A231204 -id:A231204 - cp's OEIS frontend

A359677 Zero-based weighted sum of the reversed (weakly decreasing) prime indices of n.

0, 0, 0, 1, 0, 1, 0, 3, 2, 1, 0, 3, 0, 1, 2, 6, 0, 4, 0, 3, 2, 1, 0, 6, 3, 1, 6, 3, 0, 4, 0, 10, 2, 1, 3, 7, 0, 1, 2, 6, 0, 4, 0, 3, 6, 1, 0, 10, 4, 5, 2, 3, 0, 9, 3, 6, 2, 1, 0, 7, 0, 1, 6, 15, 3, 4, 0, 3, 2, 5, 0, 11, 0, 1, 7, 3, 4, 4, 0, 10, 12, 1, 0, 7, 3

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 reversed prime indices of 12 are (2,1,1), so a(12) = 0*2 + 1*1 + 2*1 = 3.

Positions of 0's are A008578.
Positions of 1's are A100484.
The version for standard compositions is A231204, reverse of A124757.
The one-based version is A318283, unreversed A304818.
The one-based version for standard compositions is A359042, rev of A029931.
This is the reverse version of A359674.
First position of n is A359679(n), reverse of A359675.
Positions of first appearances are A359680, reverse of A359676.
A053632 counts compositions by weighted sum.
A112798 lists prime indices, length A001222, sum A056239.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A001248, A055932, A243055, A359043, A358194, A359678, A359681, A359683, A359754, A359755.

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]}];
Table[wts[Reverse[primeMS[n]]],{n,100}]

A359681 Least positive integer whose reversed (weakly decreasing) prime indices have zero-based weighted sum (A359677) equal to n.

Original entry on oeis.org

1, 4, 9, 8, 18, 50, 16, 36, 100, 54, 32, 72, 81, 108, 300, 64, 144, 400, 216, 600, 243, 128, 288, 800, 432, 486, 1350, 648, 256, 576, 729, 864, 2400, 3375, 1296, 3600, 512, 1152, 1944, 1728, 4800, 9000, 2187, 2916, 8100, 1024, 2304, 6400, 3456, 4374, 12150

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 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}
    9: {2,2}
    8: {1,1,1}
   18: {1,2,2}
   50: {1,3,3}
   16: {1,1,1,1}
   36: {1,1,2,2}
  100: {1,1,3,3}
   54: {1,2,2,2}
   32: {1,1,1,1,1}
   72: {1,1,1,2,2}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  300: {1,1,2,3,3}

The unreversed version is A359676.
First position of n in A359677, reverse A359674.
The one-based version is A359679, sorted A359754.
The sorted version is A359680, reverse A359675.
The unreversed one-based version is A359682, sorted A359755.
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.
Cf. A001248, A029931, A055932, A089633, A243055, A359043, A358194, A359360, A359361, A359497, A359683.

nn=20;
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,Prime[nn]^2}];
Table[Position[seq,k][[1,1]],{k,0,nn}]

A359495 Sum of positions of 1's in binary expansion minus sum of positions of 1's in reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jan 05 2023

Also the sum of partial sums of reversed binary expansion minus sum of partial sums of binary expansion.

			The binary expansion of 158 is (1,0,0,1,1,1,1,0), with positions of 1's {1,4,5,6,7} with sum 23, reversed {2,3,4,5,8} with sum 22, so a(158) = 1.

Alois P. Heinz, Table of n, a(n) for n = 0..16383

Indices of positive terms are A359401.
Indices of 0's are A359402.
A030190 gives binary expansion, reverse A030308.
A070939 counts binary digits.
A230877 adds up positions of 1's in binary expansion, reverse A029931.
Cf. A000120, A048793, A053632, A222955, A231204, A291166, A326669, A326672, A326673, A359042, A359043.

a:= n-> (l-> add(i*(l[-i]-l[i]), i=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..127);  # Alois P. Heinz, Jan 09 2023

sap[q_]:=Sum[q[[i]]*(2i-Length[q]-1),{i,Length[q]}];
Table[sap[IntegerDigits[n,2]],{n,0,100}]

def A359495(n):
    k = n.bit_length()-1
    return sum((i<<1)-k for i, j in enumerate(bin(n)[2:]) if j=='1') # Chai Wah Wu, Jan 09 2023

a(n) = A029931(n) - A230877(n).

If n = Sum_{i=1..k} q_i * 2^(i-1), then a(n) = Sum_{i=1..k} q_i * (2i-k-1).

A222955 Number of nX1 0..1 arrays with every row and column least squares fitting to a zero slope straight line, with a single point array taken as having zero slope.

Original entry on oeis.org

2, 2, 4, 4, 8, 8, 20, 18, 52, 48, 152, 138, 472, 428, 1520, 1392, 5044, 4652, 17112, 15884, 59008, 55124, 206260, 193724, 729096, 688008, 2601640, 2465134, 9358944, 8899700, 33904324, 32342236, 123580884, 118215780, 452902072, 434314138, 1667837680

Offset: 1

R. H. Hardin, Mar 10 2013

nonn

Column 1 of A222959

Conjecture: A binary word is counted iff it has the same sum of positions of 1's as its reverse, or, equivalently, the same sum of partial sums as its reverse. - Gus Wiseman, Jan 07 2023

			All solutions for n=4
..0....1....1....0
..0....1....0....1
..0....1....0....1
..0....1....1....0
From _Gus Wiseman_, Jan 07 2023: (Start)
The a(1) = 2 through a(7) = 20 binary words with least squares fit a line of zero slope are:
  (0)  (00)  (000)  (0000)  (00000)  (000000)  (0000000)
  (1)  (11)  (010)  (0110)  (00100)  (001100)  (0001000)
             (101)  (1001)  (01010)  (010010)  (0010100)
             (111)  (1111)  (01110)  (011110)  (0011100)
                            (10001)  (100001)  (0100010)
                            (10101)  (101101)  (0101010)
                            (11011)  (110011)  (0110001)
                            (11111)  (111111)  (0110110)
                                               (0111001)
                                               (0111110)
                                               (1000001)
                                               (1000110)
                                               (1001001)
                                               (1001110)
                                               (1010101)
                                               (1011101)
                                               (1100011)
                                               (1101011)
                                               (1110111)
                                               (1111111)
(End)

R. H. Hardin, Table of n, a(n) for n = 1..210

These words appear to be ranked by A359402.
A011782 counts compositions.
A359042 adds up partial sums of standard compositions, reversed A029931.
Cf. A053632, A070925, A231204, A318283, A359043.

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[[#]]]&]

A359401 Nonnegative integers whose sum of positions of 1's in their binary expansion is greater than the sum of positions of 1's in their reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

Original entry on oeis.org

11, 19, 23, 35, 37, 39, 43, 47, 55, 67, 69, 71, 75, 77, 79, 83, 87, 91, 95, 103, 111, 131, 133, 134, 135, 137, 139, 141, 142, 143, 147, 149, 151, 155, 157, 158, 159, 163, 167, 171, 173, 175, 179, 183, 187, 191, 199, 203, 207, 215, 223, 239, 259, 261, 262, 263

Offset: 1

Gus Wiseman, Jan 05 2023

First differs from A161601 in having 134, with binary expansion (1,0,0,0,0,1,1,0), positions of 1's 1 + 6 + 7 = 14, reversed 2 + 3 + 8 = 13.

Indices of positive terms in A359495; indices of 0's are A359402.
A030190 gives binary expansion, reverse A030308.
A070939 counts binary digits.
A230877 adds up positions of 1's in binary expansion, reverse A029931.
A326669 lists numbers with integer mean position of a 1 in binary expansion.
Cf. A000120, A048793, A051293, A053632, A222955, A231204, A291166, A304818, A326672, A326673, A359043.

sap[q_]:=Sum[q[[i]]*(2i-Length[q]-1),{i,Length[q]}];
Select[Range[0,100],sap[IntegerDigits[#,2]]>0&]

A230877(a(n)) > A029931(a(n)).

A359756 First position of n in the sequence of zero-based weighted sums of standard compositions (A124757), if we start with position 0.

Original entry on oeis.org

0, 3, 6, 7, 13, 14, 15, 27, 29, 30, 31, 55, 59, 61, 62, 63, 111, 119, 123, 125, 126

Offset: 0

Gus Wiseman, Jan 17 2023

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.
Is this sequence strictly increasing?

			The terms together with their standard compositions begin:
    0: ()
    3: (1,1)
    6: (1,2)
    7: (1,1,1)
   13: (1,2,1)
   14: (1,1,2)
   15: (1,1,1,1)
   27: (1,2,1,1)
   29: (1,1,2,1)
   30: (1,1,1,2)
   31: (1,1,1,1,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

The one-based version is A089633, for prime indices A359682.
First index of n in A124757, reverse A231204.
The version for prime indices is A359676, reverse A359681.
A053632 counts compositions by zero-based weighted sum.
A066099 lists standard compositions.
A304818 gives weighted sums of prime indices, reverse A318283.
A320387 counts multisets by weighted sum, zero-based A359678.
Cf. A000120, A029931, A059893, A070939, A083329, A359043, A359674.

nn=10;
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
seq=Table[wts[stc[n]],{n,0,2^(nn-1)}];
Table[Position[seq,k][[1,1]]-1,{k,0,nn}]

Appears to be the complement of A083329 in A089633.

A372687 Number of prime numbers whose binary indices sum to n. Number of strict integer partitions y of n such that Sum_i 2^(y_i-1) is prime.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 2, 1, 2, 0, 3, 3, 1, 4, 1, 6, 5, 8, 4, 12, 8, 12, 7, 20, 8, 16, 17, 27, 19, 38, 19, 46, 33, 38, 49, 65, 47, 67, 83, 92, 94, 113, 103, 130, 146, 127, 215, 224, 176, 234, 306, 270, 357, 383, 339, 393, 537, 540, 597, 683, 576, 798, 1026, 830, 1157

Offset: 0

Gus Wiseman, May 15 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Note the inverse of A048793 (binary indices) takes a set s to Sum_i 2^(s_i-1).

			The a(2) = 1 through a(17) = 8 prime numbers:
  2  3  5  .  17  11  19  .  257  131  73  137  97  521  4099  1031
              7       13     67   41       71       263  2053  523
                             37   23       43       139  1033  269
                                           29       83   193   163
                                                    53   47    149
                                                    31         101
                                                               89
                                                               79
The a(2) = 1 through a(11) = 3 strict partitions:
  (2)  (2,1)  (3,1)  .  (5,1)    (4,2,1)  (4,3,1)  .  (9,1)    (6,4,1)
                        (3,2,1)           (5,2,1)     (6,3,1)  (8,2,1)
                                                      (7,2,1)  (5,3,2,1)

For all positive integers (not just prime) we get A000009.
Number of prime numbers p with A029931(p) = n.
For odd instead of prime we have A096765, even A025147, non-strict A087787
Number of times n appears in A372429.
Number of rows of A372471 with sum n.
The non-strict version is A372688 (or A372887), ranks A277319 (or A372850).
These (strict) partitions have Heinz numbers A372851.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A038499 counts partitions of prime length, strict A085756.
A048793 lists binary indices:
- length A000120
- min A001511
- sum A029931
- max A070939
- reverse A272020
A058698 counts partitions of prime numbers, strict A064688.
A096111 gives product of binary indices.
A372689 lists numbers whose binary indices sum to a prime.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A023506, A029837, A035100, A071814, A230877, A231204, A359359, A372436, A372441.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&PrimeQ[Total[2^#]/2]&]],{n,0,30}]

cp's OEIS Frontend

A359677 Zero-based weighted sum of the reversed (weakly decreasing) prime indices of n.

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A359681 Least positive integer whose reversed (weakly decreasing) prime indices have zero-based weighted sum (A359677) equal to n.

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A359495 Sum of positions of 1's in binary expansion minus sum of positions of 1's in reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

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A222955 Number of nX1 0..1 arrays with every row and column least squares fitting to a zero slope straight line, with a single point array taken as having zero slope.

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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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A359401 Nonnegative integers whose sum of positions of 1's in their binary expansion is greater than the sum of positions of 1's in their reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

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A359756 First position of n in the sequence of zero-based weighted sums of standard compositions (A124757), if we start with position 0.

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