A038502 -id:A038502 - cp's OEIS frontend

A078412 a(0) = 5, a(1) = 8; for n >1, a(n)=(a(n-1)+a(n-2))/3^n, where 3^n is the highest power of 3 dividing a(n-1)+a(n-2).

5, 8, 13, 7, 20, 1, 7, 8, 5, 13, 2, 5, 7, 4, 11, 5, 16, 7, 23, 10, 11, 7, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2

Offset: 0

Yasutoshi Kohmoto, Dec 28 2002

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A038502, A078414.

Join[{5, 8, 13, 7, 20, 1, 7, 8, 5, 13, 2, 5, 7, 4, 11, 5, 16, 7, 23, 10, 11, 7},LinearRecurrence[{0, 0, 1},{2, 1, 1},79]] (* Ray Chandler, Aug 25 2015 *)

a(3n-1)=2, a(3n)=1, a(3n+1)=1 for n>=8. - Sascha Kurz, Jan 04 2003

More terms from Sascha Kurz, Jan 04 2003

A122257 Characteristic function of Pierpont primes (A005109).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Aug 29 2006

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Pierpont Prime.
Index entries for characteristic functions.

Cf. A005109, A010051, A065333, A122258 (partial sums).

smooth3Q[n_] := n == 2^IntegerExponent[n, 2]*3^IntegerExponent[n, 3];
a[n_] := Boole[PrimeQ[n] && smooth3Q[n - 1]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 16 2021 *)

is3smooth(n) = my(m = n >> valuation(n, 2)); m == 3^valuation(m, 3);
a(n) = isprime(n) && is3smooth(n-1); \\ Amiram Eldar, May 14 2025

(define (A122257 n) (if (= 1 n) 0 (if (= 1 (A065333 (- n 1))) (A010051 n) 0)))
(define (A065333 n) (if (= 1 (A038502 (A000265 n))) 1 0))
;; Antti Karttunen, Dec 07 2017

a(n) = A010051(n) * A065333(n-1).
a(n) = if (n is prime) and (n-1 is 3-smooth) then 1 else 0.
a(n) = if n=1 then 0 else A122258(n) - A122258(n-1);
a(A122259(n)) = 0, a(A005109(n)) = 1.

A264983 Odd bisection of A263273.

Original entry on oeis.org

1, 3, 7, 5, 9, 19, 13, 21, 25, 11, 15, 23, 17, 27, 55, 37, 57, 73, 31, 39, 67, 49, 63, 61, 43, 75, 79, 29, 33, 65, 47, 45, 59, 41, 69, 77, 35, 51, 71, 53, 81, 163, 109, 165, 217, 91, 111, 199, 145, 171, 181, 127, 219, 235, 85, 93, 193, 139, 117, 175, 121, 201, 229, 103, 147, 211

Offset: 0

Antti Karttunen, Dec 05 2015

Indranil Ghosh, Table of n, a(n) for n = 0..10000

Cf. A263273, A264984, A264985.

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 = Select[f /@ Range@ 130, OddQ] (* 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 a(n): return a263273(2*n + 1) # Indranil Ghosh, May 22 2017

(define (A264983 n) (A263273 (+ 1 n n)))

a(n) = A263273(2n + 1).

A264986 Even bisection of A263272; terms of A264974 doubled.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 32, 18, 20, 38, 24, 26, 28, 30, 16, 34, 36, 22, 40, 42, 68, 86, 96, 50, 104, 54, 56, 110, 60, 74, 92, 114, 44, 98, 72, 62, 116, 78, 80, 82, 84, 46, 100, 90, 64, 118, 48, 70, 88, 102, 52, 106, 108, 58, 112, 66, 76, 94, 120, 122, 284, 126, 176, 338, 204, 230, 248, 258, 140, 302, 288

Offset: 0

Antti Karttunen, Dec 05 2015

nonn

Cf. A263272, A264974, A264987.

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(4*n)/2 # Indranil Ghosh, May 23 2017

Scheme
```
(define (A264986 n) (A263272 (+ n n)))
```

a(n) = A263272(2*n).
a(n) = 2 * A264974(n).
a(n) = A263273(4*n)/2.

A264987 Odd bisection of A263272.

Original entry on oeis.org

1, 3, 5, 11, 9, 7, 13, 15, 23, 29, 33, 17, 35, 27, 19, 37, 21, 25, 31, 39, 41, 95, 45, 59, 113, 69, 77, 83, 87, 47, 101, 99, 65, 119, 51, 71, 89, 105, 53, 107, 81, 55, 109, 57, 73, 91, 111, 43, 97, 63, 61, 115, 75, 79, 85, 93, 49, 103, 117, 67, 121, 123, 203, 257, 285, 149, 311, 135, 167, 329, 177, 221, 275

Offset: 0

Antti Karttunen, Dec 05 2015

nonn

Cf. A263272, A264986, A264989.

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*(2*n + 1))/2 # Indranil Ghosh, May 23 2017

(define (A264987 n) (A263272 (+ 1 n n)))

a(n) = A263272((2*n)+1).

A336457 a(n) = A065330(sigma(n)), where A065330 is fully multiplicative with a(2) = a(3) = 1, and a(p) = p for primes p > 3.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 5, 13, 1, 1, 7, 7, 1, 1, 31, 1, 13, 5, 7, 1, 1, 1, 5, 31, 7, 5, 7, 5, 1, 1, 7, 1, 1, 1, 91, 19, 5, 7, 5, 7, 1, 11, 7, 13, 1, 1, 31, 19, 31, 1, 49, 1, 5, 1, 5, 5, 5, 5, 7, 31, 1, 13, 127, 7, 1, 17, 7, 1, 1, 1, 65, 37, 19, 31, 35, 1, 7, 5, 31, 121, 7, 7, 7, 1, 11, 5, 5, 5, 13, 7, 7, 1, 1, 5, 7, 49

Offset: 1

Antti Karttunen, Jul 24 2020

Sequence removes prime factors 2 and 3 from the prime factorization of the sum of divisors of n.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A038502, A065330, A161942, A336458.

Cf. also A336455.

Array[Times @@ Map[#1^#2 & @@ # &, DeleteCases[FactorInteger[DivisorSigma[1, #]], ?(First@ # <= 3 &)]] &, 97] (* _Michael De Vlieger, Jul 24 2020 *)

A065330(n) = (n>>valuation(n, 2)/3^valuation(n, 3));
A336457(n) = A065330(sigma(n));

a(n) = A065330(A000203(n)) = A038502(A161942(n)).

Multiplicative with a(p^e) = A065330(1 + p + p^2 + ... + p^e).

A336504 3-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_3[x].

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 26, 27, 30, 32, 36, 39, 40, 42, 44, 45, 48, 52, 54, 56, 60, 63, 64, 66, 72, 78, 80, 81, 84, 88, 90, 96, 99, 100, 104, 105, 108, 112, 117, 120, 126, 128, 130, 132, 135, 140, 144, 150, 156, 160, 162, 165, 168, 176, 180

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 3-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} A007734(d) * n_d, where A007734(d) is the multiplicative order of 3 modulo the largest divisor of d not divisible by 3, and 0 <= n_d <= phi(d)/A007734(d).
The number of terms not exceeding 10^k for k = 1, 2, ... are 7, 41, 258, 1881, 15069, 127350, 1080749, ...

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. A038502, A007734, A007949.

Cf. A000010, A260653, A336503, A336505.

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[#, 3] &]

A348932 Numbers k congruent to 1 or 5 mod 6, for which A348930(k) > k.

Original entry on oeis.org

7, 13, 19, 25, 31, 37, 43, 61, 67, 73, 79, 91, 97, 103, 109, 121, 127, 133, 139, 151, 157, 163, 175, 181, 193, 199, 211, 217, 223, 229, 241, 247, 259, 271, 277, 283, 289, 301, 307, 313, 325, 331, 337, 343, 349, 367, 373, 379, 397, 403, 409, 421, 427, 433, 439, 457, 463, 469, 475, 481, 487, 499, 511, 523, 529, 541

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

See comments in A348930.

Index entries for sequences related to sigma(n)

Cf. A007310, A348930, A348931.

Cf. also A348742, A348754.

s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[550], 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));
isA348932(n) = ((n%2)&&(n%3)&&(A348930(n)>n));

A348933 Numbers k congruent to 1 or 5 mod 6, for which A348930(k^2) < k^2.

Original entry on oeis.org

7, 13, 19, 31, 35, 37, 43, 61, 65, 67, 73, 77, 79, 91, 95, 97, 103, 109, 119, 127, 133, 139, 143, 151, 155, 157, 161, 163, 175, 181, 185, 193, 199, 203, 209, 211, 215, 217, 221, 223, 229, 241, 247, 259, 271, 277, 283, 287, 299, 301, 305, 307, 313, 323, 325, 329, 331, 335, 337, 341, 349, 365, 367, 371, 373, 377, 379

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

Any hypothetical odd term y of A005820 must by necessity be a square. If y is also a nonmultiple of 3, then the square root x = A000196(y) of such a number y must satisfy the condition that for all nontrivial unitary divisor pairs d and x/d [with gcd(d,x/d) = 1, 1 < d < x], the other divisor should reside in this sequence, and the other divisor in A348934. The explanation is similar to the one given in A348738. See also comments in A348935.

Index entries for sequences related to sigma(n)

Cf. A000196, A005820, A348738, A348930, A348931, A348934, A348935.

s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[400], MemberQ[{1, 5}, Mod[#, 6]] && s[DivisorSigma[1, #^2]] < #^2 &] (* Amiram Eldar, Nov 04 2021 *)

A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));
isA348933(n) = ((n%2)&&(n%3)&&(A348930(n^2)<(n^2)));

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

cp's OEIS Frontend

A078412 a(0) = 5, a(1) = 8; for n >1, a(n)=(a(n-1)+a(n-2))/3^n, where 3^n is the highest power of 3 dividing a(n-1)+a(n-2).

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A122257 Characteristic function of Pierpont primes (A005109).

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A264983 Odd bisection of A263273.

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A264986 Even bisection of A263272; terms of A264974 doubled.

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A264987 Odd bisection of A263272.

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A336457 a(n) = A065330(sigma(n)), where A065330 is fully multiplicative with a(2) = a(3) = 1, and a(p) = p for primes p > 3.

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A336504 3-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_3[x].

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A348932 Numbers k congruent to 1 or 5 mod 6, for which A348930(k) > k.

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