A377386 -id:A377386 - cp's OEIS frontend

A377457 Numbers k such that k and k+1 are both terms in A377386.

1, 12563307224, 15897851550, 30412355999, 37706988600, 52576459775, 67673545631, 118533901904, 244316235000, 297265003100, 332110595000, 340800265728, 349358409503, 375624917760, 378624889440, 416375389115, 450026519903, 561162864248, 596004199840, 728643460544

Offset: 1

Amiram Eldar, Oct 29 2024

			12563307224 is a term since both 12563307224 and 12563307225 are in A377386: 12563307224/A034968(12563307224) = 369509036, 369509036/A034968(369509036) = 9723922 and 9723922/A034968(9723922) = 373997 are integers, and 12563307225/A034968(12563307225) = 358951635, 358951635/A034968(358951635) = 7976703 and 7976703/A034968(7976703) = 257313 are integers.

Amiram Eldar, Table of n, a(n) for n = 1..39

Cf. A034968.
Subsequence of A118363, A328205, A377385, A377386 and A377455.
Analogous sequences: A376795 (binary), A377272 (Zeckendorf).

fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
is1(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));}
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A377385 Factorial-base Niven numbers (A118363) k such that k/f(k) is also a factorial-base Niven number, 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, 27, 36, 40, 48, 54, 72, 80, 96, 108, 120, 135, 144, 168, 175, 180, 192, 208, 210, 240, 280, 288, 336, 360, 384, 420, 432, 468, 480, 490, 572, 576, 594, 600, 630, 720, 732, 740, 750, 780, 784, 819, 840, 846, 861, 864, 888, 900, 924, 936, 945, 980, 984

Offset: 1

Amiram Eldar, Oct 27 2024

			8 is a term since 8/f(8) = 4 is an integer and also 4/f(4) = 2 is an integer.

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

Subsequence of A118363.
Subsequences: A000142, A377386.
Cf. A034968, A377384.
Analogous sequences: A376616 (binary), A377209 (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]}, Divisible[k, f] && Divisible[k/f, fdigsum[k/f]]]; Select[Range[1000], 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)); !(k % f) && !((k/f) % fdigsum(k/f));}

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).

cp's OEIS Frontend

A377457 Numbers k such that k and k+1 are both terms in A377386.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A377385 Factorial-base Niven numbers (A118363) k such that k/f(k) is also a factorial-base Niven number, where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula