Altug Alkan - cp's OEIS frontend

A346094 a(n) = n / A275823(n), where A275823(n) is the least k such that n divides phi(k^2).

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 6, 5, 2, 3, 2, 1, 4, 1, 2, 3, 4, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 3

Offset: 1

Antti Karttunen and Altug Alkan, Jul 21 2021

nonn

a(n) = n divided by the least k such that A002618(k) [= phi(k^2) = k*phi(k)] is a multiple of n.

It is easy to see that such k is always a divisor of n since k contains only some of prime factors of n and there cannot be other prime factor that does not divide n. In order to see this, let us assume p divides k (where p is prime that does not divide n) and (p-1) contribute the division in A275823. At this case there is definitely smaller option to do this instead of p-1 since it is always possible that k could contain necessary prime powers from factorization of p-1 instead of p. At the same time, obviously A275823(n) <= n. So terms of this sequence are always integers.

Cf. A002618, A275823.

Array[#/Block[{k = 1}, While[! Mod[EulerPhi[k^2], #] == 0, k++]; k] &, 105] (* Michael De Vlieger, Jul 22 2021 *)

A346094(n) = { my(k=1); while((k*eulerphi(k)) % n, k++); (n/k); };

A346708 a(n) is the least k > 1 such that p(n) divides p(n^k), or 0 if no such k exists (p = A000041).

Original entry on oeis.org

2, 3, 2, 3, 3, 6, 7, 2, 10

Offset: 1

Altug Alkan, Jul 30 2021

			a(8) = 2 because p(8) = 22 divides p(8^2) = 1741630.

Cf. A000041.

a[n_] := Module[{k = 2, p = PartitionsP[n]}, While[! Divisible[PartitionsP[n^k], p], k++]; k]; Array[a, 9] (* Amiram Eldar, Aug 04 2021 *)

a(n)=my(t=2); while(numbpart(n^t)%numbpart(n), t++); t

A346696 a(n) is the least positive k such that A000041(n) divides A000041(n+k), or 0 if no such k exists.

Original entry on oeis.org

1, 1, 6, 4, 3, 5, 2, 2, 7, 88, 16, 64, 4, 343, 25, 81, 23, 22, 21, 245, 450, 755, 75, 688, 225, 740, 4432, 307, 671, 1055, 18881, 7119, 1415, 4571, 1365, 411, 36005, 5799, 3466, 1410, 4319, 5993, 646, 60775, 4470, 90780, 34595, 36805, 77125, 11051, 2514, 46045, 32713, 114479, 109221, 19322, 571126

Offset: 0

Altug Alkan, Jul 29 2021

nonn

Conjecture: a(n) > 0 for all n.

			a(7) = 2 because A000041(7) = 15 divides A000041(9) = 30.

Cf. A000041, A079031.

a[n_]:=(k=1;While[!Divisible[PartitionsP[n+k],PartitionsP@n],k++];k);Array[a,30,0] (* Giorgos Kalogeropoulos, Jul 29 2021 *)

a(n)=my(t=1); while(numbpart(n+t)%numbpart(n), t++); t

A345877 a(1) = 1, a(n) = a(n-1)/2 if a(n-1) is even, otherwise a(n) = n - a(n-1).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 1, 7, 2, 1, 10, 5, 8, 4, 2, 1, 16, 8, 4, 2, 1, 21, 2, 1, 24, 12, 6, 3, 26, 13, 18, 9, 24, 12, 6, 3, 34, 17, 22, 11, 30, 15, 28, 14, 7, 39, 8, 4, 2, 1, 50, 25, 28, 14, 7, 49, 8, 4, 2, 1, 60, 30, 15, 49, 16, 8, 4, 2, 1, 69, 2, 1, 72, 36, 18, 9, 68, 34, 17, 63, 18, 9, 74, 37, 48, 24, 12, 6, 3

Offset: 1

Altug Alkan, Jun 28 2021

Let a_i(1) = 1 and a_i(n) = a_i(n-1)/(i+1) if a_i(n-1) is divisible by i+1, otherwise a_i(n) = n - a_i(n-1). This sequence is a_1(n) and A345886 is a_2(n).

Conjecture: a_i(n) hits every positive integers infinitely many times for all i >= 1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A090895, A135287, A345886.

a[1] = 1; a[n_] := a[n] = If[EvenQ[a[n - 1]], a[n - 1]/2, n - a[n - 1]]; Array[a, 100] (* Amiram Eldar, Jun 29 2021 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[a],a/2,n+1-a]}; NestList[nxt,{1,1},90][[;;,2]] (* Harvey P. Dale, Aug 31 2023 *)

q=vector(100); q[1]=1; for(n=2, #q, q[n] = if(q[n-1]%2, n-q[n-1], q[n-1]/2)); q

A345886 a(1) = 1, a(n) = a(n-1)/3 if a(n-1) is divisible by 3, otherwise a(n) = n - a(n-1).

Original entry on oeis.org

1, 1, 2, 2, 3, 1, 6, 2, 7, 3, 1, 11, 2, 12, 4, 12, 4, 14, 5, 15, 5, 17, 6, 2, 23, 3, 1, 27, 9, 3, 1, 31, 2, 32, 3, 1, 36, 12, 4, 36, 12, 4, 39, 13, 32, 14, 33, 11, 38, 12, 4, 48, 16, 38, 17, 39, 13, 45, 15, 5, 56, 6, 2, 62, 3, 1, 66, 22, 47, 23, 48, 16, 57, 19, 56, 20, 57, 19, 60, 20, 61, 21, 7, 77, 8

Offset: 1

Altug Alkan, Jun 28 2021

See comments section of A345877.

Cf. A345877.

a[1] = 1; a[n_] := a[n] = If[Divisible[a[n - 1], 3], a[n - 1]/3, n - a[n - 1]]; Array[a, 100] (* Amiram Eldar, Jun 29 2021 *)

q=vector(100); q[1]=1; for(n=2, #q, q[n] = if(q[n-1]%3, n-q[n-1], q[n-1]/3)); q

A343491 Number of representations of n! as a sum of 3 tetrahedral numbers (A000292).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 3, 5, 2, 3, 6, 5, 8, 8, 7, 2, 7, 8, 3, 11, 2, 2

Offset: 1

Altug Alkan, Apr 17 2021

Conjecture I: There are infinitely many n such that a(n) >= 1.
Conjecture II: Natural density of numbers n such that a(n) >= 1 is 1.
Conjecture III: Numbers n such that a(n) = 0 is a finite sequence.
Conjecture IV: a(n) >= 1 for all n.
See Links section for some solutions.

			a(4) = 2 because 4! = 0 + 4 + 20 = 4 + 10 + 10.
a(24) = 2 because 24! = f(11393630) + f(118661018) + f(127041924) = f(81298034) + f(61098204) + f(143537134) where f = A000292.

Altug Alkan, A note on sequence

Cf. A000142, A000292, A102796, A104246, A102799, A267414, A308852, A341794.

Table[Length[Solve[{i*(i + 1)*(i + 2) + j*(j + 1)*(j + 2) + k*(k + 1)*(k + 2) == 6*n!, i >= 0, j >= 0, k >= 0, i <= j, j <= k, k < (6*n!)^(1/3)}, Integers]], {n, 1, 10}] (* Vaclav Kotesovec, Apr 19 2021 *)

A337366 Number of representations of A036691(n) as a sum of 3 nonnegative cubes.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 1, 2, 1, 4, 6, 3, 8, 8, 14, 7

Offset: 0

Altug Alkan, Aug 25 2020

Conjecture I: a(n) = 0 only for n = 1. That is, any product of first n > 1 composite numbers is a sum of at most 3 positive cubes. For example,
A036691(100) = 2563573191821442299652988946477367093137353211904000000000^3 + 21431289850849406740917647451954098598503667204096000000000^3 + 26409890400237152457638095665189553529771293409280000000000^3.
Conjecture II: For any term t >= 1, there are only finitely many values of n such that a(n) = t.

			a(4) = 2 because A036691(4) = 1728 = 12^3 = 6^3 + 8^3 + 10^3.

Cf. A025447, A036691, A226955, A267414.

A036691 = Join[{1}, FoldList[Times, Select[Range[20], CompositeQ]]];
Table[Length@ PowersRepresentations[A036691[[n]], 3, 3], {n, 10}] (* Robert Price, Sep 08 2020 *)

a(n) = A025447(A036691(n)).

A333841 Integers n such that n! = x^2 + y^3 + z^4 where x, y and z are nonnegative integers, is soluble.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24

Offset: 1

Altug Alkan, Aug 14 2020

			6! = 11^2+7^3+4^4; 8! = 192^2+15^3+3^4;  9! = 443^2+55^3+4^4; 10! = 1888^2+40^3+4^4; 11! = 5896^2+172^3+16^4, so 6, 8, 9, 10 and 11 are in the sequence. - _R. J. Mathar_, Dec 15 2020

Cf. A123053, A267414, A337046.

{k: k! in A123053}. - R. J. Mathar, Dec 15 2020

A337046 Integers n such that n! = x^2 + y^3 + z^6 where x, y and z are nonnegative integers, is soluble.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 10, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25

Offset: 1

Altug Alkan, Aug 12 2020

Conjecture I: Natural density of this sequence is 1.
Conjecture II: Any sufficiently large n is in the sequence.
Conjecture III: There is a fixed value of t such that all integers >= t are terms.
If k is of the form x^2 + y^3 + z^6 then so is k*m^6 = (x*m^3)^2 + (y*m^2)^3 + (z*m)^6. - David A. Corneth, Aug 13 2020

			6 is a term since 6! = 12^2 + 8^3 + 2^6.

David A. Corneth, PARI program

Cf. A267414, A273553 (subsequence).

\\ See Corneth link. David A. Corneth, Aug 13 2020

a(12)-a(18) from David A. Corneth, Aug 12 2020

A336205 Numbers k that can be expressed as x^3 + y^3 + z^3 with x^2 + y^2 + z^2 <= k where x, y, z are integers.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 10, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 43, 45, 46, 48, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 69, 71, 72, 73, 80, 81, 83, 88, 90, 91, 92, 97, 98, 99, 100, 101, 106, 109, 116, 117, 118, 119, 120, 123, 124, 125, 126, 127, 128, 129, 132

Offset: 1

Altug Alkan, Jul 12 2020

See A336240 for border case x^2 + y^2 + z^2 = x^3 + y^3 + z^3.
What is the natural density of this sequence?
There are infinitely many infinite parametric families of solutions which have negative values in (x,y,z). For example, 8*(3*a-1)^2*m^6 + 12*(3*a-1)*(a-1)*m^4 - 6*(2*a-1)*m^2 + 2*a^3 + 1 are terms for all a >= 0, m >= 0. (x = 1 - (6*a-2)*m^2, y = a - m*(1-(6*a-2)*m^2), z = a + m*(1-(6*a-2)*m^2)). - Altug Alkan, Jul 17 2020
By definition, corresponding (x,y,z) variables are produced by equation x^3 + y^3 + z^3 = x^2 + y^2 + z^2 + t with t >= 0. That is, x^2*(x-1) + y^2*(y-1) + z^2*(z-1) >= 0. Conjecture: Every even integer can be represented as x^2*(x-1) + y^2*(y-1) + z^2*(z-1) where x, y, z are integers. - Altug Alkan, Jul 19 2020

			11 is not a term because there is no (x,y,z) with x^2 + y^2 + z^2 <= 11 when x^3 + y^3 + z^3 = 11.
18 is a term because (-1)^3 + (-2)^3 + 3^3 = 18 and (-1)^2 + (-2)^2 + 3^2 <= 18.
61 is a term because (-4)^3 + 0^3 + 5^3 = 61 and (-4)^2 + 0^2 + 5^2 <= 61.
354 is a term because (-11)^3 + (-8)^3 + 13^3 = (-11)^2 + (-8)^2 + 13^2 = 354.

Robert Israel, Table of n, a(n) for n = 1..8000
Rémy Sigrist, C program for A336205

Cf. A004825 (subsequence), A060464 (supersequence), A336240.

C
```
See Links section.
```

filter:= proc(n) local x,y,z,e1,e2;
  for x from 0 while 3*x^2 <= n do
    for y from 0 while x^2 + 2*y^2 <= n do
      for e1 in [-1,1] do for e2 in [-1,1] do
        z:= surd(n + e1*x^3 + e2*y^3,3);
        if z::integer and x^2 + y^2 + z^2 <= n then return true fi;
  od od od od;
  false
end proc:
select(filter, [$0..200]); # Robert Israel, Jul 12 2020

filter[n_] := Module[{x, y, z, e1, e2},
  For[x = 0, 3*x^2 <= n, x++,
    For[y = 0, x^2 + 2*y^2 <= n, y++,
      For[e1 = -1, e1 <= 1, e1 += 2, For[e2 = -1, e2 <= 1, e2 += 2,
        z = (n + e1*x^3 + e2*y^3)^(1/3);
        If[IntegerQ[z] && x^2 + y^2 + z^2 <= n, Return[True]]
  ]]]]; False];
Select[Range[0, 200], filter] (* Jean-François Alcover, Aug 11 2023, after Robert Israel *)

cp's OEIS Frontend

User: Altug Alkan

A346094 a(n) = n / A275823(n), where A275823(n) is the least k such that n divides phi(k^2).

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A346708 a(n) is the least k > 1 such that p(n) divides p(n^k), or 0 if no such k exists (p = A000041).

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A346696 a(n) is the least positive k such that A000041(n) divides A000041(n+k), or 0 if no such k exists.

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A345877 a(1) = 1, a(n) = a(n-1)/2 if a(n-1) is even, otherwise a(n) = n - a(n-1).

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A345886 a(1) = 1, a(n) = a(n-1)/3 if a(n-1) is divisible by 3, otherwise a(n) = n - a(n-1).

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A343491 Number of representations of n! as a sum of 3 tetrahedral numbers (A000292).

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A337366 Number of representations of A036691(n) as a sum of 3 nonnegative cubes.

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A333841 Integers n such that n! = x^2 + y^3 + z^4 where x, y and z are nonnegative integers, is soluble.

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A337046 Integers n such that n! = x^2 + y^3 + z^6 where x, y and z are nonnegative integers, is soluble.

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