A038502 -id:A038502 - cp's OEIS frontend

A265354 Permutation of nonnegative integers: a(n) = A263273(A264985(n)).

0, 1, 3, 2, 4, 9, 6, 10, 12, 7, 5, 19, 8, 13, 27, 18, 28, 36, 21, 11, 57, 24, 37, 30, 15, 31, 39, 22, 16, 64, 23, 14, 55, 20, 46, 58, 25, 17, 73, 26, 40, 81, 54, 82, 108, 63, 29, 171, 72, 109, 90, 45, 85, 117, 66, 34, 192, 69, 38, 165, 60, 100, 174, 75, 35, 219, 78, 118, 84, 33, 91, 93, 48, 32, 138, 51, 112, 111, 42, 94, 120, 67

Offset: 0

Antti Karttunen, Dec 07 2015

Composition of A263273 with the permutation obtained from its odd bisection.

Antti Karttunen, Table of n, a(n) for n = 0..9841
Index entries for sequences that are permutations of the natural numbers

Inverse: A265353.
Cf. A263273, A264985.
Cf. also A265352, A265355, A265356.

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a264985(n): return (a263273(2*n + 1) - 1)/2
def a(n): return a263273(a264985(n)) # Indranil Ghosh, May 22 2017

(define (A265354 n) (A263273 (A264985 n)))

a(n) = A263273(A264985(n)).

A242603 Largest divisor of n not divisible by 7. Remove factors 7 from n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 11, 12, 13, 2, 15, 16, 17, 18, 19, 20, 3, 22, 23, 24, 25, 26, 27, 4, 29, 30, 31, 32, 33, 34, 5, 36, 37, 38, 39, 40, 41, 6, 43, 44, 45, 46, 47, 48, 1, 50, 51, 52, 53, 54, 55, 8, 57, 58, 59, 60, 61, 62, 9, 64, 65, 66, 67, 68, 69, 10, 71, 72, 73, 74, 75, 76, 11

Offset: 1

Wolfdieter Lang, Jun 18 2014

This is member p = 7 in the p-family of sequences (p a prime).
See A000265, A038502 and A132739 for primes 2, 3 and 5, also for formulas, programs and references.
As well as being multiplicative, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 21 2019

			From _Indranil Ghosh_, Jan 31 2017: (Start)
For n = 12, the divisors of 12 are 1,2,3,4,6 and 12. The largest divisor not divisible by 7 is 12. So, a(12) = 12.
For n = 14, the divisors of 14 are 1,2,7 and 14. The largest divisor not divisible by 7 is 2. So, a(14) = 2. (End)
From _Peter Bala_, Feb 21 2019: (Start)
Sum_{n >= 1} n*a(n)*x^n = G(x) - (6*7)*G(x^7) - (6*49)*G(x^49) - (6*343)*G(x^343) - ..., where G(x) = x*(1 + x)/(1 - x)^3.
Sum_{n >= 1} (1/n)*a(n)*x^n = H(x) - (6/7)*H(x^7) - (6/49)*H(x^49) - (6/343)*H(x^343) - ..., where H(x) = x/(1 - x).
Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (6/7^2)*L(x^7) - (6/49^2)*L(x^49) - (6/343^2)*L(x^343) - ..., where L(x) = Log(1/(1 - x)).
Also, Sum_{n >= 1} (1/a(n))*x^n = L(x) + (6/7)*L(x^7) + (6/7)*L(x^49) + (6/7)*L(x^343) ... . (End)

Indranil Ghosh, Table of n, a(n) for n = 1..20000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.

Cf. A000265, A038502, A132739, A214411.

Table[n/7^IntegerExponent[n, 7], {n, 80}] (* Alonso del Arte, Jun 18 2014 *)

a(n) = f = factor(n);  for (i=1, #f~, if (f[i,1]==7, f[i, 1]=1)); factorback(f); \\ Michel Marcus, Jun 18 2014

a(n) = n \ 7^valuation(n, 7) \\ David A. Corneth, Feb 21 2019

def A242603(n):
    for i in range(n,0,-1):
        if n%i==0 and i%7!=0:
            return i # Indranil Ghosh, Jan 31 2017

Multiplicative with a(p^e) = 1 if p = 7, else p^e.
Dirichlet g.f.: zeta(s-1)*7*(7^(s-1) -  1)/(7^s - 1).
a(n) = n/A268354(n).
From Peter Bala, Feb 21 2019: (Start)
a(n) = n/gcd(n,7^n).
O.g.f.: F(x) - 6*F(x^7) - 6*F(x^49) - 6*F(x^243) - ..., where F(x) = x/(1 - x)^2 is the generating function for the positive integers. More generally, for m >= 1,
Sum_{n >= 0} (a(n)^m)*x^n = F(m,x) - (7^m - 1)( F(m,x^7) + F(m,x^49) + F(m,x^243) + ...), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m_th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.
Repeatedly applying the Euler operator x*d/dx or its inverse operator to the o.g.f. for the sequence a(n) produces generating functions for the sequences (n^m*a(n))n>=1, m in Z. Some examples are given below. (End)
Sum_{k=1..n} a(k) ~ (7/16) * n^2. - Amiram Eldar, Nov 28 2022

A265353 Permutation of nonnegative integers: a(n) = A264985(A263273(n)).

Original entry on oeis.org

0, 1, 3, 2, 4, 10, 6, 9, 12, 5, 7, 19, 8, 13, 31, 24, 28, 37, 15, 11, 33, 18, 27, 30, 21, 36, 39, 14, 16, 46, 23, 25, 73, 69, 55, 64, 17, 22, 58, 26, 40, 94, 78, 85, 112, 51, 34, 100, 72, 82, 91, 75, 109, 118, 42, 32, 96, 20, 35, 105, 60, 99, 102, 45, 29, 87, 54, 81, 84, 57, 90, 93, 48, 38, 114, 63, 108, 111, 66, 117, 120, 41

Offset: 0

Antti Karttunen, Dec 07 2015

Composition of A263273 with the permutation obtained from its odd bisection.

Antti Karttunen, Table of n, a(n) for n = 0..9841
Index entries for sequences that are permutations of the natural numbers

Inverse: A265354.
Cf. A263273, A264985.
Cf. also A265341, A265351, A265355, A265356.

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a264985(n): return (a263273(2*n + 1) - 1)/2
def a(n): return a264985(a263273(n)) # Indranil Ghosh, May 22 2017

(define (A265353 n) (A264985 (A263273 n)))

a(n) = A264985(A263273(n)).

A343430 Part of n composed of prime factors of the form 3k-1.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 11, 4, 1, 2, 5, 16, 17, 2, 1, 20, 1, 22, 23, 8, 25, 2, 1, 4, 29, 10, 1, 32, 11, 34, 5, 4, 1, 2, 1, 40, 41, 2, 1, 44, 5, 46, 47, 16, 1, 50, 17, 4, 53, 2, 55, 8, 1, 58, 59, 20, 1, 2, 1, 64, 5, 22, 1, 68, 23, 10, 71, 8, 1, 2, 25, 4, 11, 2, 1, 80, 1, 82, 83, 4, 85

Offset: 1

Peter Munn, Jun 08 2021

Largest term of A004612 that divides n.
Modulo 6, the prime numbers are partitioned into 4 nonempty sets: {2}, {3}, primes of the form 6k-1 (A007528) and primes of the form 6k+1 (A002476). The modulo 3 partition is nearly the same, but unites the only even prime, 2, with primes of the form 6k-1 in the set of primes we use here.
A positive integer m is a Loeschian number (a term of A003136) if and only if a(A007913(m)) = 1, that is the squarefree part of m has no prime factors of the form 3k-1.

			n = 60 has prime factorization 60 = 2 * 2 * 3 * 5. Factors 2 = 3*1 - 1 and 5 = 3*2 - 1 have form 3k-1, whereas 3 does not (having form 3k). Multiplying the factors of form 3k-1, we get 2 * 2 * 5 = 20. So a(60) = 20.

Equivalent sequences for prime factors of other forms: A006519 (2 only), A000265 (2k+1), A038500 (3 only), A038502 (3k+/-1), A170818 (4k+1), A097706 (4k-1), A248909 (6k+1), A343431 (6k-1).
Range of terms: A004612 (closure under multiplication of A003627).
Cf. A002476, A007528, squarefree part (A007913) of terms of A003136.
First 28 terms are the same as A247503.

f[p_, e_] := If[Mod[p, 3] == 2, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 11 2021 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 1] + 1) % 3, f[i, 1] = 1); ); factorback(f); } \\ after Michel Marcus at A248909

from math import prod
from sympy import factorint
def A343430(n): return prod(p**e for p, e in factorint(n).items() if p%3==2) # Chai Wah Wu, Dec 23 2022

Completely multiplicative with a(p) = p if p is of the form 3k-1, otherwise a(p) = 1.
For k >= 1, a(n) = a(k*n) / gcd(k, a(k*n)).
a(n) = A006519(n) * A343431(n).
a(n) = (n / A038500(n)) / A248909(n) = A038502(n) / A248909(n).
A006519(a(n)) = a(A006519(n)) = A006519(n).
A343431(a(n)) = a(A343431(n)) = A343431(n).
A038500(a(n)) = a(A038500(n)) = 1.
A248909(a(n)) = a(A248909(n)) = 1.

A067622 Consider the power series (x + 1)^(1/3) = 1 + x/3-x^2/9 + 5x^3/81 + ...; sequence gives numerators of coefficients.

Original entry on oeis.org

1, 1, -1, 5, -10, 22, -154, 374, -935, 21505, -55913, 147407, -1179256, 3174920, -8617640, 70664648, -194327782, 537259162, -13431479050, 37466757350, -104906920580, 884215473460, -2491879970660, 7042269482300, -59859290599550

Offset: 0

Benoit Cloitre, Feb 02 2002

a(n) is also the numerator of the binomial coefficient C(k,n) evaluated at k=1/3, e.g. a(4) = (1/24)k(k-1)(k-2)(k-3), plug in k=1/3 and take numerator. - James R. Buddenhagen, Aug 16 2014

Denominators are A067623.

s := convert(taylor((x+1)^(1/3), x, 50), polynom): for n from 0 to 50 do printf(`%a,`, abs(numer(coeff(s, x, n)))) od;
seq(numer(subs(k=1/3,expand(binomial(k,n)))),n=0..50) # James R. Buddenhagen, Aug 16 2014

a(n) =(-1)^n*A004990(n)*A067623(n)/A000244(n); ignoring signs, a(n) =A038502(A004990(n)) =A038502(A034164(n-2)). a(n)'s sign is (-1)^(n+1) if n>0.

Edited by Henry Bottomley and James Sellers, Feb 11 2002

A265904 Self-inverse permutation of nonnegative integers: a(n) = A263272(A263273(A263272(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 11, 6, 29, 8, 9, 10, 5, 12, 13, 38, 33, 92, 17, 18, 83, 20, 87, 110, 35, 24, 89, 26, 27, 28, 7, 30, 37, 32, 15, 86, 23, 36, 31, 14, 39, 40, 119, 114, 281, 44, 99, 254, 65, 276, 335, 98, 51, 260, 71, 54, 245, 56, 249, 326, 101, 60, 263, 74, 261, 272, 47, 330, 353, 116, 105, 278, 53, 72, 251, 62, 267, 332, 107, 78, 269, 80, 81, 82, 19

Offset: 0

Antti Karttunen, Jan 02 2016

A263273 conjugated with the permutation obtained from its even bisection.

Antti Karttunen, Table of n, a(n) for n = 0..6561
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A263272, A263273, A265351, A265352.
Cf. also A265902.
Cf. A265369, A266190, A266401, A266403 (other conjugates or similar derivations of A263273).

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@g[n] h[n]]]; t = Table[f[2 n]/2, {n, 0, 1000}]; Table[t[[f[t[[n + 1]]] + 1]], {n, 0, 83}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 *)

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a263272(n): return a263273(2*n)/2
def a(n): return a263272(a263273(a263272(n))) # Indranil Ghosh, May 25 2017

(define (A265904 n) (A263272 (A263273 (A263272 n))))

a(n) = A263272(A263273(A263272(n))).
As a composition of related permutations:
a(n) = A263272(A265352(n)).
a(n) = A265351(A263272(n)).
Other identities. For all n >= 0:
a(3*n) = 3*a(n).
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A244414 Remove highest power of 6 from n.

Original entry on oeis.org

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

A264984 Even bisection of A263273; terms of A263262 doubled.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 22, 16, 18, 20, 14, 24, 26, 28, 30, 64, 46, 36, 58, 40, 66, 76, 34, 48, 70, 52, 54, 56, 38, 60, 74, 32, 42, 68, 50, 72, 62, 44, 78, 80, 82, 84, 190, 136, 90, 172, 118, 192, 226, 100, 138, 208, 154, 108, 166, 112, 174, 220, 94, 120, 202, 148, 198, 184, 130

Offset: 0

Antti Karttunen, Dec 05 2015

nonn

Cf. A000035, A010673, A010873, A263272, A263273, A264983, A264975.

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a(n): return a263273(2*n) # Indranil Ghosh, May 22 2017

Scheme
```
(define (A264984 n) (A263273 (+ n n)))
```

a(n) = 2 * A263272(n).
a(n) = A263273(2*n).
Other identities. For all n >= 0:
A010873(a(n)) = 2 * A000035(n) = A010673(n).

A265342 Permutation of even numbers: a(n) = 2 * A265351(n).

Original entry on oeis.org

0, 2, 4, 6, 8, 22, 12, 10, 16, 18, 20, 58, 24, 26, 76, 66, 64, 70, 36, 14, 40, 30, 28, 34, 48, 46, 52, 54, 56, 166, 60, 62, 184, 174, 172, 178, 72, 74, 220, 78, 80, 238, 228, 226, 232, 198, 68, 202, 192, 190, 196, 210, 208, 214, 108, 38, 112, 42, 44, 130, 120, 118, 124, 90, 32, 94, 84, 82, 88, 102, 100, 106, 144

Offset: 0

Antti Karttunen, Dec 07 2015

nonn

Iterating this sequence as 1, a(1), a(a(1)), a(a(a(1))), ... yields A264980.

Antti Karttunen, Table of n, a(n) for n = 0..9841

Cf. A265351.

Cf. also A265341, A263273, A264980.

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a263272(n): return a263273(2*n)/2
def a(n): return 2*a263272(a263273(n)) # Indranil Ghosh, May 25 2017

Scheme
```
(define (A265342 n) (* 2 (A265351 n)))
```

a(n) = 2 * A265351(n).

A348931 Numbers k congruent to 1 or 5 mod 6, for which A348930(k) < k.

Original entry on oeis.org

5, 11, 17, 23, 29, 35, 41, 47, 49, 53, 55, 59, 65, 71, 77, 83, 85, 89, 95, 101, 107, 113, 115, 119, 125, 131, 137, 143, 145, 149, 155, 161, 167, 169, 173, 179, 185, 187, 191, 197, 203, 205, 209, 215, 221, 227, 233, 235, 239, 245, 251, 253, 257, 263, 265, 269, 275, 281, 287, 293, 295, 299, 305, 311, 317, 319, 323

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

See comments in A348930.

Index entries for sequences related to sigma(n)

Cf. A007310, A348930, A348932, A348933.

Cf. also A348741, A348753.

s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[350], MemberQ[{1, 5}, Mod[#, 6]] && s[DivisorSigma[1, #]] < # &] (* Amiram Eldar, Nov 04 2021 *)

A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));
isA348931(n) = ((n%2)&&(n%3)&&(A348930(n)

cp's OEIS Frontend

A265354 Permutation of nonnegative integers: a(n) = A263273(A264985(n)).

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A242603 Largest divisor of n not divisible by 7. Remove factors 7 from n.

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A265353 Permutation of nonnegative integers: a(n) = A264985(A263273(n)).

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A343430 Part of n composed of prime factors of the form 3k-1.

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A067622 Consider the power series (x + 1)^(1/3) = 1 + x/3-x^2/9 + 5x^3/81 + ...; sequence gives numerators of coefficients.

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A265904 Self-inverse permutation of nonnegative integers: a(n) = A263272(A263273(A263272(n))).

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A244414 Remove highest power of 6 from n.

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