A043289 -id:A043289 - cp's OEIS frontend

A043290 Maximal run length in base 16 representation of n.

1, 1, 1, 1, 1, 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, 2, 1, 1, 1, 1, 1, 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, 2, 1, 1, 1, 1, 1, 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, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Clark Kimberling

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A043276, A043277, A043278, A043279, A043280, A043281, A043282, A043283, A043284, A043285, A043286, A043287, A043288, A043289 for base-2 to base-15 analogs.

A043290[n_]:=Max[Map[Length,Split[IntegerDigits[n,16]]]];Array[A043290,100] (* Paolo Xausa, Sep 27 2023 *)

A043290(n,b=16)={my(m,c=1);while(n>0,n%b==(n\=b)%b && c++ && next;m=max(m,c);c=1);m} \\ Use optional 2nd arg to get sequences A043276 through A043289. - M. F. Hasler, Jul 23 2013

from itertools import groupby
def A043290(n): return max(len(list(g)) for k, g in groupby(hex(n)[2:])) # Chai Wah Wu, Mar 09 2023

More terms from Antti Karttunen, Sep 21 2018

A384914 The number of unordered factorizations of n into numbers of the form p^(k^2) where p is prime and k >= 0 (A323520).

Original entry on oeis.org

1, 1, 1, 1, 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, 2, 1, 1, 1, 1, 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, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 12 2025

First differs from A203640, A295658 and A365333 at n = 64, from A043289 and A053164 at n = 81, and from A063775 at n = 512.

			a(16) = 2 since 4 has 2 factorizations: 2^1 * 2^1 * 2^1 * 2^1 and 2^4, with exponents 1 and 4 that are squares.

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

Cf. A000290, A001156, A197680, A323520.
Cf. A043289, A053164, A063775, A203640, A295658, A365333.
Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384915, A384916.

s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * Boole[IntegerQ[Sqrt[d]]], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n];
f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*issquare(d)) * s(n-j))/n);
a(n) = vecprod(apply(s, factor(n)[, 2]));

Multiplicative with a(p^e) = A001156(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.08451356983124311685..., where f(x) = (1-x) / Product_{k>=1} (1-x^(k^2)).

A043567 Number of runs in base-15 representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Clark Kimberling

Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences.

			For n = 226, its base-15 representation is "101" as 226 = 1*(15^2) + 0*(15^1) + 1*(15^0). "101" has three runs, thus a(226) = 3.
For n = 482, its base-15 representation is "222" as 482 = 2*(15^2) + 2*(15^1) + 2*(15^0). "222" has just one run, thus a(482) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A043289, A043542, A297783 (number of distinct runs), A297770.

Table[Length@ Split@ IntegerDigits[n, 15], {n, 0, 105}] (* Michael De Vlieger, Oct 10 2017 *)

(define (A043567 n) (let loop ((n n) (runs 1) (pd (modulo n 15))) (if (zero? n) runs (let ((d (modulo n 15))) (loop (/ (- n d) 15) (+ runs (if (not (= d pd)) 1 0)) d))))) ;; Antti Karttunen, Oct 10 2017

More terms from Antti Karttunen, Oct 10 2017

Updated by Clark Kimberling, Feb 04 2018

A203640 Length of the cycle reached for the map x->A203639(x), starting at n.

Original entry on oeis.org

1, 1, 1, 1, 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, 2, 1, 1, 1, 1, 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, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

R. J. Mathar, Jan 04 2012

Differs from A063775 at n = 64, 81, 128, 144, 162, 192, ... Differs from A053164 at n = 64, 81, 128, 144, 162, 192, 216, 243,... Differs from A043289 at n = 64, 128, 144, 192, 216, 225,... - R. J. Mathar, Jan 11 2012

			Starting from n = 12, the derived sequence is A203639(12) = 4, A203639(4) = 4, A203639(4) = 4, etc, and the sequence 12, 4, 4, 4, 4, ... has cycle length 1. Therefore a(12) = 1.
From _Antti Karttunen_, Sep 13 2017: (Start)
Starting from n = 16, the derived sequence is A203639(16) = 32, A203639(32) = 80, A203639(80) = 32, etc, and the sequence 16, 32, 80, 32, 80, ... has cycle length 2. Therefore a(16) = 2.
Starting from n = 2916, the derived sequence is A203639(2916) = 5832, A203639(5832) = 17496, A203639(17496) = 61236, A203639(61236) = 20412, A203639(20412) = 5832,  etc, and the sequence 2916, 5832, 17496, 61236, 20412, 5832, 17496, 61236, 20412, ... has cycle length 4. Therefore a(2916) = 4. This is also the first point where the sequence attains a value larger than 2. (End)

Antti Karttunen, Table of n, a(n) for n = 1..65537

idx := proc(L,n) for i from 1 to nops(L) do if op(i,L)=n then return i ; end if; end do: return -1; end proc:
A203640 := proc(n) local s,dr,d; s := [n] ;dr :=n ; for d from 1 do dr := A203639(dr) ; ii := idx(s,dr) ; if ii >0 then return nops(s)-ii+1 ; else s := [op(s),dr] ; end if ; end do: end proc:

(define (A203640 n) (let loop ((visited (list n)) (i 1)) (let ((next (A203639 (car visited)))) (cond ((member next visited) => (lambda (prepath) (+ 1 (- i (length prepath))))) (else (loop (cons next visited) (+ 1 i)))))))
;; (Code for A203639 given under that entry.) - Antti Karttunen, Sep 13 2017

A043287 Maximal run length in base-13 representation of n.

Original entry on oeis.org

1, 1, 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, 2, 1, 1, 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, 2, 1, 1, 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, 2, 1, 1, 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, 2, 1, 1, 1, 1, 1

Offset: 1

Clark Kimberling

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A043276, A043277, A043278, A043279, A043280, A043281, A043282, A043283, A043284, A043285, A043286, A043288, A043289, A043290 for base-2 to base-16 analogs.

A043287[n_]:=Max[Map[Length,Split[IntegerDigits[n,13]]]];Array[A043287,100] (* Paolo Xausa, Sep 27 2023 *)

A043287(n,b=13)={my(m,c=1);while(n>0,n%b==(n\=b)%b&&c++&&next;m=max(m,c);c=1);m} \\ M. F. Hasler, Jul 23 2013

More terms from Antti Karttunen, Sep 21 2018

A043288 Maximal run length in base-14 representation of n.

Original entry on oeis.org

1, 1, 1, 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, 2, 1, 1, 1, 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, 2, 1, 1, 1, 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, 2, 1, 1, 1, 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, 2

Offset: 1

Clark Kimberling

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A043276, A043277, A043278, A043279, A043280, A043281, A043282, A043283, A043284, A043285, A043286, A043287, A043289, A043290 for base-2 to base-16 analogs.

A043288[n_]:=Max[Map[Length,Split[IntegerDigits[n,14]]]];Array[A043288,100] (* Paolo Xausa, Sep 27 2023 *)

A043288(n,b=14)={my(m,c=1);while(n>0,n%b==(n\=b)%b&&c++&&next;m=max(m,c);c=1);m} \\ M. F. Hasler, Jul 23 2013

More terms from Antti Karttunen, Sep 21 2018

A043542 Number of distinct base-15 digits of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Clark Kimberling

Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A043289, A043567.

Table[Length[Union[IntegerDigits[n,15]]],{n,110}] (* Harvey P. Dale, Jul 14 2020 *)

A043542(n) = #vecsort(digits(n, 15), , 8); \\ Antti Karttunen, Oct 08 2017

More terms from Antti Karttunen, Oct 08 2017

A365333 The number of exponentially odd coreful divisors of the largest square dividing n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 01 2023

First differs from A043289, A053164, A063775, A203640 and A295658 at n = 64.
The number of squares dividing the largest exponentially odd divisor of n is A325837(n).
The sum of the exponentially odd divisors of the largest square dividing n is A365334(n). [corrected, Sep 08 2023]
The number of exponentially odd divisors of the largest square dividing n is the same as the number of squares dividing n, A046951(n). - Amiram Eldar, Sep 08 2023

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

Cf. A008833, A046100, A046951, A325837, A365334.

Cf. A043289, A053164, A063775, A203640, A295658.

f[p_, e_] := Max[1, Floor[e/2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> max(1, x\2), factor(n)[, 2]));

a(n) = A325837(A008833(n)).
a(n) = 1 if and only if n is a biquadratefree number (A046100).
Multiplicative with a(p^e) = max(1, floor(e/2)).
Dirichlet g.f.: zeta(s) * zeta(4*s) * zeta(6*s) / zeta(12*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15015/(1382*Pi^2) = 1.100823... .

Name corrected by Amiram Eldar, Sep 08 2023

cp's OEIS Frontend

A043290 Maximal run length in base 16 representation of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A384914 The number of unordered factorizations of n into numbers of the form p^(k^2) where p is prime and k >= 0 (A323520).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A043567 Number of runs in base-15 representation of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Scheme

Extensions

A203640 Length of the cycle reached for the map x->A203639(x), starting at n.

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

Scheme

A043287 Maximal run length in base-13 representation of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A043288 Maximal run length in base-14 representation of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A043542 Number of distinct base-15 digits of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A365333 The number of exponentially odd coreful divisors of the largest square dividing n.

Views

Author

Keywords