A132739 -id:A132739 - cp's OEIS frontend

A244414 Remove highest power of 6 from n.

1, 2, 3, 4, 5, 1, 7, 8, 9, 10, 11, 2, 13, 14, 15, 16, 17, 3, 19, 20, 21, 22, 23, 4, 25, 26, 27, 28, 29, 5, 31, 32, 33, 34, 35, 1, 37, 38, 39, 40, 41, 7, 43, 44, 45, 46, 47, 8, 49, 50, 51, 52, 53, 9, 55, 56, 57, 58, 59, 10, 61, 62, 63, 64, 65, 11

Offset: 1

Wolfdieter Lang, Jun 27 2014

This is instance g = 6 of the g-family of sequences, call it r(g,n), where for g >= 2 the highest power of g is removed from n. See the crossrefs.

The present sequence is not multiplicative: a(6) = 1 not a(2)*a(3) = 6. In the prime factor decomposition one has to consider a(2^e2*3^e^3) as one entity, also for e2 >= 0, e3 >= 0 with a(1) = 1, and apply the rule given in the formula section. With this rule the sequence will be multiplicative in an unusual sense. - Wolfdieter Lang, Feb 12 2018

			a(1) = 1 = 1/6^A122841(1) = 1/6^0.
a(9) = a(2^0*3^2), min(0,2) = 0, a(9) = 2^(0-0)*3^(2-0) = 1*9 = 9.
a(12) = a(2^2*3^1), m = min(2,1) = 1, a(12) = 2^(2-1)*3^(1-1) = 2^1*1 = 2.
a(30) = a(2*3*5) = a(2^1*3^1)*a(5) = 1*a(5) = 5.

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

Cf. A122841, A000265, A038502, A065883, A132739, A242603.

A007310, A007913, A008833 are used to express relationship between terms of this sequence.

a[n_] := n/6^IntegerExponent[n, 6]; Array[a, 66] (* Robert G. Wilson v, Feb 12 2018 *)

a(n) = n/6^valuation(n,6); \\ Joerg Arndt, Jun 28 2014

a(n) = n/6^A122841(n), n >= 1.
For n >= 2, a(n) is sort of multiplicative if a(2^e2*3^e3) = 2^(e2 - m)*3^(e3 - m) with m = m(e2, e3) = min(e2, e3), for e2, e3 >= 0, a(1) = 1, and a(p^e) = p^e for primes p >= 5.
From Peter Munn, Jun 04 2020: (Start)
Proximity to being multiplicative may be expressed as follows:
a(n * A007310(k)) = a(n) * a(A007310(k));
a(n^2) = a(n)^2;
a(n) = a(A007913(n)) * a(A008833(n)).
(End)
Sum_{k=1..n} a(k) ~ (3/7) * n^2. - Amiram Eldar, Nov 20 2022

Incorrect multiplicity claim corrected by Wolfdieter Lang, Feb 12 2018

A277555 Numbers k such that k/5^m == 3 (mod 5), where 5^m is the greatest power of 5 that divides k.

Original entry on oeis.org

3, 8, 13, 15, 18, 23, 28, 33, 38, 40, 43, 48, 53, 58, 63, 65, 68, 73, 75, 78, 83, 88, 90, 93, 98, 103, 108, 113, 115, 118, 123, 128, 133, 138, 140, 143, 148, 153, 158, 163, 165, 168, 173, 178, 183, 188, 190, 193, 198, 200, 203, 208, 213, 215, 218, 223, 228

Offset: 1

Clark Kimberling, Oct 20 2016

Positions of 3 in A277543. Numbers that have 3 as their rightmost nonzero digit when written in base 5.

This is one sequence in a 4-way splitting of the positive integers; the other three are indicated in the Mathematica program.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A132739, A277543, A277550, A277551, A277548.

z = 200; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[5, 1] (* A277550 *)
p[5, 2] (* A277551 *)
p[5, 3] (* A277555 *)
p[5, 4] (* A277548 *)

isok(n) = n/5^valuation(n, 5) % 5 == 3; \\ Michel Marcus, Oct 20 2016

A221919 Triangle of numerators of sum of two unit fractions: 1/n + 1/m, n >= m >= 1.

Original entry on oeis.org

2, 3, 1, 4, 5, 2, 5, 3, 7, 1, 6, 7, 8, 9, 2, 7, 2, 1, 5, 11, 1, 8, 9, 10, 11, 12, 13, 2, 9, 5, 11, 3, 13, 7, 15, 1, 10, 11, 4, 13, 14, 5, 16, 17, 2, 11, 3, 13, 7, 3, 4, 17, 9, 19, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2, 13, 7, 5, 1, 17, 1, 19, 5, 7, 11, 23, 1

Offset: 1

Wolfdieter Lang, Feb 21 2013

The triangle of the corresponding denominators is given in A221918.
See A221918 for comments on resistance, reduced mass and radius of the twin circles in Archimedes's arbelos, as well as references.
The column sequences give A000027(n+1), A060819(n+2), A106610(n+3), A106617(n+4), A132739(n+5), A222464 for n >= m = 1,2,..., 6.

			The triangle a(n,m) begins:
n\m   1   2    3   4    5   6    7    8   9   10  11  12 ...
1:    2
2:    3   1
3:    4   5    2
4:    5   3    7   1
5:    6   7    8   9    2
6:    7   2    1   5   11   1
7:    8   9   10  11   12  13    2
8:    9   5   11   3   13   7   15    1
9:   10  11    4  13   14   5   16   17
10:  11   3   13   7    3   4   17    9  19    1
11:  12  13   14  15   16  17   18   15  20   21   2
12:  13   7    5   1   17   1   19    5   7   11  23   1
...
a(n,1) = n + 1 because R(n,1) = n/(n+1), gcd(n,n+1) = 1, hence denominator(R(n,m)) = n + 1.
a(5,4) = 9 because R(5,4) = 20/9, gcd(20,9) = 1, hence denominator( R(5,4)) = 9.
a(6,3) = 1 because R(6,3) = 18/9 = 2/1.
For the rationals R(n,m) see A221918.

Cf. A221918 (companion triangle).

a[n_, m_] := Numerator[1/n + 1/m]; Table[a[n, m], {n, 1, 12}, {m, 1, n}] // Flatten  (* Jean-François Alcover, Feb 25 2013 *)

a(n,m) = numerator(2/n + 1/m), n >= m >= 1, and 0 otherwise.

A221918(n,m)/a(n,m) = R(n,m) = n*m/(n+m). 1/R(n,m) = 1/n + 1/m.

A336505 5-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_5[x].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 18, 20, 24, 25, 30, 32, 35, 36, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 62, 64, 65, 66, 70, 72, 75, 78, 80, 84, 88, 90, 93, 96, 100, 104, 105, 108, 110, 112, 117, 120, 124, 125, 126, 128, 130, 132, 135, 140, 144, 150

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

For a rational prime number p, a "p-practical number" is a number m such that the polynomial x^m - 1 has a divisor of every degree <= m in F_p[x], the prime field of order p.
A number m is 5-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} A007736(d) * n_d, where A007736(d) is the multiplicative order of 5 modulo the largest divisor of d not divisible by 5, and 0 <= n_d <= phi(d)/A007736(d).
The number of terms not exceeding 10^k for k = 1, 2, ... are 7, 46, 286, 2179, 16847, 141446, 1223577, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack and Lola Thompson, On the degrees of divisors of T^n-1>, New York Journal of Mathematics, Vo. 19 (2013), pp. 91-116, preprint, arXiv:1206.2084 [math.NT], 2012.
Lola Thompson, Products of distinct cyclotomic polynomials, Ph.D. thesis, Dartmouth College, 2012.
Lola Thompson, On the divisors of x^n - 1 in F_p[x], International Journal of Number Theory, Vol. 9, No. 2 (2013), pp. 421-430.
Lola Thompson, Variations on a question concerning the degrees of divisors of x^n - 1, Journal de Théorie des Nombres de Bordeaux, Vol. 26, No. 1 (2014), pp. 253-267.
Eric Weisstein's World of Mathematics, Finite Field.
Wikipedia, Finite field.

Cf. A132739, A007736, A112765.

Cf. A000010, A260653, A336503, A336504.

rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; mo[n_, p_] := MultiplicativeOrder[p, n/p^IntegerExponent[n, p]]; ppQ[n_, p_] := Module[{d = Divisors[n]}, m = mo[#, p] & /@ d; ns = EulerPhi[d]/m; r = rep[m, ns]; Min @ Rest @ CoefficientList[Series[Product[1 + x^r[[i]], {i, Length[r]}], {x, 0, n}], x] >  0]; Select[Range[200], ppQ[#, 5] &]

A165725 Largest divisor of n coprime to 30. I.e., a(n) = max { k | gcd(n, k) = k and gcd(k, 30) = 1 }.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 11, 1, 13, 7, 1, 1, 17, 1, 19, 1, 7, 11, 23, 1, 1, 13, 1, 7, 29, 1, 31, 1, 11, 17, 7, 1, 37, 19, 13, 1, 41, 7, 43, 11, 1, 23, 47, 1, 49, 1, 17, 13, 53, 1, 11, 7, 19, 29, 59, 1, 61, 31, 7, 1, 13, 11, 67, 17, 23, 7, 71, 1, 73, 37, 1, 19, 77, 13, 79, 1, 1, 41, 83, 7

Offset: 1

Barry Wells (wells.barry(AT)gmail.com), Sep 25 2009

This is the sequence of the largest divisor of n which is coprime to 30. The product of the first 3 prime numbers is 2*3*5=30. This sequence gives the largest factor of n which does not include 2, 3 or 5 in its prime factorization.

			The largest factor of 1, 2, 3, 4, 5 and 6 not including the primes 2, 3 and 5 is 1. 7 is prime and therefore its sequence value is 7. For p > 5, p prime, gives a(p) = p. As 14 = 2*7, a(14)= 7. As 98 = 2*7*7, a(98)= 49.

Barry Wells, Table of n, a(n) for n = 1..1024 [a(392) and a(704) corrected by Sean A. Irvine]

A051037 gives the smooth five numbers, numbers whose prime divisor only include 2, 3 and 5. A132740 gives the largest divisor of n coprime to 10. A065330 gives a(n) = max { k | gcd(n, k) = k and gcd(k, 6) = 1 }.
Largest divisor of n coprime to a prime factor of 30: A000265 (2), A038502 (3), A132739 (5).
Cf. A355582.

a[n_] := n / Times @@ ({2, 3, 5}^IntegerExponent[n, {2, 3, 5}]); Array[a, 100] (* Amiram Eldar, Jul 10 2022 *)

a(n)=n>>valuation(n,2)/3^valuation(n,3)/5^valuation(n,5) \\ Charles R Greathouse IV, Jul 16 2017

From Amiram Eldar, Jul 10 2022: (Start)
Multiplicative with a(p^e) = p^e if p >= 7 and 1 otherwise.
a(n) = n/A355582(n). (End)
Sum_{k=1..n} a(k) ~ (5/24) * n^2. - Amiram Eldar, Nov 28 2022
Dirichlet g.f.: zeta(s-1)*(2^s-2)*(3^s-3)*(5^s-5)/((2^s-1)*(3^s-1)*(5^s-1)). - Amiram Eldar, Jan 04 2023

A193304 Squarefree numbers multiplied by powers of 5.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 122, 123, 125, 127, 129, 130, 131, 133, 134, 137, 138, 139, 141, 142, 143, 145, 146, 149, 150, 151, 154, 155, 157, 158, 159, 161, 163, 165, 166, 167, 170

Offset: 1

José María Grau Ribas, Jul 21 2011

nonn

Numbers k such that A008683(A132739(k)) is not zero, where A008683 is the Moebius mu function. - Antti Karttunen, Jun 21 2014

Antti Karttunen, Table of n, a(n) for n = 1..10000

Differs from A240370 for the first time at n=109, where A240370(109)=169, while here it is missing, and a(109)=170.

Cf. A008683, A132739.

N:= 1000: # to get all terms <= N
sf:= select(numtheory:-issqrfree,{$1..N}):
map(t -> seq(t*5^i, i=0..floor(log[5](N/t))), sf);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Apr 16 2015

lim = 102; sf = Select[Range[lim], SquareFreeQ]; lim5 = 5^Range[0, Log[5, lim]]; Select[Union[Flatten[Outer[Times, sf, lim5]]], # <= lim &]

is(n)=issquarefree(n/5^valuation(n,5)) \\ Charles R Greathouse IV, Jul 31 2011

;; With Antti Karttunen's IntSeq-library.
(define A193304 (NONZERO-POS 1 1 (COMPOSE A008683 A132739)))
;; Reflecting essentially the above Pari-program, Antti Karttunen, Jun 21 2014

a(n) ~ kn with k = 10*Pi^2/63. - Charles R Greathouse IV, Apr 16 2015

A222464 a(n) = (n+6)/gcd(n*6,n+6), n >= 6.

Original entry on oeis.org

1, 13, 7, 5, 4, 17, 1, 19, 5, 7, 11, 23, 2, 25, 13, 3, 7, 29, 5, 31, 8, 11, 17, 35, 1, 37, 19, 13, 10, 41, 7, 43, 11, 5, 23, 47, 4, 49, 25, 17, 13, 53, 3, 55, 14, 19, 29, 59, 5, 61, 31, 7, 16, 65, 11, 67, 17, 23, 35, 71, 2, 73, 37, 25, 19, 77, 13, 79, 20, 9, 41

Offset: 6

Wolfdieter Lang, Feb 21 2013

This is the sixth column (m=6) of the triangle A221919.

			a(10) = numerator(16/60) = numerator(4/15) = 4 = 16/gcd(60,16) = 16/gcd(36,16).

Cf. A221919, A000027(n+1) (m=1), A060819(n+2) (m=2), A106610(n+3) (m=3), A106617(n+4) (m=4), A132739(n+5) (m=5).

a(n) = A221919(n,6) = denominator(n*6/(n+6)) = numerator(n+6/n*6) = (n+6)/gcd(n*6,n+6) = (n+6)/gcd(36,n+6), n >= 6.

A370757 a(n) is the least k > 0 such that 1/n and 1/k have equivalent repeating decimal digits.

Original entry on oeis.org

1, 1, 3, 1, 1, 6, 7, 1, 9, 1, 11, 3, 13, 7, 6, 1, 17, 18, 19, 1, 21, 22, 23, 6, 1, 26, 27, 7, 29, 3, 31, 1, 33, 17, 7, 36, 37, 19, 39, 1, 41, 42, 43, 44, 45, 23, 47, 3, 49, 1, 51, 13, 53, 54, 55, 7, 57, 29, 59, 6, 61, 31, 63, 1, 26, 66, 67, 17, 69, 7, 71, 72

Offset: 1

Rémy Sigrist, Feb 29 2024

In other words, a(n) is the least k > 0 such that the fractional parts of (10^i)/n and (10^j)/k are equal for some integers i, j.

a(n) is not always a divisor of n. For example, a(65) = 26 is not a divisor of 65. - David A. Corneth, Mar 01 2024

			The first terms, alongside the decimal expansion of 1/n with its repeating decimal digits in parentheses, are:
  n   a(n)  1/n
  --  ----  -----------
   1     1  1.(0)
   2     1  0.5(0)
   3     3  0.(3)
   4     1  0.25(0)
   5     1  0.2(0)
   6     6  0.1(6)
   7     7  0.(142857)
   8     1  0.125(0)
   9     9  0.(1)
  10     1  0.1(0)
  11    11  0.(09)
  12     3  0.08(3)
  13    13  0.(076923)
  14     7  0.07(142857)
  15     6  0.0(6)

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences related to decimal expansion of 1/n

Cf. A000265 (base-2 analog), A038502 (base-3 analog), A132739 (base-5 analog), A242603 (base-7 analog).

Cf. A003592, A007732, A132740.

PARI
```
\\ See Links section.
```

from itertools import count
from sympy import multiplicity, n_order
def A370757(n):
    m2, m5 = (~n & n-1).bit_length(), multiplicity(5,n)
    r = max(m2,m5)
    w, m = 10**r, 10**(t:=n_order(10,n2) if (n2:=(n>>m2)//5**m5)>1 else 1)-1
    c = w//n
    s = str(m*w//n-c*m).zfill(t)
    l = len(s)
    for k in count(1):
        m2, m5 = (~k & k-1).bit_length(), multiplicity(5,k)
        r = max(m2,m5)
        w, m = 10**r, 10**(t:=n_order(10,k2) if (k2:=(k>>m2)//5**m5)>1 else 1)-1
        c = w//k
        if any(s[i:]+s[:i] == str(m*w//k-c*m).zfill(t) for i in range(l)):
            return k # Chai Wah Wu, Mar 03 2024

a(n) = 1 iff n belongs to A003592.
a(10*n) = a(n).
A007732(a(n)) = A007732(n).

cp's OEIS Frontend

A244414 Remove highest power of 6 from n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A277555 Numbers k such that k/5^m == 3 (mod 5), where 5^m is the greatest power of 5 that divides k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A221919 Triangle of numerators of sum of two unit fractions: 1/n + 1/m, n >= m >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A336505 5-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_5[x].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A165725 Largest divisor of n coprime to 30. I.e., a(n) = max { k | gcd(n, k) = k and gcd(k, 30) = 1 }.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A193304 Squarefree numbers multiplied by powers of 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

Formula

A222464 a(n) = (n+6)/gcd(n*6,n+6), n >= 6.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A370757 a(n) is the least k > 0 such that 1/n and 1/k have equivalent repeating decimal digits.

Views

Author

Keywords