A376617 -id:A376617 - cp's OEIS frontend

A376795 Numbers k such that k and k+1 are both in A376617.

1, 10624, 13824, 1114112, 2625664, 4563999, 6554624, 16843904, 17266688, 17368064, 20003840, 27137024, 32375160, 32679360, 42993664, 44643599, 63732096, 69222464, 69424640, 70083584, 80778752, 84783104, 85458944, 90256383, 92478000, 116469899, 118063231, 121900544

Offset: 1

Amiram Eldar, Oct 04 2024

			10624 is a term since both 10624 and 10625 are in A376617: 10624/A000120(10624) = 2656, 2656/A000120(2656) = 664, and 664/A000120(664) = 166 are integers, and 10625/A000120(10625) = 2125, 2125/A000120(2125) = 425, and 425/A000120(425) = 85 are integers.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Subsequence of A330931, A376617 and A376793.

Cf. A000120.

q[k_] := 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[1.2*10^6], q[#] && q[#+1] &]

s(n) = {my(w = hammingweight(n)); if(w == 1, 0, if(n % w, 1, 1 + s(n/w)));}
is1(k) = {my(sk = s(k)); sk == 0 || sk >= 4;}
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

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));}

A376615 a(n) is the number of iterations that n requires to reach a noninteger under the map x -> x / wt(x), where wt(k) is the binary weight of k (A000120); a(n) = 0 if n is a power of 2.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 3, 1, 1, 1, 0, 1, 2, 1, 3, 2, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 3, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 2, 1, 3, 2, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 30 2024

The powers of 2 are fixed points of the map, since wt(2^k) = 1 for all k >= 0. Therefore they are arbitrarily assigned the value a(2^k) = 0.

Each number n starts a chain of a(n) integers: n, n/wt(n), (n/wt(n))/wt(n/wt(n)), ..., of them the first a(n)-1 integers are binary Niven numbers (A049445).

			a(6) = 2 since 6/wt(6) = 3 and 3/wt(3) = 3/2 is a noninteger that is reached after 2 iterations.
a(20) = 3 since 20/wt(20) = 10, 10/wt(10) = 5 and 5/wt(5) = 5/2 is a noninteger that is reached after 3 iterations.

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

Cf. A000005, A000079, A000120, A049445, A065878, A376616, A376617, A376619.

a[n_] := a[n] = Module[{bw = DigitCount[n, 2, 1]}, If[bw == 1, 0, If[!Divisible[n, bw], 1, 1 + a[n/bw]]]]; Array[a, 100]

a(n) = {my(w = hammingweight(n)); if(w == 1, 0, if(n % w, 1, 1 + a(n/w)));}

a(n) = 0 if and only if n is in A000079 (by definition).
a(n) = 1 if and only if n is in A065878.
a(n) >= 2 if and only if n is in A049445 \ A000079 (i.e., n is a binary Niven number that is not a power of 2).
a(n) >= 3 if and only if n is in A376616 \ A000079.
a(n) >= 4 if and only if n is in A376617 \ A000079.
a(2*n) >= a(n).
a(3*2^n) = n+1 for n >= 0.
a(n) < A000005(n).

A377210 Zeckendorf-Niven numbers (A328208) k such that m = k/z(k) and m/z(m) are also Zeckendorf-Niven numbers, where z(k) = A007895(k) is the number of terms in the Zeckendorf representation of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 16, 21, 24, 26, 30, 34, 42, 48, 55, 60, 68, 78, 89, 110, 120, 126, 144, 178, 180, 192, 204, 233, 243, 264, 270, 288, 300, 312, 324, 330, 360, 377, 466, 480, 534, 540, 576, 600, 610, 621, 672, 720, 754, 768, 864, 987, 1020, 1056

Offset: 1

Amiram Eldar, Oct 20 2024

			24 is a term since 24/z(24) = 12, 12/z(12) = 4 and 4/z(4) = 2 are all integers.

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

Cf. A000045 (a subsequence), A007895, A376617 (binary analog).

Subsequence of A328208 and A377209.

zeck[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* Alonso del Arte at A007895 *)
q[k_] := Module[{z = zeck[k], z2, m, n}, IntegerQ[m = k/z] && Divisible[m, z2 = zeck[m]] && Divisible[n = m/z2, zeck[n]]]; Select[Range[1000], q]

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
is(k) = {my(z = zeck(k), z2, m); if(k % z, return(0)); m = k/z; z2 = zeck(m); !(m % z2) && !((m/z2) % zeck(m/z2)); }

A377386 Factorial-base Niven numbers (A118363) k such that m = k/f(k) and m/f(m) are also factorial-base Niven numbers, where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 36, 40, 48, 54, 72, 80, 96, 108, 120, 135, 144, 180, 192, 240, 280, 288, 360, 384, 432, 480, 576, 594, 600, 720, 840, 864, 1200, 1215, 1225, 1296, 1344, 1440, 1680, 1728, 1800, 2160, 2240, 2352, 2400, 2520, 2592, 2704, 2730, 2880, 3000

Offset: 1

Amiram Eldar, Oct 27 2024

			16 is a term since 16/f(16) = 4 is an integer, 4/f(4) = 2 is an integer, and 2/f(2) = 2 is an integer.

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

Subsequence of A118363 and A377385.
A000142 is a subsequence.
Cf. A034968, A377384.
Analogous sequences: A376617 (binary), A377210 (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]; q[k_] := Module[{f = fdigsum[k], f2, m, n}, IntegerQ[m = k/f] && Divisible[m, f2 = fdigsum[m]] && Divisible[n = m/f2, fdigsum[n]]]; Select[Range[3000], q]

fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
is(k) = {my(f = fdigsum(k), f2, m); if(k % f, return(0)); m = k/f; f2 = fdigsum(m); !(m % f2) && !((m/f2) % fdigsum(m/f2)); }

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

Original entry on oeis.org

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;}

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A376795 Numbers k such that k and k+1 are both in A376617.

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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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A376615 a(n) is the number of iterations that n requires to reach a noninteger under the map x -> x / wt(x), where wt(k) is the binary weight of k (A000120); a(n) = 0 if n is a power of 2.

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A377210 Zeckendorf-Niven numbers (A328208) k such that m = k/z(k) and m/z(m) are also Zeckendorf-Niven numbers, where z(k) = A007895(k) is the number of terms in the Zeckendorf representation of k.

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A377386 Factorial-base Niven numbers (A118363) k such that m = k/f(k) and m/f(m) are also factorial-base Niven numbers, where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k.

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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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