A376615 -id:A376615 - cp's OEIS frontend

A376619 a(n) is the least odd number k such that A376615(k) = n, or -1 if no such number exists.

3, 21, 345, 10625, 74375, 860625, 84189105, 1599592995, 23993894925

Offset: 1

Amiram Eldar, Sep 30 2024

Without the restriction to odd numbers the corresponding sequence is 3*2^(n-1) = A007283(n-1).
All the terms above 3 are odd binary Niven numbers (A144302).
a(10) > 10^13, if it exists.

			  n | The n iterations
  --+------------------------------------------------------
  1 | 3 -> 3/2
  2 | 21 -> 7 -> 7/3
  3 | 345 -> 69 -> 23 -> 23/4
  4 | 10625 -> 2125 -> 425 -> 85 -> 85/4
  5 | 74375 -> 10625 -> 2125 -> 425 -> 85 -> 85/4
  6 | 860625 -> 95625 -> 10625 -> 2125 -> 425 -> 85 -> 85/4

Cf. A007283, A049445, A144302, A376615, A376616, A376617.

s[n_] := s[n] = Module[{bw = DigitCount[n, 2, 1]}, If[bw == 1, 0, If[!Divisible[n, bw], 1, 1 + s[n/bw]]]]; seq[len_] := Module[{v = Table[0, {len}], c = 0, k = 3, i}, While[c < len, i = s[k]; If[v[[i]] == 0, c++; v[[i]] = k]; k += 2]; v]; seq[5]

s(n) = {my(w = hammingweight(n)); if(w == 1, 0, if(n % w, 1, 1 + s(n/w)));}
lista(len) = {my(v = vector(len), c = 0, k = 3, i); while(c < len, i = s(k); if(v[i] == 0, c++; v[i] = k); k += 2); v;}

A376616 Binary Niven numbers (A049445) k such that k/wt(k) is also a binary Niven number, where wt(k) = A000120(k) is the binary weight of k.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 20, 24, 32, 36, 40, 48, 64, 68, 72, 80, 96, 126, 128, 132, 136, 144, 160, 192, 240, 252, 256, 260, 264, 272, 276, 288, 320, 324, 345, 368, 384, 405, 414, 432, 460, 464, 480, 486, 504, 512, 516, 520, 528, 544, 552, 576, 624, 640, 648, 688, 690

Offset: 1

Amiram Eldar, Sep 30 2024

Numbers k such that A376615(k) = 0 or A376615(k) >= 3.

If m is a term then 2^k * m is a term for all k >= 0.

			12 is a term since 12/wt(12) = 6 is an integer and also 6/wt(6) = 3 is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A049445.
Subsequences: A000079, A376617, A376618.
Cf. A000120, A376615.

q[k_] := Module[{w = DigitCount[k, 2, 1]}, Divisible[k, w] && Divisible[k/w, DigitCount[k/w, 2, 1]]]; Select[Range[1000], q]

is(k) = {my(w = hammingweight(k)); !(k % w) && !((k/w) % hammingweight(k/w));}

A376617 Binary Niven numbers (A049445) k such that m = k/wt(k) and m/wt(m) are also binary Niven numbers, where wt(k) = A000120(k) is the binary weight of k.

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 32, 40, 48, 64, 72, 80, 96, 128, 136, 144, 160, 192, 256, 264, 272, 288, 320, 384, 512, 520, 528, 544, 576, 640, 756, 768, 960, 1024, 1032, 1040, 1056, 1088, 1104, 1152, 1280, 1296, 1380, 1472, 1512, 1536, 1620, 1656, 1728, 1840, 1856, 1920, 1944

Offset: 1

Amiram Eldar, Sep 30 2024

Numbers k such that A376615(k) = 0 or A376615(k) >= 4.

			24 is a term since 24/wt(24) = 12 is an integer, 12/wt(12) = 6 is an integer, and 6/wt(6) = 3 is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A049445 and A376616.
A000079 is a subsequence.
Cf. A000120, A376615.

q[k_] := Module[{w = DigitCount[k, 2, 1], w2, m, n}, IntegerQ[m = k/w] && Divisible[m, w2 = DigitCount[m, 2, 1]] && Divisible[n = m/w2, DigitCount[n, 2, 1]]]; Select[Range[2000], q]

s(n) = {my(w = hammingweight(n)); if(w == 1, 0, if(n % w, 1, 1 + s(n/w)));}
is(k) = {my(sk = s(k)); sk == 0 || sk >= 4;}

A377384 a(n) is the number of iterations that n requires to reach a noninteger or a factorial number under the map x -> x / f(x), where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k; a(n) = 0 if n is a factorial number.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 0, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Oct 27 2024

The factorial numbers are fixed points of the map, since f(k!) = 1 for all k >= 0. Therefore they are arbitrarily assigned the value a(k!) = 0.

Each number n starts a chain of a(n) integers: n, n/f(n), (n/f(n))/f(n/f(n)), ..., of them the first a(n)-1 integers are factorial-base Niven numbers (A118363).

			a(8) = 2 since 8/f(8) = 4 and 4/f(4) = 2 is a factorial number that is reached after 2 iterations.
a(27) = 3 since 27/f(27) = 9, 9/f(9) = 3 and 3/f(3) = 3/2 is a noninteger that is reached after 3 iterations.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000142, A034968, A118363, A286607, A377385, A377386.

Analogous sequences: A376615 (binary), A377208 (Zeckendorf).

fdigsum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; a[n_] := a[n] = Module[{s = fdigsum[n]}, If[s == 1, 0, If[!Divisible[n, s], 1, 1 + a[n/s]]]]; Array[a, 100]

fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
a(n) = {my(f = fdigsum(n)); if(f == 1, 0, if(n % f, 1, 1 + a(n/f)));}

def f(n, p=2): return n if nMichael S. Branicky, Oct 27 2024

a(n) = 0 if and only if n is in A000142 (by definition).
a(n) = 1 if and only if n is in A286607.
a(n) >= 2 if and only if n is in A118363 \ A000142 (i.e., n is a factorial-base Niven number that is not a factorial number).
a(n) >= 3 if and only if n is in A377385 \ A000142.
a(n) >= 4 if and only if n is in A377386 \ A000142.
a(n) < A000005(n).

A377208 a(n) is the number of iterations that n requires to reach a noninteger or a Fibonacci number under the map x -> x / z(x), where z(k) = A007895(k) is the number of terms in the Zeckendorf representation of k; a(n) = 0 if n is a Fibonacci number.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 2, 0, 2, 1, 1, 1, 2, 1, 1, 0, 2, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 0, 3, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1

Offset: 1

Amiram Eldar, Oct 20 2024

The Fibonacci numbers are the fixed points of the map, since z(Fibonacci(k)) = 1 for all k >= 1. Therefore they are arbitrarily assigned the value a(Fibonacci(k)) = 0.

Each number n starts a chain of a(n) integers: n, n/z(n), (n/z(n))/z(n/z(n)), ..., of them the first a(n)-1 integers are Zeckendorf-Niven numbers (A328208).

			a(12) = 2 since 12/z(12) = 4 and 4/z(4) = 2 is a Fibonacci number that is reached after 2 iterations.
a(36) = 3 since 36/z(36) = 18, 18/z(18) = 9 and 9/z(9) = 9/2 is a noninteger that is reached after 3 iterations.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000045, A007895, A328208, A376615 (binary analog), A377209, A377210.

zeck[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* Alonso del Arte at A007895 *)
a[n_] := a[n] = Module[{z = zeck[n]}, If[z == 1, 0, If[!Divisible[n, z], 1, 1 + a[n/z]]]]; Array[a, 100]

zeck(n) = if(n<4, n>0, my(k=2, s, t); while(fibonacci(k++)<=n, ); while(k && n, t=fibonacci(k); if(t<=n, n-=t; s++); k--); s) \\ Charles R Greathouse IV at A007895
a(n) = {my(z = zeck(n)); if(z == 1, 0, if(n % z, 1, 1 + a(n/z)));}

a(n) = 0 if and only if n is in A000045 (by definition).
a(n) >= 2 if and only if n is in A328208 \ A000079 (i.e., n is a Zeckendorf-Niven number that is not a Fibonacci number).
a(n) >= 3 if and only if n is in A377209 \ A000079.
a(n) >= 4 if and only if n is in A377210 \ A000079.
a(n) < A000005(n).

cp's OEIS Frontend

A376619 a(n) is the least odd number k such that A376615(k) = n, or -1 if no such number exists.

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A376616 Binary Niven numbers (A049445) k such that k/wt(k) is also a binary Niven number, where wt(k) = A000120(k) is the binary weight of k.

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A376617 Binary Niven numbers (A049445) k such that m = k/wt(k) and m/wt(m) are also binary Niven numbers, where wt(k) = A000120(k) is the binary weight of k.

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A377384 a(n) is the number of iterations that n requires to reach a noninteger or a factorial number under the map x -> x / f(x), where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k; a(n) = 0 if n is a factorial number.

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A377208 a(n) is the number of iterations that n requires to reach a noninteger or a Fibonacci number under the map x -> x / z(x), where z(k) = A007895(k) is the number of terms in the Zeckendorf representation of k; a(n) = 0 if n is a Fibonacci number.

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