A286371 -id:A286371 - cp's OEIS frontend

A286372 a(n) = A286366(A064216(n)).

4, 6, 8, 13, 4, 9, 8, 11, 13, 12, 14, 8, 28, 6, 9, 13, 11, 21, 9, 11, 13, 12, 8, 8, 40, 14, 9, 65, 14, 13, 8, 13, 64, 13, 11, 8, 9, 30, 20, 12, 4, 9, 21, 11, 8, 21, 14, 20, 12, 9, 12, 13, 23, 9, 8, 11, 13, 64, 8, 84, 28, 14, 116, 12, 14, 9, 85, 11, 8, 12, 11, 65, 65, 42, 8, 13, 13, 21, 9, 11, 21, 13, 66, 8, 28, 12, 9, 49, 14, 13, 8, 11, 65, 20, 14, 13, 9, 66

Offset: 1

Antti Karttunen, May 09 2017

nonn

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

Cf. A064216, A278223, A286243, A286366, A286371, A286373.

(define (A286372 n) (A286366 (A064216 n)))

a(n) = A286366(A064216(n)).

A286369 Compound filter: a(n) = 2*A286364(n) + floor(A072400(n)/4).

Original entry on oeis.org

2, 2, 4, 2, 7, 5, 5, 2, 14, 6, 4, 4, 7, 5, 11, 2, 6, 14, 4, 7, 33, 5, 5, 5, 20, 6, 58, 5, 7, 11, 5, 2, 32, 6, 10, 14, 7, 5, 11, 6, 6, 32, 4, 4, 25, 5, 5, 4, 14, 20, 10, 7, 7, 59, 11, 5, 32, 6, 4, 11, 7, 5, 135, 2, 42, 32, 4, 6, 33, 11, 5, 14, 6, 6, 28, 4, 33, 11, 5, 7, 242, 6, 4, 33, 43, 5, 11, 5, 6, 24, 10, 5, 33, 5, 11, 5, 6, 14, 134, 20, 7, 11, 5, 6, 46, 6

Offset: 1

Antti Karttunen, May 09 2017

nonn

This sequence contains, in addition to the information contained in A286364 (which packs the values of A286361(n) and A286363(n) to a single value with the pairing function A000027) also the bit-2 of A072400(n) (its third least significant bit), which is here stored as the least significant bit of a(n). In contrast to A286366, the parity of the highest power of 2 dividing n is not stored.
Thus we have (among other such identities) the following two identities related to equivalence class partitioning:
For all odd i, odd j: a(i) = a(j) <=> A286366(i) = A286366(j).
For all odd i, odd j: a(i) = a(j) => A010877(i) = A010877(j). [On odd numbers the information contained in a(n) is sufficient to determine the value of n modulo 8, one of the 1, 3, 5 or 7.]

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

Cf. A072400, A286364, A286366, A286370, A286371.

from sympy.ntheory.factor_ import digits
from sympy import factorint
from operator import mul
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def A(n, k):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==k else i**f[i] for i in f])
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a286364(n): return T(a046523(n/A(n, 1)), a046523(n/A(n, 3)))
def a072400(n): return int(str(int(''.join(map(str, digits(n, 4)[1:]))[::-1]))[::-1], 4)%8
def a(n): return 2*a286364(n) + int(a072400(n)/4) # Indranil Ghosh, May 09 2017

(define (A286369 n) (+ (* 2 (A286364 n)) (floor->exact (/ (A072400 n) 4))))

a(n) = 2*A286364(n) + floor(A072400(n)/4).

A286373 a(n) = A286366(A048673(n)).

Original entry on oeis.org

4, 6, 8, 13, 4, 6, 11, 11, 13, 8, 9, 9, 28, 12, 30, 12, 14, 11, 8, 6, 9, 13, 21, 12, 40, 14, 269, 42, 4, 13, 8, 14, 64, 13, 21, 12, 65, 20, 8, 21, 11, 8, 11, 8, 11, 8, 116, 20, 13, 14, 8, 65, 23, 9, 11, 13, 14, 9, 9, 11, 14, 11, 66, 85, 21, 30, 28, 11, 12, 13, 13, 42, 14, 11, 20, 14, 30, 6, 66, 9, 12, 84, 49, 11, 8, 9, 23, 8, 28, 9, 11, 8, 65, 13, 484, 11, 20

Offset: 1

Antti Karttunen, May 09 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python program to generate the sequence

Cf. A048673, A278224, A286366, A286371, A286372.

(define (A286373 n) (A286366 (A048673 n)))

a(n) = A286366(A048673(n)).

A286370 a(n) = A286369(A139391(n)).

Original entry on oeis.org

2, 2, 7, 2, 2, 4, 4, 2, 5, 7, 6, 4, 7, 5, 5, 2, 7, 14, 7, 7, 2, 4, 10, 4, 4, 7, 6, 5, 4, 11, 5, 2, 20, 6, 7, 14, 5, 4, 4, 7, 5, 33, 42, 4, 6, 5, 5, 4, 7, 20, 33, 7, 7, 58, 4, 5, 4, 7, 6, 11, 5, 5, 11, 2, 14, 32, 7, 6, 7, 10, 4, 14, 11, 7, 6, 4, 7, 11, 11, 7, 7, 6, 73, 33, 2, 4, 4, 4, 4, 25, 6, 5, 10, 5, 11, 4, 6, 14, 7, 20, 4, 10, 10, 7, 5, 7, 32, 58, 6, 11

Offset: 1

Antti Karttunen, May 09 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python program to generate the sequence
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A139391, A286369, A286371.

(define (A286370 n) (A286369 (A139391 n)))

a(n) = A286369(A139391(n)).

A286461 Compound filter (2-adic valuation of n & 4k+1,4k+3 prime-signature combination of 2n-1): a(n) = P(A001511(n), A286364((2*n)-1)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 4, 9, 22, 5, 4, 32, 4, 5, 121, 9, 46, 437, 4, 20, 121, 17, 4, 24, 4, 5, 67, 14, 22, 17, 4, 24, 121, 5, 4, 2562, 211, 5, 121, 9, 4, 107, 121, 14, 7261, 5, 211, 24, 4, 17, 121, 41, 4, 2280, 4, 9, 254, 5, 4, 32, 4, 17, 67, 24, 22, 17, 631, 35, 121, 5, 121, 783, 4, 5, 121, 32, 211, 2280, 4, 9, 67, 17, 4, 41, 121, 5, 254, 9, 46, 2280, 4, 140, 121, 5, 4, 24

Offset: 1

Antti Karttunen, May 10 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function
Indranil Ghosh, Python program to generate the sequence

Cf. A000027, A001511, A046523, A278223, A286161, A286364, A286371, A286372, A286463, A286465.

(define (A286461 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A286364 (+ n n -1))) 2) (- (A001511 n)) (- (* 3 (A286364 (+ n n -1)))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A286364((2*n)-1))^2) - A001511(n) - 3*A286364((2*n)-1)).

cp's OEIS Frontend

A286372 a(n) = A286366(A064216(n)).

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A286369 Compound filter: a(n) = 2*A286364(n) + floor(A072400(n)/4).

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A286373 a(n) = A286366(A048673(n)).

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Formula

A286370 a(n) = A286369(A139391(n)).

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Programs

Scheme

Formula

A286461 Compound filter (2-adic valuation of n & 4k+1,4k+3 prime-signature combination of 2n-1): a(n) = P(A001511(n), A286364((2*n)-1)), where P(n,k) is sequence A000027 used as a pairing function.

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