A286260 -id:A286260 - cp's OEIS frontend

A286460 Compound filter (2-adic valuation & sum of the divisors): a(n) = P(A001511(n), A000203(n)), where P(n,k) is sequence A000027 used as a pairing function.

1, 8, 7, 39, 16, 80, 29, 157, 79, 173, 67, 438, 92, 302, 277, 600, 154, 782, 191, 949, 497, 668, 277, 1957, 466, 905, 781, 1656, 436, 2630, 497, 2284, 1129, 1487, 1129, 4281, 704, 1832, 1541, 4282, 862, 4658, 947, 3658, 3004, 2630, 1129, 8133, 1597, 4373, 2557, 4953, 1432, 7262, 2557, 7507, 3161, 4097, 1771, 14368, 1892, 4658, 5357, 8785, 3487, 10442, 2279

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
Index entries for sequences related to sigma(n)

Cf. A000027, A286360.

Cf. A000593, A146076 (sequences matching to this filter), also A000203, A161942, A286260, A286357.

A000203(n) = sigma(n);
A001511(n) = (1+valuation(n,2));
A286460(n) = (1/2)*(2 + ((A001511(n)+A000203(n))^2) - A001511(n) - 3*A000203(n));
for(n=1, 10000, write("b286460.txt", n, " ", A286460(n)));

from sympy import divisor_sigma as D
def a001511(n): return bin(n)[2:][::-1].index("1") + 1
def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def a(n): return T(a001511(n), D(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 12 2017

(define (A286460 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A000203 n)) 2) (- (A001511 n)) (- (* 3 (A000203 n))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A000203(n))^2) - A001511(n) - 3*A000203(n)).

A286034 Compound filter: a(n) = P(A046523(n), A161942(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 8, 3, 49, 8, 34, 3, 239, 124, 97, 8, 165, 30, 34, 34, 1051, 47, 1237, 17, 508, 21, 97, 8, 727, 565, 331, 74, 165, 122, 733, 3, 4403, 34, 502, 34, 7911, 192, 196, 72, 2302, 233, 526, 68, 508, 1237, 97, 8, 3051, 1774, 5368, 97, 1782, 380, 727, 97, 727, 51, 1231, 122, 3220, 498, 34, 288, 18019, 331, 733, 155, 2713, 34, 733, 47, 35317, 705, 1897, 873, 1047, 34

Offset: 1

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
MathWorld, Pairing Function

Cf. A000027, A046523, A161942, A286260.

A000265(n) = (n >> valuation(n, 2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A161942(n) = A000265(sigma(n));
A286034(n) = (2 + ((A046523(n)+A161942(n))^2) - A046523(n) - 3*A161942(n))/2;
for(n=1, 16384, write("b286034.txt", n, " ", A286034(n)));

from sympy import factorint, divisors, divisor_sigma
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 a000265(n): return max(list(filter(lambda i: i%2 == 1, divisors(n))))
def a161942(n): return a000265(divisor_sigma(n))
def a(n): return T(a046523(n), a161942(n)) # Indranil Ghosh, May 07 2017

(define (A286034 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A161942 n)) 2) (- (A046523 n)) (- (* 3 (A161942 n))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A161942(n))^2) - A046523(n) - 3*A161942(n)).

A286358 Compound filter: a(n) = P(A286357(n), A161942(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 4, 6, 22, 8, 13, 10, 106, 79, 47, 13, 39, 30, 19, 19, 466, 47, 742, 24, 233, 21, 58, 19, 139, 466, 233, 32, 49, 122, 70, 21, 1954, 26, 380, 26, 4096, 192, 139, 49, 1037, 233, 34, 81, 256, 782, 70, 26, 531, 1597, 4279, 70, 1227, 380, 157, 70, 157, 41, 1037, 139, 280, 498, 34, 124, 8002, 256, 83, 174, 2018, 34, 83, 70, 18916, 705, 1655, 531, 669, 34, 280, 41

Offset: 1

Antti Karttunen, May 10 2017

nonn

Partitions natural numbers to the same equivalence classes as A000203. That is, for all i, j: a(i) = a(j) <=> A000203(i) = A000203(j). This follows because both A161942(n) and A286357(n) can be (are) defined as functions of A000203, and on the other hand, A000203(n) can be uniquely reconstructed from A161942(n) and A286357(n), thus from a(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function
Index entries for sequences related to sigma(n)

Cf. A000027, A000203, A161942, A286034, A286260, A286357, A286359, A286360, A286460.

A001511(n) = (1+valuation(n,2));
A000265(n) = (n >> valuation(n, 2));
A161942(n) = A000265(sigma(n));
A286357(n) = A001511(sigma(n));
A286358(n) = (1/2)*(2 + ((A286357(n)+A161942(n))^2) - A286357(n) - 3*A161942(n));
for(n=1, 10000, write("b286358.txt", n, " ", A286358(n)));

(define (A286358 n) (* (/ 1 2) (+ (expt (+ (A286357 n) (A161942 n)) 2) (- (A286357 n)) (- (* 3 (A161942 n))) 2)))

a(n) = (1/2)*(2 + ((A286357(n)+A161942(n))^2) - A286357(n) - 3*A161942(n)).

A286451 Compound filter (2-adic valuation of sigma(n) & 2-adic valuation of n): a(n) = P(A286357(n), A001511(n)), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = 0 by an explicit convention.

Original entry on oeis.org

0, 2, 6, 4, 3, 9, 10, 7, 1, 5, 6, 13, 3, 14, 10, 11, 3, 2, 6, 8, 21, 9, 10, 18, 1, 5, 10, 19, 3, 14, 21, 16, 15, 5, 15, 4, 3, 9, 10, 12, 3, 27, 6, 13, 3, 14, 15, 24, 1, 2, 10, 8, 3, 14, 10, 25, 15, 5, 6, 19, 3, 27, 10, 22, 6, 20, 6, 8, 21, 20, 10, 7, 3, 5, 6, 13, 21, 14, 15, 17, 1, 5, 6, 34, 6, 9, 10, 18, 3, 5, 15, 19, 36, 20, 10, 31, 3, 2, 6, 4, 3, 14, 10

Offset: 1

Antti Karttunen, May 13 2017

nonn

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

Cf. A000027, A001511, A286357, A286260, A286358.

A001511(n) = (1+valuation(n,2));
A286357(n) = A001511(sigma(n));
A286451(n) = if(1==n,0,(1/2)*(2 + ((A286357(n)+A001511(n))^2) - A286357(n) - 3*A001511(n)));
for(n=1, 10000, write("b286451.txt", n, " ", A286451(n)));

from sympy import divisor_sigma as D
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a001511(n): return bin(n)[2:][::-1].index("1") + 1
def a(n): return 0 if n==1 else T(a001511(D(n)), a001511(n)) # Indranil Ghosh, May 14 2017

(define (A286451 n) (if (= 1 n) 0 (* (/ 1 2) (+ (expt (+ (A286357 n) (A001511 n)) 2) (- (A286357 n)) (- (* 3 (A001511 n))) 2))))

a(1) = 0; for n > 1, a(n) = (1/2)*(2 + ((A286357(n)+A001511(n))^2) - A286357(n) - 3*A001511(n)).

A286595 Compound filter (2-adic valuation & deficiency/abundance): a(n) = P(A001511(n), A286449(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 2, 6, 4, 12, 11, 10, 16, 5, 22, 48, 37, 8, 11, 15, 46, 68, 67, 108, 22, 107, 106, 175, 137, 30, 154, 18, 172, 138, 191, 21, 67, 173, 106, 256, 232, 57, 106, 329, 277, 138, 301, 13, 37, 353, 352, 501, 407, 467, 191, 24, 466, 138, 497, 634, 562, 632, 631, 744, 704, 192, 106, 28, 352, 138, 742, 39, 301, 38, 781, 950, 862, 597, 596, 58, 631, 138, 904, 1133, 407

Offset: 1

Antti Karttunen, May 21 2017

nonn

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

Cf. A000027, A001511, A033879, A033880, A286449, A286260, A286451, A286460, A286592, A286593.

(define (A286595 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A286449 n)) 2) (- (A001511 n)) (- (* 3 (A286449 n))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A286449(n))^2) - A001511(n) - 3*A286449(n)).

A286259 Compound filter: a(n) = P(A001511(n), A049820(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 1, 6, 4, 5, 11, 25, 16, 23, 37, 31, 56, 57, 56, 110, 106, 80, 137, 123, 137, 173, 211, 175, 232, 255, 254, 279, 352, 255, 407, 471, 407, 467, 466, 409, 596, 597, 596, 599, 742, 597, 821, 783, 742, 905, 991, 866, 1036, 992, 1082, 1131, 1276, 1083, 1276, 1279, 1379, 1487, 1597, 1228, 1712, 1713, 1597, 1960, 1831, 1713, 2081, 2019, 2081, 1955, 2347, 1957

Offset: 1

Antti Karttunen, May 07 2017

nonn

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

Cf. A000027, A001511, A049820, A286161, A286162, A286260, A286034.

A001511(n) = (1+valuation(n,2));
A049820(n) = (n-numdiv(n));
A286259(n) = (2 + ((A001511(n)+A049820(n))^2) - A001511(n) - 3*A049820(n))/2;
for(n=1, 10000, write("b286259.txt", n, " ", A286259(n)));

from sympy import divisor_count as d
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), n - d(n)) # Indranil Ghosh, May 07 2017

(define (A286259 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A049820 n)) 2) (- (A001511 n)) (- (* 3 (A049820 n))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A049820(n))^2) - A001511(n) - 3*A049820(n)).

cp's OEIS Frontend

A286460 Compound filter (2-adic valuation & sum of the divisors): a(n) = P(A001511(n), A000203(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286034 Compound filter: a(n) = P(A046523(n), A161942(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286358 Compound filter: a(n) = P(A286357(n), A161942(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286451 Compound filter (2-adic valuation of sigma(n) & 2-adic valuation of n): a(n) = P(A286357(n), A001511(n)), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = 0 by an explicit convention.

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A286595 Compound filter (2-adic valuation & deficiency/abundance): a(n) = P(A001511(n), A286449(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286259 Compound filter: a(n) = P(A001511(n), A049820(n)), where P(n,k) is sequence A000027 used as a pairing function.

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