A304818 -id:A304818 - cp's OEIS frontend

A363621 Positive integers whose prime indices have reverse-weighted alternating sum 0.

1, 6, 21, 40, 50, 54, 65, 132, 133, 154, 210, 224, 319, 340, 351, 360, 374, 392, 450, 481, 486, 507, 546, 598, 624, 644, 731, 825, 855, 969, 1007, 1029, 1054, 1144, 1210, 1254, 1320, 1364, 1386, 1403, 1408, 1520, 1558, 1653, 1750, 1785, 1827, 1836, 1890, 1960

Offset: 1

Gus Wiseman, Jun 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.

We define the reverse-weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(k-i) i * y_{k-i+1}.

			The prime indices of 360 are {1,1,1,2,2,3}, with reverse-weighted alternating sum 1*3 - 2*2 + 3*2 - 4*1 + 5*1 - 6*1 = 0, so 360 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     6: {1,2}
    21: {2,4}
    40: {1,1,1,3}
    50: {1,3,3}
    54: {1,2,2,2}
    65: {3,6}
   132: {1,1,2,5}
   133: {4,8}
   154: {1,4,5}
   210: {1,2,3,4}
   224: {1,1,1,1,1,4}
   319: {5,10}
   340: {1,1,3,7}
   351: {2,2,2,6}
   360: {1,1,1,2,2,3}

The unweighted version is A000290.
Partitions of this type are counted by A363532.
Positions of zeros in A363620 and A363624, reverse A363619 and A363625.
Compositions of this type are counted by A363626.
A053632 counts compositions by weighted sum.
A055396 gives minimum prime index, maximum A061395.
A112798 lists prime indices, length A001222, sum A056239.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices.
A318283 gives weighted sum of reversed prime indices.
A320387 counts multisets by weighted sum.
A344616 gives reverse-alternating sum of prime indices.
A363623 counts partitions by reverse-weighted alternating sum.
Cf. A046660, A106529, A124010, A181819, A261079, A316524, A359674, A363622.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
revaltwtsum[y_]:=Sum[(-1)^(Length[y]-k)*k*y[[-k]],{k,1,Length[y]}];
Select[Range[1000],revaltwtsum[prix[#]]==0&]

A363624 Weighted alternating sum of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 2, -1, 3, 0, 4, 2, -2, 1, 5, 3, 6, 2, -1, -2, 7, 1, 8, 4, 0, 3, 9, -1, -3, 4, 4, 5, 10, 2, 11, 3, 1, 5, -2, -3, 12, 6, 2, 0, 13, 3, 14, 6, 5, 7, 15, 4, -4, 0, 3, 7, 16, 0, -1, 1, 4, 8, 17, -2, 18, 9, 6, -3, 0, 4, 19, 8, 5, 1, 20, 2, 21, 10, 3, 9, -3, 5

Offset: 1

Gus Wiseman, Jun 13 2023

sign

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

We define the weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(i - 1) * i * y_i.

			The partition with Heinz number 600 is (3,3,2,1,1,1), with weighted alternating sum 1*3 - 2*3 + 3*2 - 4*1 + 5*1 - 6*1 = -2, so a(600) = -2.

The non-alternating version is A318283, reverse A304818.
The unweighted version is A344616, reverse A316524.
For multisets instead of partitions we have A363619.
Positions of zeros are A363621, counted by A363532.
The triangle for this rank statistic is A363622, reverse A363623.
The reverse version is A363625, for multisets A363620.
A055396 gives minimum prime index, maximum A061395.
A112798 lists prime indices, length A001222, sum A056239.
A264034 counts partitions by weighted sum, reverse A358194.
A320387 counts multisets by weighted sum, reverse A007294.
A359677 gives zero-based weighted sum of prime indices, reverse A359674.
A363626 counts compositions with reverse-weighted alternating sum 0.
Cf. A046660, A053632, A106529, A124010, A215366, A261079, A358136, A359361, A359755.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]],{k,1,Length[y]}];
Table[altwtsum[Reverse[prix[n]]],{n,100}]

A363625 Reverse-weighted alternating sum of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 2, 2, 5, 5, 5, 6, 7, 4, 2, 7, 3, 8, 8, 6, 9, 9, 6, 3, 11, 4, 11, 10, 6, 11, 3, 8, 13, 5, 3, 12, 15, 10, 10, 13, 9, 14, 14, 7, 17, 15, 8, 4, 4, 12, 17, 16, 5, 7, 14, 14, 19, 17, 7, 18, 21, 10, 3, 9, 12, 19, 20, 16, 7, 20, 4, 21, 23, 5, 23

Offset: 1

Gus Wiseman, Jun 15 2023

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

We define the reverse-weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(k-i) * i * y_{k-i+1}.

			The partition with Heinz number 600 is (3,3,2,1,1,1), so a(600) = -1*1 + 2*1 - 3*1 + 4*2 - 5*3 + 6*3 = 9.

The non-alternating version is A304818, reverse A318283.
The unweighted version is A316524, reverse A344616.
For multisets instead of partitions we have A363620.
The triangle for this rank statistic is A363623, reverse A363622.
The reverse version is A363624, for multisets A363619.
A055396 gives minimum prime index, maximum A061395.
A112798 lists prime indices, length A001222, sum A056239.
A264034 counts partitions by weighted sum, reverse A358194.
A320387 counts multisets by weighted sum, reverse A007294.
Cf. A046660, A053632, A106529, A124010, A261079, A359361, A359755, A363532, A363621, A363626.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
revaltwtsum[y_]:=Sum[(-1)^(Length[y]-k)*k*y[[-k]],{k,1,Length[y]}];
Table[revaltwtsum[Reverse[prix[n]]],{n,100}]

A372432 Positive integers k such that the prime indices of k are not disjoint from the binary indices of k.

Original entry on oeis.org

3, 5, 6, 14, 15, 18, 20, 22, 27, 28, 30, 39, 42, 45, 51, 52, 54, 55, 56, 60, 63, 66, 68, 70, 75, 77, 78, 85, 87, 88, 90, 91, 95, 99, 100, 102, 104, 105, 110, 111, 114, 117, 119, 121, 123, 125, 126, 133, 135, 138, 140, 147, 150, 152, 154, 159, 162, 165, 168

Offset: 1

Gus Wiseman, May 03 2024

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.

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 binary indices of 18 are {2,5}, and the prime indices are {1,2,2}, so 18 is in the sequence.
The terms together with their prime indices begin:
    3: {2}
    5: {3}
    6: {1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   27: {2,2,2}
   28: {1,1,4}
   30: {1,2,3}
The terms together with their binary expansions and binary indices begin:
    3:      11 ~ {1,2}
    5:     101 ~ {1,3}
    6:     110 ~ {2,3}
   14:    1110 ~ {2,3,4}
   15:    1111 ~ {1,2,3,4}
   18:   10010 ~ {2,5}
   20:   10100 ~ {3,5}
   22:   10110 ~ {2,3,5}
   27:   11011 ~ {1,2,4,5}
   28:   11100 ~ {3,4,5}
   30:   11110 ~ {2,3,4,5}

For subset instead of overlap we have A372430.
The complement is A372431.
Equal lengths: A071814, zeros of A372441.
Equal sums: A372427, zeros of A372428.
Equal maxima: A372436, zeros of A372442.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A001221, A059893, A096111, A230877, A243055, A304818, A355536, A358136, A372429.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[bix[#],prix[#]]!={}&]

A372586 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 12, 15, 16, 17, 20, 21, 29, 32, 36, 42, 43, 45, 46, 47, 48, 51, 53, 54, 55, 59, 60, 61, 63, 64, 65, 66, 67, 68, 71, 73, 78, 79, 80, 81, 84, 89, 91, 93, 94, 95, 97, 99, 101, 105, 110, 111, 113, 114, 115, 116, 118, 119, 121, 122, 125, 127

Offset: 1

Gus Wiseman, May 14 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.
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 even version is A372587.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
            {1}   1  ()
            {2}   2  (1)
          {1,2}   3  (2)
            {3}   4  (1,1)
          {1,3}   5  (3)
            {4}   8  (1,1,1)
          {1,4}   9  (2,2)
          {3,4}  12  (2,1,1)
      {1,2,3,4}  15  (3,2)
            {5}  16  (1,1,1,1)
          {1,5}  17  (7)
          {3,5}  20  (3,1,1)
        {1,3,5}  21  (4,2)
      {1,3,4,5}  29  (10)
            {6}  32  (1,1,1,1,1)
          {3,6}  36  (2,2,1,1)
        {2,4,6}  42  (4,2,1)
      {1,2,4,6}  43  (14)
      {1,3,4,6}  45  (3,2,2)
      {2,3,4,6}  46  (9,1)
    {1,2,3,4,6}  47  (15)
          {5,6}  48  (2,1,1,1,1)

Positions of odd terms in A372428, zeros A372427.
For minimum (A372437) we have A372439, complement A372440.
For length (A372441, zeros A071814) we have A372590, complement A372591.
For maximum (A372442, zeros A372436) we have A372588, complement A372589.
The complement is A372587.
For just binary indices:
- length: A000069, complement A001969
- sum: A158705, complement A158704
- minimum: A003159, complement A036554
- maximum: A053738, complement A053754
For just prime indices:
- length: A026424 (count A027193), complement A028260 (count A027187)
- sum: A300063 (count A058695), complement A300061 (count A058696)
- minimum: A340932 (count A026804), complement A340933 (count A026805)
- maximum: A244991 (count A027193), complement A244990 (count A027187)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A066208, A160786, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],OddQ[Total[bix[#]]+Total[prix[#]]]&]

Numbers k such that A029931(k) + A056239(k) is odd.

A372589 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is even.

Original entry on oeis.org

3, 4, 5, 9, 12, 13, 14, 16, 17, 20, 22, 23, 25, 30, 31, 35, 36, 37, 38, 39, 42, 43, 48, 49, 52, 53, 54, 56, 57, 58, 61, 63, 64, 66, 67, 68, 69, 73, 75, 77, 80, 82, 83, 85, 88, 90, 92, 93, 94, 97, 99, 100, 102, 103, 109, 110, 115, 118, 119, 120, 121, 123, 124

Offset: 1

Gus Wiseman, May 14 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.
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 odd version is A372588.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {1,2}   3  (2)
          {3}   4  (1,1)
        {1,3}   5  (3)
        {1,4}   9  (2,2)
        {3,4}  12  (2,1,1)
      {1,3,4}  13  (6)
      {2,3,4}  14  (4,1)
          {5}  16  (1,1,1,1)
        {1,5}  17  (7)
        {3,5}  20  (3,1,1)
      {2,3,5}  22  (5,1)
    {1,2,3,5}  23  (9)
      {1,4,5}  25  (3,3)
    {2,3,4,5}  30  (3,2,1)
  {1,2,3,4,5}  31  (11)
      {1,2,6}  35  (4,3)
        {3,6}  36  (2,2,1,1)
      {1,3,6}  37  (12)
      {2,3,6}  38  (8,1)
    {1,2,3,6}  39  (6,2)
      {2,4,6}  42  (4,2,1)
    {1,2,4,6}  43  (14)

For sum (A372428, zeros A372427) we have A372587, complement A372586.
For minimum (A372437) we have A372440, complement A372439.
For length (A372441, zeros A071814) we have A372591, complement A372590.
Positions of even terms in A372442, zeros A372436.
The complement is A372588.
For just binary indices:
- length: A001969, complement A000069
- sum: A158704, complement A158705
- minimum:  A036554, complement A003159
- maximum: A053754, complement A053738
For just prime indices:
- length: A026424 A028260 (count A027187), complement (count A027193)
- sum: A300061 (count A058696), complement A300063 (count A058695)
- minimum: A340933 (count A026805), complement A340932 (count A026804)
- maximum: A244990 (count A027187), complement A244991 (count A027193)
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031215 lists even-indexed primes, odd A031368.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A006141, A066207, A243055, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[2,100],EvenQ[IntegerLength[#,2]+PrimePi[FactorInteger[#][[-1,1]]]]&]

Numbers k such that A070939(k) + A061395(k) is even.

A372590 Numbers whose binary weight (A000120) plus bigomega (A001222) is odd.

Original entry on oeis.org

1, 3, 4, 5, 12, 14, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 29, 30, 35, 38, 43, 45, 48, 49, 53, 55, 56, 62, 63, 64, 66, 68, 69, 71, 72, 74, 75, 78, 80, 81, 82, 83, 84, 87, 88, 89, 91, 92, 93, 94, 99, 100, 101, 102, 104, 105, 108, 113, 114, 115, 116, 118, 120

Offset: 1

Gus Wiseman, May 14 2024

nonn

The even version is A372591.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {1}   1  ()
      {1,2}   3  (2)
        {3}   4  (1,1)
      {1,3}   5  (3)
      {3,4}  12  (2,1,1)
    {2,3,4}  14  (4,1)
        {5}  16  (1,1,1,1)
      {1,5}  17  (7)
      {2,5}  18  (2,2,1)
      {3,5}  20  (3,1,1)
    {1,3,5}  21  (4,2)
    {2,3,5}  22  (5,1)
  {1,2,3,5}  23  (9)
    {1,4,5}  25  (3,3)
    {2,4,5}  26  (6,1)
  {1,2,4,5}  27  (2,2,2)
  {1,3,4,5}  29  (10)
  {2,3,4,5}  30  (3,2,1)
    {1,2,6}  35  (4,3)
    {2,3,6}  38  (8,1)
  {1,2,4,6}  43  (14)
  {1,3,4,6}  45  (3,2,2)

For sum (A372428, zeros A372427) we have A372586, complement A372587.
For minimum (A372437) we have A372439, complement A372440.
Positions of odd terms in A372441, zeros A071814.
For maximum (A372442, zeros A372436) we have A372588, complement A372589.
The complement is A372591.
For just binary indices:
- length: A000069, complement A001969
- sum: A158705, complement A158704
- minimum: A003159, complement A036554
- maximum: A053738, complement A053754
For just prime indices:
- length: A026424 (count A027193), complement A028260 (count A027187)
- sum: A300063 (count A058695), complement A300061 (count A058696)
- minimum: A340932 (count A026804), complement A340933 (count A026805)
- maximum: A244991 (count A027193), complement A244990 (count A027187)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A066208, A160786, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[100],OddQ[DigitCount[#,2,1]+PrimeOmega[#]]&]

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.

A362560 Number of integer partitions of n whose weighted sum is not divisible by n.

Original entry on oeis.org

0, 1, 1, 4, 5, 8, 12, 19, 25, 38, 51, 70, 93, 124, 162, 217, 279, 360, 462, 601, 750, 955, 1203, 1502, 1881, 2336, 2892, 3596, 4407, 5416, 6623, 8083, 9830, 11943, 14471, 17488, 21059, 25317, 30376, 36424, 43489, 51906, 61789, 73498, 87186, 103253, 122098

Offset: 1

Gus Wiseman, Apr 28 2023

nonn

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.

Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.

			The weighted sum of y = (3,3,1) is 1*3+2*3+3*1 = 12, which is not a multiple of 7, so y is counted under a(7).
The a(2) = 1 through a(7) = 12 partitions:
  (11)  (21)  (22)    (32)    (33)      (43)
              (31)    (41)    (42)      (52)
              (211)   (221)   (51)      (61)
              (1111)  (311)   (321)     (322)
                      (2111)  (411)     (331)
                              (2211)    (421)
                              (21111)   (511)
                              (111111)  (2221)
                                        (4111)
                                        (22111)
                                        (31111)
                                        (211111)

For median instead of mean we have A322439 aerated, complement A362558.
The complement is counted by A362559.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A264034 counts partitions by weighted sum.
A304818 = weighted sum of prime indices, row-sums of A359361.
A318283 = weighted sum of reversed prime indices, row-sums of A358136.
Cf. A001227, A051293, A067538, A240219, A261079, A326622, A349156, A360068, A360069, A360241, A362051.

Table[Length[Select[IntegerPartitions[n],!Divisible[Total[Accumulate[Reverse[#]]],n]&]],{n,30}]

A363531 Heinz numbers of integer partitions such that 3*(sum) = (reverse-weighted sum).

Original entry on oeis.org

1, 32, 144, 216, 243, 672, 1008, 1350, 2176, 2250, 2520, 2673, 3125, 3969, 4160, 4200, 5940, 6240, 6615, 7344, 7424, 7744, 8262, 9261, 9800, 9900, 10400, 11616, 12250, 12312, 12375, 13104, 13720, 14720, 14742, 16767, 16807, 17150, 19360, 21840, 22080, 23100

Offset: 1

Gus Wiseman, Jun 12 2023

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. The reverse-weighted sum is the weighted sum of the reverse, also the sum of partial sums. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18 and the reverse-weighted sum is 4*4 + 3*2 + 2*2 + 1*1 = 27.

			The terms together with their prime indices begin:
      1: {}
     32: {1,1,1,1,1}
    144: {1,1,1,1,2,2}
    216: {1,1,1,2,2,2}
    243: {2,2,2,2,2}
    672: {1,1,1,1,1,2,4}
   1008: {1,1,1,1,2,2,4}
   1350: {1,2,2,2,3,3}
   2176: {1,1,1,1,1,1,1,7}
   2250: {1,2,2,3,3,3}
   2520: {1,1,1,2,2,3,4}
   2673: {2,2,2,2,2,5}
   3125: {3,3,3,3,3}
   3969: {2,2,2,2,4,4}
   4160: {1,1,1,1,1,1,3,6}

These partitions are counted by A363526.
The non-reverse version is A363530, counted by A363527.
A053632 counts compositions by weighted sum.
A055396 gives minimum prime index, maximum A061395.
A112798 lists prime indices, length A001222, sum A056239.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, row-sums of A359361.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
A320387 counts multisets by weighted sum, zero-based A359678.
Cf. A000041, A000720, A001221, A046660, A106529, A118914, A124010, A181819, A215366, A359362, A359755.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],3*Total[prix[#]]==Total[Accumulate[prix[#]]]&]

A056239(a(n)) = A318283(a(n))/3.

cp's OEIS Frontend

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A372586 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is odd.

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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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A362560 Number of integer partitions of n whose weighted sum is not divisible by n.

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