A159973 -id:A159973 - cp's OEIS frontend

A368625 Characteristic function of non-refactorable numbers (A159973).

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

Offset: 1

Wesley Ivan Hurt, Jan 01 2024

Antti Karttunen, Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Refactorable Number
Index entries for characteristic functions

Cf. A000005, A033950 (refactorable numbers), A054008, A159973, A336040.

Table[(Ceiling[n/DivisorSigma[0, n]] - Floor[n/DivisorSigma[0, n]]), {n, 100}]

A368625(n) = !!(n%numdiv(n)); \\ Antti Karttunen, Jan 17 2025

a(n) = ceiling(n/d(n))-floor(n/d(n)), where d(n) is the number of divisors of n (A000005).
a(n) = 1 - A336040(n).
a(n) = [A054008(n) > 0], where [ ] is the Iverson bracket. - Antti Karttunen, Jan 17 2025

Data section extended up to a(105) by Antti Karttunen, Jan 17 2025

A374540 a(1) = 0; for n >= 2, a(n) is the number of iterations needed for the map x -> x/A000005(x) to reach a least integer, when starting from x = A033950(n).

Original entry on oeis.org

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

Offset: 1

Ctibor O. Zizka, Jul 11 2024

nonn

The refactorability "depth" for refactorable numbers. Numbers from A159973 have the refactorability "depth" 0. Records reached for A033950(A360806(n)), i.e. the growth of the sequence is very slow.

			n = 2: A033950(2) = 2, 2/A000005(2) = 1, thus a(2) = 1.
n = 3: A033950(3) = 8, 8/A000005(8) = 2 --> 2/A000005(2) = 1, thus a(3) = 2.
n = 13: A033950(13) = 80, 80/A000005(80) = 8 --> 8/A000005(8) = 2 --> 2/A000005(2) = 1, thus a(13) = 3.

Cf. A000005, A033950, A036762, A159973, A360806.

f[n_] := Module[{v = NestWhileList[# / DivisorSigma[0, #] &, n, IntegerQ[#] && # > 1 &], len}, len = Length[v]; If[IntegerQ[v[[2]]], If[v[[-1]] == 1, len - 1, len - 2], Nothing]]; f[1] = 0; Array[f, 1200] (* Amiram Eldar, Jul 11 2024 *)

A368822 Number of compositions of n into two non-refactorable parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 4, 3, 4, 5, 4, 6, 8, 8, 8, 11, 10, 9, 10, 12, 12, 15, 14, 13, 14, 17, 16, 19, 20, 19, 20, 24, 22, 23, 22, 23, 24, 27, 24, 27, 28, 27, 28, 34, 32, 31, 32, 35, 34, 37, 36, 37, 38, 41, 38, 41, 40, 41, 40, 45, 44, 43, 44, 49, 48, 47, 48, 52, 50, 53, 50, 53, 52

Offset: 1

Wesley Ivan Hurt, Jan 07 2024

			a(12) = 3 since there are 3 ordered ways to write 12 as the sum of two non-refactorable numbers: 5 + 7 = 6 + 6 = 7 + 5.

Index entries for sequences related to compositions

Cf. A159973, A368625.

Table[Sum[(Ceiling[k/DivisorSigma[0, k]] - Floor[k/DivisorSigma[0, k]]) (Ceiling[(n - k)/DivisorSigma[0, (n - k)]] - Floor[(n - k)/DivisorSigma[0, (n - k)]]), {k, n - 1}], {n, 100}]

a(n) = Sum_{k=1..n-1} c(k) * c(n-k), where c = A368625.

cp's OEIS Frontend

A368625 Characteristic function of non-refactorable numbers (A159973).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A374540 a(1) = 0; for n >= 2, a(n) is the number of iterations needed for the map x -> x/A000005(x) to reach a least integer, when starting from x = A033950(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A368822 Number of compositions of n into two non-refactorable parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula