A243055 -id:A243055 - cp's OEIS frontend

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

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[[#]]]&]

A355531 Minimal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 1, 1, 5, 1, 6, 1, 2, 1, 7, 1, 8, 1, 2, 1, 9, 1, 1, 1, 1, 1, 10, 1, 11, 1, 2, 1, 2, 1, 12, 1, 2, 1, 13, 1, 14, 1, 1, 1, 15, 1, 1, 1, 2, 1, 16, 1, 3, 1, 2, 1, 17, 1, 18, 1, 1, 1, 3, 1, 19, 1, 2, 1, 20, 1, 21, 1, 1, 1, 2, 1, 22, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 14 2022

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 augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

			The reversed prime indices of 825 are (5,3,3,2), with augmented differences (3,1,2,2), so a(825) = 1.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are A008578.
Positions of 1's are 2 followed by A013929.
The non-augmented maximal version is A286470, also A355526.
The non-augmented version is A355524, also A355525.
Row minima of A355534, which has Heinz number A325351.
The maximal version is A355535.
A001222 counts prime indices.
A112798 lists prime indices, sum A056239.
Cf. A124010, A129654, A243055, A243056, A307824, A325366, A325394, A355528, A355533, A355536.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aug[y_]:=Table[If[i

A329343 Difference between the indices of the smallest and the largest primorial in the greedy sum of primorials adding to A108951(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 0, 0, 2

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

The greedy sum is also the sum with the minimal number of primorials, used for example in the primorial base representation.

Positions of the records (and conjecturally, the positions of the first occurrences of each n) begin as 1, 8, 27, 162, 289, 529, 841, 1369, 1681, 2209, 2809, 3481, 4489, 5041, 5329, 6889, ..., that after 162 all seem to be squares of certain primes. See also A329051.

			For n = 18 = 2 * 3^2, A108951(18) = A034386(2) * A034386(3)^2 = 2 * 6^2 = 72 = 30 + 30 + 6 + 6, and as the largest primorial in the sum is 30 = A002110(3), and the least primorial is 6 = A002110(2), we have a(18) = 3-2 = 1.

Cf. A002110, A034386, A108951, A243055, A276086, A324886, A324888, A329040, A329051.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A243055(n) = if(1==n,0,my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); (primepi(gpf)-primepi(lpf)));
A329343(n) = A243055(A324886(n));

a(n) = A243055(A324886(n)).

A372588 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is odd.

Original entry on oeis.org

2, 6, 7, 8, 10, 11, 15, 18, 19, 21, 24, 26, 27, 28, 29, 32, 33, 34, 40, 41, 44, 45, 46, 47, 50, 51, 55, 59, 60, 62, 65, 70, 71, 72, 74, 76, 78, 79, 81, 84, 86, 87, 89, 91, 95, 96, 98, 101, 104, 105, 106, 107, 108, 111, 112, 113, 114, 116, 117, 122, 126, 128

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 A372589.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {2}   2  (1)
      {2,3}   6  (2,1)
    {1,2,3}   7  (4)
        {4}   8  (1,1,1)
      {2,4}  10  (3,1)
    {1,2,4}  11  (5)
  {1,2,3,4}  15  (3,2)
      {2,5}  18  (2,2,1)
    {1,2,5}  19  (8)
    {1,3,5}  21  (4,2)
      {4,5}  24  (2,1,1,1)
    {2,4,5}  26  (6,1)
  {1,2,4,5}  27  (2,2,2)
    {3,4,5}  28  (4,1,1)
  {1,3,4,5}  29  (10)
        {6}  32  (1,1,1,1,1)
      {1,6}  33  (5,2)
      {2,6}  34  (7,1)
      {4,6}  40  (3,1,1,1)
    {1,4,6}  41  (13)
    {3,4,6}  44  (5,1,1)
  {1,3,4,6}  45  (3,2,2)

For sum (A372428, zeros A372427) we have A372586.
For minimum (A372437) we have A372439, complement A372440.
For length (A372441, zeros A071814) we have A372590, complement A372591.
Positions of odd terms in A372442, zeros A372436.
The complement is A372589.
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, A006141, A066208, A160786, A243055, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

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

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

A260442 Sequence A260443 sorted into ascending order.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 15, 17, 18, 19, 23, 29, 30, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 75, 77, 79, 83, 89, 90, 97, 101, 103, 105, 107, 109, 113, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 210, 211, 221, 223, 227, 229, 233, 239, 241, 245, 251, 257, 263, 269, 270, 271, 277, 281, 283, 293, 307, 311

Offset: 0

Antti Karttunen, Jul 29 2015

nonn

Each term is a prime factorization encoding of one of the Stern polynomials. See A260443 for details.

Numbers n for which A260443(A048675(n)) = n. - Antti Karttunen, Oct 14 2016

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A003961, A048675, A260443.
Subsequence of A073491.
From 2 onward the positions of nonzeros in A277333.
Various subsequences: A000040, A002110, A070826, A277317, A277200 (even terms).  Also all terms of A277318 are included here.
Cf. also A277323, A277324 and permutation pair A277415 & A277416.

allocatemem(2^30);
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2))));
isA260442(n) = (A260443(A048675(n)) == n);  \\ The most naive version.
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])) \\ Charles R Greathouse IV, Apr 23 2015
A061395(n) =  if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
isA260442(n) = ((1==n) || isprime(n) || ((omega(n) == 1+(A061395(n)-A055396(n))) && (A260443(A048675(n)) == n))); \\ Somewhat optimized.
i=0; n=0; while(i < 10001, n++; if(isA260442(n), write("b260442.txt", i, " ", n); i++));
\\ Antti Karttunen, Oct 14 2016

from sympy import factorint, prime, primepi
from operator import mul
from functools import reduce
def a048675(n):
    F=factorint(n)
    return 0 if n==1 else sum([F[i]*2**(primepi(i) - 1) for i in F])
def a003961(n):
    F=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
def a(n): return n + 1 if n<2 else a003961(a(n//2)) if n%2==0 else a((n - 1)//2)*a((n + 1)//2)
print([n for n in range(301) if a(a048675(n))==n]) # Indranil Ghosh, Jun 21 2017

;; With Antti Karttunen's IntSeq-library.
(define A260442 (FIXED-POINTS 0 1 (COMPOSE A260443 A048675)))
;; An optimized version:
(define A260442 (MATCHING-POS 0 1 (lambda (n) (or (= 1 n) (= 1 (A010051 n)) (and (not (< (A001221 n) (+ 1 (A243055 n)))) (= n (A260443 (A048675 n))))))))
;; Antti Karttunen, Oct 14 2016

A358135 Difference of first and last parts of the n-th composition in standard order.

Original entry on oeis.org

0, 0, 0, 0, -1, 1, 0, 0, -2, 0, -1, 2, 0, 1, 0, 0, -3, -1, -2, 1, -1, 0, -1, 3, 0, 1, 0, 2, 0, 1, 0, 0, -4, -2, -3, 0, -2, -1, -2, 2, -1, 0, -1, 1, -1, 0, -1, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, -5, -3, -4, -1, -3, -2, -3, 1, -2, -1, -2, 0, -2

Offset: 1

Gus Wiseman, Oct 31 2022

sign

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.

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

See link for sequences related to standard compositions.
The first and last parts are A065120 and A001511.
This is the first minus last part of row n of A066099.
The version for Heinz numbers of partitions is A243055.
Row sums of A358133.
The partial sums of standard compositions are A358134, adjusted A242628.
A011782 counts compositions.
A333766 and A333768 give max and min in standard compositions, diff A358138.
A351014 counts distinct runs in standard compositions.
Cf. A000120, A001511, A029837, A029931, A048896, A058891, A070939, A133494, A329395, A357187.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[-First[stc[n]]+Last[stc[n]],{n,1,100}]

a(n) = A001511(n) - A065120(n).

A359360 Length times minimum part of the integer partition with Heinz number n. Least prime index of n times number of prime indices of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 3, 4, 2, 5, 3, 6, 2, 4, 4, 7, 3, 8, 3, 4, 2, 9, 4, 6, 2, 6, 3, 10, 3, 11, 5, 4, 2, 6, 4, 12, 2, 4, 4, 13, 3, 14, 3, 6, 2, 15, 5, 8, 3, 4, 3, 16, 4, 6, 4, 4, 2, 17, 4, 18, 2, 6, 6, 6, 3, 19, 3, 4, 3, 20, 5, 21, 2, 6, 3, 8, 3, 22, 5, 8, 2

Offset: 1

Gus Wiseman, Dec 28 2022

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A prime index of n is a number m such that prime(m) divides n.

			The partition with Heinz number 7865 is (6,5,5,3), so a(7865) = 4*3 = 12.

Difference of A056239 and A359358.
The opposite version is A326846.
A055396 gives minimum prime index, maximum A061395.
A112798 list prime indices, length A001222, sum A056239.
A243055 subtracts the least prime index from the greatest.
A358195 gives Heinz numbers of rows of A358172, even bisection A241916.
Cf. A006141, A124010, A246277, A268192, A316413, A325352, A326836, A326837, A326844, A355534, A356958.

Table[PrimeOmega[n]*PrimePi[FactorInteger[n][[1,1]]],{n,100}]

a(n) = if (n==1, 0, my(f=factor(n)); bigomega(f)*primepi(f[1, 1])); \\ Michel Marcus, Dec 28 2022

a(n) = A001222(n) * A055396(n).

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[#]]!={}&]

cp's OEIS Frontend

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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A355531 Minimal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

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A329343 Difference between the indices of the smallest and the largest primorial in the greedy sum of primorials adding to A108951(n).

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A372588 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is odd.

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A260442 Sequence A260443 sorted into ascending order.

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A358135 Difference of first and last parts of the n-th composition in standard order.

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