A200947 -id:A200947 - cp's OEIS frontend

A121559 Final result (0 or 1) under iterations of {r mod (max prime p <= r)} starting at r = n.

1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

Kerry Mitchell, Aug 07 2006

Previous name: Find r1 = n modulo p1, where p1 is the largest prime not greater than n. Then find r2 = r1 modulo p2, where p2 is the largest prime not greater than r1. Repeat until the last r is either 1 or 0; a(n) is the last r value.
The sequence has the form of blocks of 0's between 1's. See sequence A121560 for the lengths of the blocks of zeros.
The function r mod (max prime p <= r), which appears in the definition, equals r - (max prime p <= r) = A064722(r), because p <= r < 2*p by Bertrand's postulate, where p is the largest prime less than or equal to r. - Pontus von Brömssen, Jul 31 2022

			a(9) = 0 because 7 is the largest prime not larger than 9, 9 mod 7 = 2, 2 is the largest prime not greater than 2 and 2 mod 2 = 0.

Kerry Mitchell, Table of n, a(n) for n = 1..7919

Cf. A007917 and A064722 (both for the iterations).

Cf. A121560, A121561, A121562, A175077, A200947.

Abs[Table[FixedPoint[Mod[#,NextPrime[#+1,-1]]&,n],{n,110}]] (* Harvey P. Dale, Mar 17 2023 *)

a(n) = if (n==1, return (1)); na = n; while((nb = (na % precprime(na))) > 1, na = nb); return(nb); \\ Michel Marcus, Aug 22 2014

a(p) = 0 when p is prime. - Michel Marcus, Aug 22 2014
a(n) = A175077(n+1) - 1. - Pontus von Brömssen, Jul 31 2022
a(n) = A200947(n) mod 2. - Alois P. Heinz, Jun 12 2023

New name from Michel Marcus, Aug 22 2014

A007924 The number n written using the greedy algorithm in the base where the values of the places are 1 and primes.

Original entry on oeis.org

0, 1, 10, 100, 101, 1000, 1001, 10000, 10001, 10010, 10100, 100000, 100001, 1000000, 1000001, 1000010, 1000100, 10000000, 10000001, 100000000, 100000001, 100000010, 100000100, 1000000000, 1000000001, 1000000010, 1000000100, 1000000101

Offset: 0

R. Muller

Any nonnegative number can be written as a sum of distinct primes + e, where e is 0 or 1.
Terms contain only digits 0 and 1.
Without the "greedy" condition there is ambiguity - for example 5 = 3+2 has two representations.

			4 = 3 + 1, so a(4) = 101.

S. S. Pillai, "An arithmetical function concerning primes", Annamalai University Journal (1930), pp. 159-167.

John Cerkan, Table of n, a(n) for n = 0..5000
K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, 1996, 50 pages. See page 33.
K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, 1996, 50 pages. [Cached copy] See page 33.
Florian Luca & Ravindranathan Thangadurai, On an arithmetic function considered by Pillai, Journal de théorie des nombres de Bordeaux 21:3 (2009), pp. 695-701.
C. Rivera, Prime puzzle 78
F. Smarandache, Only Problems, Not Solutions!
F. Smarandache, Definitions, Solved and Unsolved Problems, Conjectures, and Theorems in Number Theory and Geometry, edited by M. Perez, Xiquan Publishing House 2000.

Cf. A200947, A066352.

Subsequence of A007088.

cprime[n_Integer] := (If[n==0, 1, Prime[n]]);gentable[n_Integer] := (m=n; ptable={};While[m!=0, (i=0; While[cprime[i]<=m, i++]; j=0;While[jFrank M Jackson, Jan 06 2012 *)

a(n)=if(n>1, my(p=precprime(n)); 10^primepi(p)+a(n-p), n) \\ Charles R Greathouse IV, Feb 01 2013

a(n) is the binary representation of b(n) = 2^pi(n) + b(n-p(pi(n))) for n > 0 and a(0) = b(0)= 0, where pi(k) = number of primes <= k (A000720) and p(k) = k-th prime (A008578). - Frank Ellermann, Dec 18 2001

Additional references from Felice Russo, Sep 14 2001
Antedated to 1930 by Charles R Greathouse IV, Aug 28 2010
Definition clarified by Frank M Jackson and N. J. A. Sloane, Dec 30 2011

A075058 Lexicographically earliest infinite sequence of distinct positive numbers with the property that every positive integer is a sum of distinct terms (see algorithm below).

Original entry on oeis.org

1, 2, 3, 7, 13, 23, 47, 97, 193, 383, 769, 1531, 3067, 6133, 12269, 24533, 49069, 98129, 196247, 392503, 785017, 1570007, 3140041, 6280067, 12560147, 25120289, 50240587, 100481167, 200962327, 401924639, 803849303, 1607698583, 3215397193, 6430794373

Offset: 0

Amarnath Murthy, Sep 07 2002

nonn

This sequence starts with a(0)=1, subsequent terms a(n) for n > 0 being obtained by selecting the greatest prime <= 1 + Sum_{i=0..n-1} a(i). This ensures that the sequence has the required property because Sum_{i=0..n-1} a(i) >= a(n) - 1, for all n >= 0 and a(0)=1, is a necessary and sufficient condition for it to hold.

			Given that the first 7 terms of the sequence are 1,2,...,23,47 then a(8)=(greatest prime) <= (1+2+...+23,47) + 1 = 97, hence a(8)=97.

Charles R Greathouse IV, Table of n, a(n) for n = 0..1000
Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Cf. A068524, A007924, A066352, A200947.

prevprime[n_Integer] := (j=n; While[!PrimeQ[j], j--]; j) aprime[0]=1; aprime[n_Integer] := (aprime[n] = prevprime[Sum[aprime[m], {m, 0, n - 1}] + 1]); Table[aprime[p], {p, 0, 50}]
a[0] = 1; a[n_] := a[n] = NextPrime[Sum[a[k], {k, 0, n-1}]+2, -1]; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Sep 30 2013 *)

print1(s=1);for(n=1,20,k=precprime(s+1);print1(", "k);s+=k) \\ Charles R Greathouse IV, Apr 05 2013

a(n) = (greatest prime) <= 1 + Sum_{i=0..n-1} a(i).

a(n) ~ k*2^n, with k roughly 0.748643. - Charles R Greathouse IV, Apr 05 2013

Entry revised by Frank M Jackson, Dec 03 2011

Edited by N. J. A. Sloane, May 20 2023

A185101 The number n written using the minimum number of terms in the base where the values of the places are 1 and primes (noncomposites). For multiple solutions the smallest binary value is chosen.

Original entry on oeis.org

0, 1, 10, 100, 101, 1000, 1001, 10000, 1100, 10010, 10100, 100000, 11000, 1000000, 100100, 1000010, 101000, 10000000, 110000, 100000000, 1010000, 100000010, 10001000, 1000000000, 1100000, 1000000010, 100010000, 1100100, 10100000, 10000000000

Offset: 0

Frank M Jackson, Jan 23 2012

nonn

There are many ways of generating binary vectors a(n) for selecting noncomposites that when summed give n. A007924 uses the greedy algorithm. The above sequence uses the strong Goldbach conjecture that any integer is the sum of at most three distinct summands. It generates a(n) to select the minimum number of distinct noncomposites. Where there are multiple solutions, it chooses the smallest binary vector.

			n=57 which is > 6 and odd, so m = (nextprime > 57/3) = 23 and n-m = 34 is even, thus A082467(17) = 6 and algorithm selects {23,11,23}. These are not distinct primes, so m = nextprime(nextprime > n/3) = 29 and A082467(14)=3, thus a(n) selects {29,11,17} as the binary vector 10010100000.

Eric Weisstein's World of Mathematics, Goldbach Conjecture.

Cf. A007924, A066352, A200947.

nextprime[j_] := Module[{k}, If[j==0, 1, (k=Floor[j]+1; While[!PrimeQ[k], k++]; k)]]; primetable[n_] := Module[{p, q}, Which[n==1, {0, 2, 0}, n==2, {1, 3, 0}, n==3, {1, 5, 0}, True, (p=n+1; q=2n-p; While[q>0&&!(PrimeQ[p]&&PrimeQ[q]), p++; q--]; {0, q, p})]]; fintable[m_] := Module[{temptable}, Which[m==0, {0, 0, 0}, m==1, {1, 0, 0}, PrimeQ[m], {0, m, 0}, PrimeQ[m-2]&&m>4, {0, 2, m-2}, EvenQ[m], primetable[m/2], True, (temptable=primetable[(m-nextprime[m/3])/2]; If[temptable[[3]]==nextprime[m/3], (temptable=primetable[(m-nextprime[nextprime[m/3]])/2]; temptable[[1]]=nextprime[nextprime[m/3]]), temptable[[1]]=nextprime[m/3]]; temptable)]]; decimal[t_] := Module[{temp2table, tempdecimal=0}, (temp2table=fintable[t]; If[temp2table[[1]]==0, Null, tempdecimal=tempdecimal+2^PrimePi[temp2table[[1]]]]; If[temp2table[[2]]==0, Null, tempdecimal=tempdecimal+2^PrimePi[temp2table[[2]]]]; If[temp2table[[3]]==0, Null, tempdecimal=tempdecimal+2^PrimePi[temp2table[[3]]]];tempdecimal)];Table[IntegerString[decimal[i], 2], {i, 0, 100}]

For n, 1 to 6, a(n) is manually defined. For n prime, a(n) selects n. For n > 6 and n-2 prime, a(n) selects 2 and n-2. For n > 6 and even, use A082467(n/2) to give k, then a(n) selects n/2+k, n/2-k. For n>6 and odd, let m = (nextprime > n/3), then n-m is even and A082467((n-m)/2) gives k, a(n) selects m, (n-m)/2-k, (n-m)/2+k. If m = (n-m)/2+k, then m = nextprime(nextprime > n/3) and repeat.

Name clarified by Frank M Jackson, Oct 08 2013

A345297 a(n) is the least k >= 0 such that A331835(k) = n.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 22, 23, 26, 27, 29, 30, 31, 43, 45, 46, 47, 54, 55, 58, 59, 61, 62, 63, 94, 95, 107, 109, 110, 111, 118, 119, 122, 123, 125, 126, 127, 187, 189, 190, 191, 222, 223, 235, 237, 238, 239, 246, 247, 250, 251, 253, 254, 255

Offset: 0

Rémy Sigrist, Jun 13 2021

Sequence A200947 gives the position of the last occurrence of a number in A331835.

			We have:
           n|  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18
  ----------+------------------------------------------------------------------
  A331835(n)|  0  1  2  3  3  4  5  6  5  6   7   8   8   9  10  11   7   8   9
So a(0) = 0,
   a(1) = 1,
   a(2) = 2,
   a(3) = 3,
   a(4) = 5,
   a(5) = 6,
   a(6) = 7,
   a(7) = 10,
   a(8) = 11,
   a(9) = 13,
   a(10) = 14,
   a(11) = 15.

Rémy Sigrist, Table of n, a(n) for n = 0..2000
Rémy Sigrist, C program for A345297

Cf. A014284, A200947, A331835.

C
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See Links section.
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from sympy import prime
def p(n): return prime(n) if n >= 1 else 1
def A331835(n): return sum(p(i)*int(b) for i, b in enumerate(bin(n)[:1:-1]))
def adict(klimit):
    adict = dict()
    for k in range(klimit+1):
        fk = A331835(k)
        if fk not in adict: adict[fk] = k
    n, alst = 0, []
    while n in adict: alst.append(adict[n]); n += 1
    return alst
print(adict(255)) # Michael S. Branicky, Jun 13 2021

a(A014284(n)) = 2^n - 1.

a(n) <= A200947(n).

A331835 Replace 2^k in binary expansion of n with k-th prime number for any k > 0 (and keep 2^0).

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 5, 6, 7, 8, 8, 9, 10, 11, 7, 8, 9, 10, 10, 11, 12, 13, 12, 13, 14, 15, 15, 16, 17, 18, 11, 12, 13, 14, 14, 15, 16, 17, 16, 17, 18, 19, 19, 20, 21, 22, 18, 19, 20, 21, 21, 22, 23, 24, 23, 24, 25, 26, 26, 27, 28, 29, 13, 14, 15, 16, 16

Offset: 0

Rémy Sigrist, Jan 28 2020

Every nonnegative integer appears in this sequence as A008578 is a complete sequence.

For any m >= 0, m appears A036497(m) times, the first and last occurrences being at indices A345297(m) and A200947(m), respectively. - Rémy Sigrist, Jun 13 2021

			For n = 43:
- 43 = 2^0 + 2^1 + 2^3 + 2^5,
- so a(43) = 2^0 + prime(1) + prime(3) + prime(5) = 1 + 2 + 5 + 11 = 19.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Eric Weisstein's World of Mathematics, Complete Sequence
Index entries for sequences related to binary expansion of n

Cf. A008578, A036497, A048672, A089625, A200947, A345297.

Array[Total@ Map[If[# == 0, 1, Prime[#]] &, Position[Reverse@ IntegerDigits[#, 2], 1][[All, 1]] - 1] &, 68] (* Michael De Vlieger, Jan 29 2020 *)

a(n) = my (b=Vecrev(binary(n\2))); n%2 + sum(k=1, #b, if (b[k], prime(k), 0))

from sympy import prime
def p(n): return prime(n) if n >= 1 else 1
def a(n): return sum(p(i)*int(b) for i, b in enumerate(bin(n)[:1:-1]))
print([a(n) for n in range(69)]) # Michael S. Branicky, Jun 13 2021

a(2*n) = A089625(n) for any n > 0.
a(2*n+1) = A089625(n) + 1 for any n > 0.
G.f.: x/(1 - x^2) + (1/(1 - x)) * Sum_{k>=1} prime(k) * x^(2^k) / (1 + x^(2^k)). - Ilya Gutkovskiy, May 24 2024

A368577 Irregular table T(n, k), n >= 0, k = 1..A036497(n), read by rows; the n-th row lists the numbers m such that A331835(m) = n.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Dec 31 2023

As a flat sequence, this is a permutation of the nonnegative integers (with inverse A368578).

			Table T(n, k) begins:
  n   n-th row
  --  ------------------
   0  0
   1  1
   2  2
   3  3, 4
   4  5
   5  6, 8
   6  7, 9
   7  10, 16
   8  11, 12, 17
   9  13, 18
  10  14, 19, 20
  11  15, 21, 32
  12  22, 24, 33
  13  23, 25, 34, 64
  14  26, 35, 36, 65
  15  27, 28, 37, 66
  16  29, 38, 40, 67, 68

Rémy Sigrist, Table of n, a(n) for n = 0..10888 (rows for n = 0..104 flattened)
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A036497, A200947, A331835, A345101, A345297, A368578 (inverse).

PARI
```
\\ See Links section.
```

T(n, 1) = A345297(n).

T(n, A036497(n)) = A200947(n).

A201997 a(n) is the decimal value of the binary vector used to select terms of A075058 whose sum is n.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 60, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87

Offset: 0

Frank M Jackson, Dec 07 2011

			For n=22, the binary vector when applied to A075058 is {0,1,0,1,1,0,...}, consequently 2+7+13=22. The decimal value of the binary vector (in ascending powers of 2) is 26, so a(22)=26.

Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Cf. A007924, A066352, A200947, A075058.

prevprime[n_Integer] := (j=n; If[n==1, 1, While[!PrimeQ[j], j--]; j]); aprime[n_Integer] := (aprime[n]=prevprime[Sum[aprime[m], {m, 0, n - 1}] + 1]); gentable[n_Integer] := (m=n; ptable={0}; While[m!=0, (i=0; While[aprime[i]<=m && ptable[[i + 1]]!=1, (AppendTo[ptable, 0];i++)]; ptable[[i]] = 1; m = m - aprime[i - 1])]; ptable); decimal[n_Integer] := (gentable[n]; Sum[2^(k-1)*ptable[[k]], {k, 1, Length[ptable]}]); aprime[0]=1; Table[decimal[r], {r,0,100}]

Binary(a(n)) x A075058 = n, where x is the inner product and the binary vector is in ascending powers of 2 with infinite trailing zeros.

Edited by N. J. A. Sloane, May 20 2023

A202618 a(n) is the smallest integer that is the sum of n distinct terms of A075058.

Original entry on oeis.org

0, 1, 4, 6, 19, 42, 89, 96, 289, 672, 1441, 2972, 6039, 12172, 24441, 48974, 98043, 196172, 392419, 784922, 1569939, 3139946, 6279987, 12560054, 25120201

Offset: 0

Frank M Jackson, Dec 21 2011

Any nonnegative integer can be written as a sum of distinct terms of A075058. a(n) is the smallest integer that is the sum of n distinct terms of A075058 in the same way that A066352 gives a Pillai sequence for the sequence comprising 1 followed by all the primes.

			For n=5, the binary vector at A201997(54) is the smallest binary vector containing 5 1's and when applied to A075058 selects the integer 42. Consequently because 42=23+13+3+2+1 and 1,2,3,13,23 are all terms of the complete sequence A075058, a(5)=42.

Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Cf. A007924, A066352, A200947, A075058, A201997.

prevprime[n_Integer] := (j=n;If[n==1, 1, While[!PrimeQ[j], j--]; j]);aprime[n_Integer] := (aprime[n]=prevprime[Sum[aprime[m], {m, 0, n - 1}]+1]);gentable[n_Integer] := (m=n;ptable={0};While[m!=0, (i=0;While[aprime[i]<=m && ptable[[i+1]]!=1, (AppendTo[ptable, 0];i++)];ptable[[i]] = 1;m=m-aprime[i - 1])];ptable);decimal[n_Integer] := (gentable[n];Sum[2^(k-1)*ptable[[k]], {k, 1, Length[ptable]}]);ones[n_Integer] :=(gentable[n];Sum[ptable[[k]], {k, 1, Length[ptable]}]);changeones[n_Integer] := (p = 0; While[ones[p] < n, p++]; p);aprime[0]=1;Table[changeones[r], {r, 0, 20}]

Find the smallest m such that binary(A201997(m)) x {1,1,1,...} = n, where x is the inner product, {1,1,1,1,...} is an infinite binary vector of 1's and binary(A201997(m)) a binary vector with infinite trailing zeros both in ascending powers of 2. Then a(n) = binary(m) x A075058, where x is the inner product and the binary vector is in ascending powers of 2 with infinite trailing zeros.

Edited by N. J. A. Sloane, May 20 2023

A205598 The number n written using a minimizing algorithm in the base where the values of the places are 1 and primes.

Original entry on oeis.org

0, 1, 10, 11, 101, 110, 111, 1010, 1011, 1101, 1110, 1111, 10110, 10111, 11010, 11011, 11101, 11110, 11111, 101011, 101101, 101110, 101111, 110110, 110111, 111010, 111011, 111101, 111110, 111111, 1011110, 1011111

Offset: 0

Frank M Jackson, Feb 08 2012

nonn

Any nonnegative number can be written as a sum of distinct primes + e, where e is 0 or 1 (see A007924, which uses a greedy algorithm for writing n). However in this sequence a(n) is generated by using a minimizing algorithm that gives the smallest binary vector for select members from the sequence Q = (1 union primes) that when summed gives n. Without the minimizing condition there is ambiguity -- for example, 8 = 7+1 = 5+3 = 5+2+1 has three representations.

			8 = 7+1 = 5+3 = 5+2+1, so a(8) = 1011.

Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Cf. A007924, A185101, A200947.

aprime[n_] := If[n==0, 1, Prime[n]]; seqtable[l_] := (stable=Table[aprime[j], {j, 0, l}]; stable); inttable[p_] := (itable=Reverse[IntegerDigits[p, 2]]; itable); h=1; otable={0}; ttable={}; While[h<100, (inttable[h]; seqtable[Length[itable]-1]; test=itable.stable; If[!MemberQ[ttable, test], AppendTo[otable, h], Null]; AppendTo[ttable, test]; h++)]; IntegerString[otable, 2]

Let Q be the ordered sequence of (1 union primes), then a(n) x Q = n, where x is the inner product and the binary vector a(n) is in ascending powers of 2 with infinite trailing zeros.

cp's OEIS Frontend

A121559 Final result (0 or 1) under iterations of {r mod (max prime p <= r)} starting at r = n.

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A007924 The number n written using the greedy algorithm in the base where the values of the places are 1 and primes.

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A075058 Lexicographically earliest infinite sequence of distinct positive numbers with the property that every positive integer is a sum of distinct terms (see algorithm below).

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A185101 The number n written using the minimum number of terms in the base where the values of the places are 1 and primes (noncomposites). For multiple solutions the smallest binary value is chosen.

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A345297 a(n) is the least k >= 0 such that A331835(k) = n.

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A331835 Replace 2^k in binary expansion of n with k-th prime number for any k > 0 (and keep 2^0).

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A368577 Irregular table T(n, k), n >= 0, k = 1..A036497(n), read by rows; the n-th row lists the numbers m such that A331835(m) = n.

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