A286372 -id:A286372 - cp's OEIS frontend

A286366 Compound filter: a(n) = 2*A286365(n) + floor(A072400(n)/4).

4, 6, 8, 4, 13, 11, 9, 6, 28, 14, 8, 8, 13, 11, 21, 4, 12, 30, 8, 13, 65, 11, 9, 11, 40, 14, 116, 9, 13, 23, 9, 6, 64, 14, 20, 28, 13, 11, 21, 14, 12, 66, 8, 8, 49, 11, 9, 8, 28, 42, 20, 13, 13, 119, 21, 11, 64, 14, 8, 21, 13, 11, 269, 4, 84, 66, 8, 12, 65, 23, 9, 30, 12, 14, 56, 8, 65, 23, 9, 13, 484, 14, 8, 65, 85, 11, 21, 11, 12, 50, 20, 9, 65, 11, 21, 11

Offset: 1

Antti Karttunen, May 08 2017

nonn

Each term of this sequence contains, in addition to the information contained in A286365 (which packs the values of A286361(n) and A286363(n) and parity of the exponent of the highest power of 2 dividing n) also the bit-2 of A072400(n) (its third least significant bit), which is here stored as the least significant bit of a(n). Note that the whole A072400(n) can be recovered based on the other information contained in a(n). Together all this information is enough - by Lagrange's "Four Squares theorem" - to determine what is the least number of squares that add up to n. Thus it follows that for all i, j: a(i) = a(j) => A002828(i) = A002828(j).

A286369 is similar, but without the parity of the 2-adic value present.

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

Cf. A000035, A002828, A007814, A072400, A286361, A286363, A286364, A286365, A286368, A286369, A286372, A286373.

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 a007814(n): return 1 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a286365(n): return 2*a286364(n) + a007814(n)%2
def a072400(n): return int(str(int(''.join(map(str, digits(n, 4)[1:]))[::-1]))[::-1], 4)%8
def a(n): return 2*a286365(n) + int(a072400(n)/4) # Indranil Ghosh, May 09 2017

(define (A286366 n) (+ (* 2 (A286365 n)) (floor->exact (/ (A072400 n) 4))))

a(n) = 2*A286365(n) + floor(A072400(n)/4).

A286371 a(n) = A286369(A003961(n)).

Original entry on oeis.org

2, 4, 7, 14, 5, 11, 4, 58, 20, 33, 7, 25, 6, 32, 10, 242, 4, 28, 5, 135, 11, 11, 7, 77, 14, 10, 73, 134, 5, 46, 7, 994, 42, 32, 33, 50, 6, 33, 43, 555, 4, 47, 5, 25, 29, 11, 7, 277, 14, 134, 11, 24, 4, 89, 10, 554, 10, 33, 7, 160, 4, 11, 28, 4034, 11, 54, 5, 134, 42, 873, 6, 118, 5, 10, 25, 135, 11, 55, 4, 2259, 272, 32, 6, 161, 33, 33, 10, 77, 6, 81, 10, 25

Offset: 1

Antti Karttunen, May 09 2017

nonn

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

Cf. A003961, A286369, A286370, A286372, A286373.

(define (A286371 n) (A286369 (A003961 n)))

a(n) = A286369(A003961(n)).

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)).

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)).

A286452 Compound filter (largest prime factor of n & prime signature of 2n-1): a(n) = P(A061395(n), A046523(2n-1)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 5, 2, 18, 5, 14, 16, 5, 9, 50, 5, 42, 59, 9, 2, 73, 23, 44, 31, 14, 20, 199, 5, 18, 61, 5, 40, 115, 9, 77, 67, 50, 35, 40, 5, 90, 179, 61, 9, 391, 14, 185, 50, 9, 100, 205, 23, 14, 94, 35, 27, 1006, 5, 20, 40, 44, 115, 395, 31, 228, 131, 59, 2, 61, 20, 295, 442, 54, 14, 320, 23, 346, 265, 9, 44, 125, 61, 275, 31, 23, 104, 1349, 14, 52, 314, 65, 125, 430

Offset: 1

Antti Karttunen, May 14 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A046523, A061395, A278223, A285334, A286356, A286372, A286453.

from sympy import primepi, primefactors, factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
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 a061395(n): return 0 if n==1 else primepi(primefactors(n)[-1])
def a(n): return T(a061395(n), a046523(2*n - 1)) # Indranil Ghosh, May 14 2017

(define (A286452 n) (* (/ 1 2) (+ (expt (+ (A061395 n) (A046523 (+ n n -1))) 2) (- (A061395 n)) (- (* 3 (A046523 (+ n n -1)))) 2)))

a(n) = (1/2)*(2 + ((A061395(n)+A046523(2n-1))^2) - A061395(n) - 3*A046523(2n-1)).

A286453 Compound filter: a(n) = P(A061395(n), A286465(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 5, 11, 94, 5, 14, 254, 17, 9, 195, 47, 259, 500, 9, 11, 413, 138, 44, 303, 32, 20, 2784, 47, 354, 216, 5, 329, 506, 9, 77, 3161, 356, 35, 175, 107, 202, 2709, 216, 24, 11188, 14, 420, 356, 24, 285, 450, 498, 70, 2349, 35, 51, 115937, 5, 20, 329, 74, 310, 3420, 864, 1243, 336, 500, 11, 384, 20, 580, 47285, 87, 14, 615, 498, 1296, 3015, 9, 74, 3491, 216

Offset: 1

Antti Karttunen, May 14 2017

nonn

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

Cf. A000027, A046523, A061395, A112049, A278223, A286356, A286372, A286452.

(define (A286453 n) (* (/ 1 2) (+ (expt (+ (A061395 n) (A286465 n)) 2) (- (A061395 n)) (- (* 3 (A286465 n))) 2)))

a(n) = (1/2)*(2 + ((A061395(n)+A286465(n))^2) - A061395(n) - 3*A286465(n)).

cp's OEIS Frontend

A286366 Compound filter: a(n) = 2*A286365(n) + floor(A072400(n)/4).

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A286371 a(n) = A286369(A003961(n)).

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

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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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Formula

A286452 Compound filter (largest prime factor of n & prime signature of 2n-1): a(n) = P(A061395(n), A046523(2n-1)), where P(n,k) is sequence A000027 used as a pairing function.

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Author

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Programs

Python

Scheme

Formula

A286453 Compound filter: a(n) = P(A061395(n), A286465(n)), where P(n,k) is sequence A000027 used as a pairing function.

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Formula