions of first appearances in the sequence of weighted sums of reversed prime indices (A318283).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 19, 24, 27, 32, 36, 43, 48, 59, 61, 64, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Gus Wiseman, Jan 15 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   19: {8}
   24: {1,1,1,2}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   43: {14}
   48: {1,1,1,1,2}

Positions of first appearances in A318283, unreversed A304818.
This is the sorted version of A359679.
The zero-based version is A359680, unreversed A359675.
The unreversed version is A359755, unsorted A359682.
A053632 counts compositions by weighted sum.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A029931, A089633, A124757, A243055, A358194, A359497, A359674, A359677, A359681, A359683.

nn=100;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ots[y_]:=Sum[i*y[[i]],{i,Length[y]}];
seq=Table[ots[Reverse[primeMS[n]]],{n,1,nn}];
Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]

A081118 Triangle of first n numbers per row having exactly n 1's in binary representation.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 23, 27, 29, 31, 47, 55, 59, 61, 63, 95, 111, 119, 123, 125, 127, 191, 223, 239, 247, 251, 253, 255, 383, 447, 479, 495, 503, 507, 509, 511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1023, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043

Offset: 1

Reinhard Zumkeller, Mar 06 2003

T(n,n) = A036563(n+1) = 2^(n+1) - 3.

Numbers of the form 2^t - 2^k - 1, 1 <= k < t.

			Triangle begins:
.......... 1 ......... ................ 1
........ 3...5 ....... .............. 11 101
...... 7..11..13 ..... .......... 111 1011 1101
... 15..23..27..29 ... ...... 1111 10111 11011 11101
. 31..47..55..59..61 . . 11111 101111 110111 111011 111101.

Reinhard Zumkeller, Rows n=1..150 of triangle, flattened

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691, A038461, A038462, A038463.
Cf. A181741 (primes), A208083, subsequence of A089633.
Cf. A131094.

a081118 n k = a081118_tabl !! (n-1) !! (k-1)
a081118_row n = a081118_tabl !! (n-1)
a081118_tabl  = iterate
   (\row -> (map ((+ 1) . (* 2)) row) ++ [4 * (head row) + 1]) [1]
a081118_list = concat a081118_tabl
-- Reinhard Zumkeller, Feb 23 2012

Table[2^(n+1)-2^(n-k+1)-1,{n,10},{k,n}]//Flatten (* Harvey P. Dale, Apr 09 2020 *)

T(n, k) = 2^(n+1) - 2^(n-k+1) - 1, 1<=k<=n.

a(n) = (2^A002260(n)-1)*2^A004736(n)-1; a(n)=(2^i-1)*2^j-1, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Apr 04 2013

A118462 Decimal equivalent of binary encoding of partitions into distinct parts.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 6, 9, 16, 7, 10, 17, 32, 11, 12, 18, 33, 64, 13, 19, 20, 34, 65, 128, 14, 21, 24, 35, 36, 66, 129, 256, 15, 22, 25, 37, 40, 67, 68, 130, 257, 512, 23, 26, 38, 41, 48, 69, 72, 131, 132, 258, 513, 1024, 27, 28, 39, 42, 49, 70, 73, 80, 133, 136, 259, 260, 514

Offset: 0

Franklin T. Adams-Watters, Apr 28 2006

A part of size k in the partition makes the 2^(k-1) bit of the number be 1. The partitions of n are in reverse Mathematica ordering, so that each row is in ascending order. This is a permutation of the nonnegative integers.
The sequence is the concatenation of the sets: e_n={j>=0: A029931(j)=n}, n=0,1,...: e_0={0}, e_1={1}, e_2={2}, e_3={3,4}, e_4={5,8}, e_5={6,9,16}, e_6={7,10,17,32}, e_7={11,12,18.33.64}, ... . - Vladimir Shevelev, Mar 16 2009
This permutation of the nonnegative integers A001477 has fixed points 0, 1, 2, 3, 4, 5, 325, 562, 800, 4449, ... and inverse permutation A118463. - Alois P. Heinz, Sep 06 2014
Row n lists in increasing order the binary ranks of all strict integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). - Gus Wiseman, May 21 2024

			Partition 11 is [4,2], which gives binary 1010 (2^(4-1)+2^(2-1)), or 10, so a(11)=10.
Triangle begins:
   0;
   1;
   2;
   3,  4;
   5,  8;
   6,  9, 16;
   7, 10, 17, 32;
  11, 12, 18, 33, 64;
  13, 19, 20, 34, 65, 128;
  14, 21, 24, 35, 36,  66, 129, 256;
  15, 22, 25, 37, 40,  67,  68, 130, 257, 512;
  ...
From _Gus Wiseman_, May 21 2024: (Start)
The tetrangle of strict partitions (A118457) begins:
  (1)  (2)  (2,1)  (3,1)  (3,2)  (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)
            (3)    (4)    (4,1)  (4,2)    (4,3)    (5,2,1)  (5,3,1)
                          (5)    (5,1)    (5,2)    (5,3)    (5,4)
                                 (6)      (6,1)    (6,2)    (6,2,1)
                                          (7)      (7,1)    (6,3)
                                                   (8)      (7,2)
                                                            (8,1)
                                                            (9)
(End)

Alois P. Heinz, Rows n = 0..42, flattened
V. Shevelev, A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n), arXiv:0903.1743 [math.NT], 2009. [From _Vladimir Shevelev_, Mar 17 2009]
Index entries for sequences that are permutations of the natural numbers

Cf. A118463, A118457, A000009 (row lengths).
Cf. A089633 (first column), A000079 (last in each column). - Franklin T. Adams-Watters, Mar 16 2009
Cf. A246867.
A variation encoding all partitions is A225620.
Row sums are A372888.
A048793 lists binary indices, sum A029931, length A000120.
Cf. A000041, A019565, A029837, A048675, A272020.

b:= proc(n, i) option remember; `if`(n=0, [0], `if`(i<1, [], [seq(
      map(p->p+2^(i-1)*j, b(n-i*j, i-1))[], j=0..min(1, n/i))]))
    end:
T:= n-> sort(b(n$2))[]:
seq(T(n), n=0..14);  # Alois P. Heinz, Sep 06 2014

b[n_, i_] := b[n, i] = If[n==0, {0}, If[i<1, {}, Flatten[Table[b[n-i*j, i-1 ] + 2^(i-1)*j, {j, 0, Min[1, n/i]}]]]]; T[n_] := Sort[b[n, n]]; Table[ T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 27 2015, after Alois P. Heinz *)
Table[Total[2^(#-1)]&/@Select[Reverse[IntegerPartitions[n]],UnsameQ@@#&],{n,0,10}] (* Gus Wiseman, May 21 2024 *)

A077011 Triangle read by rows: T(n,k) = A002110(n)/prime(n+1-k), k = 1..n.

Original entry on oeis.org

1, 2, 3, 6, 10, 15, 30, 42, 70, 105, 210, 330, 462, 770, 1155, 2310, 2730, 4290, 6006, 10010, 15015, 30030, 39270, 46410, 72930, 102102, 170170, 255255, 510510, 570570, 746130, 881790, 1385670, 1939938, 3233230, 4849845, 9699690, 11741730

Offset: 1

Amarnath Murthy, Oct 26 2002

Original Name was: "Triangle in which the n-th row contains all possible products of n-1 of the first n primes in ascending order."
A024451(n) gives the sum of the n-th row.
When the triangle is parsed in blocks of ascending length, as shown in the example, there is the following interpretation: The integers Z regarded as a module over themselves contain unshortenable generating sets of different lengths, in fact, infinitely many of each desired length. Each of the blocks is the minimal example of an unshortenable generating set of the respective length. For example, {6,10,15} generates Z as 1=6+10-15. However, removing one of the numbers leaves two numbers that are not relatively prime, precluding generation of Z. An analogous argument succeeds for all other blocks alike. Each block contains numbers such that there is no prime factor common to all. Taking differences sufficiently often one ends up with two coprime numbers whence the generating property follows from Bézout's theorem. If just one number is removed from the set, relative primality is lost. The minimality of the numbers used in each block is evident from the construction. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 04 2006

			Triangle begins:
      1;
      2,     3;
      6,    10,    15;
     30,    42,    70,   105;
    210,   330,   462,   770,   1155;
   2310,  2730,  4290,  6006,  10010,  15015;
  30030, 39270, 46410, 72930, 102102, 170170, 255255;
  ...

Alois P. Heinz, Rows n = 1..130, flattened

Row sums give A024451.
Reversal of A258566.
Cf. A002110, A089633, A372666.

T:= proc(n) local t;
      t:= mul(ithprime(i), i=1..n);
      seq(t/ithprime(n-i), i=0..n-1)
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Jun 04 2012

T[n_] := Module[{t = Product[Prime[i], {i, 1, n}]}, Table[t/Prime[n - i], {i, 0, n - 1}]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, May 19 2016, translated from Maple *)

T(n,k) = vecprod(primes(n))/prime(n+1-k); \\ Michel Marcus, May 19 2024

A089633(n-1) = Sum_{p | n} 2^(pi(p) - 1) for n > 1, pi(x) = A000720(x). - Michael De Vlieger, May 19 2024

More terms from Sascha Kurz, Jan 26 2003

Name changed by David James Sycamore, May 19 2024

A142151 a(n) = OR{k XOR (n-k): 0<=k<=n}.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 6, 7, 14, 13, 14, 11, 14, 13, 14, 15, 30, 29, 30, 27, 30, 29, 30, 23, 30, 29, 30, 27, 30, 29, 30, 31, 62, 61, 62, 59, 62, 61, 62, 55, 62, 61, 62, 59, 62, 61, 62, 47, 62, 61, 62, 59, 62, 61, 62, 55, 62, 61, 62, 59, 62, 61, 62, 63, 126, 125, 126, 123, 126, 125

Offset: 0

Reinhard Zumkeller, Jul 15 2008

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Logical Convolutions

Cf. A003817, A000004, A142149, A086099, A142150, A001477, A089633.

import Data.Bits (xor, (.|.))
a142151 :: Integer -> Integer
a142151 = foldl (.|.) 0 . zipWith xor [0..] . reverse . enumFromTo 1
-- Reinhard Zumkeller, Mar 31 2015

using IntegerSequences
A142151List(len) = [Bits("CIMP", n, n+1) for n in 0:len]
println(A142151List(69))  # Peter Luschny, Sep 25 2021

A142151 := n -> n + Bits:-Nor(n, n+1):
seq(A142151(n), n=0..69); # Peter Luschny, Sep 26 2019

from functools import reduce
from operator import or_
def A142151(n): return 0 if n == 0 else reduce(or_,(k^n-k for k in range(n+1))) if n % 2 else (1 << n.bit_length()-1)-1 <<1 # Chai Wah Wu, Jun 30 2022

a(2*n) = 2*(A062383(n)-1);

A023416(a(n)) <= 1.

cp's OEIS Frontend

A359497 Greatest positive integer whose weakly increasing prime indices have weighted sum (A304818) equal to n.

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A359679 Least number with weighted sum of reversed (weakly decreasing) prime indices (A318283) equal to n.

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A359675 Positions of first appearances in the sequence of zero-based weighted sums of prime indices (A359674).

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A359680 Positions of first appearances in the sequence of zero-based weighted sums of reversed prime indices (A359677).

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A359683 Greatest positive integer whose reversed (weakly decreasing) prime indices have weighted sum (A318283) equal to n.

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A359754 Positions of first appearances in the sequence of weighted sums of reversed prime indices (A318283).

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A081118 Triangle of first n numbers per row having exactly n 1's in binary representation.

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A118462 Decimal equivalent of binary encoding of partitions into distinct parts.

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