A286359 -id:A286359 - cp's OEIS frontend

A286360 Compound filter (prime signature & sum of the divisors): a(n) = P(A046523(n), A000203(n)), where P(n,k) is sequence A000027 used as a pairing function.

1, 8, 12, 49, 23, 142, 38, 239, 124, 259, 80, 753, 107, 412, 412, 1051, 173, 1237, 212, 1390, 672, 826, 302, 3427, 565, 1087, 1089, 2223, 467, 5080, 530, 4403, 1384, 1717, 1384, 7911, 743, 2086, 1836, 6352, 905, 7780, 992, 4477, 3928, 2932, 1178, 14583, 1774, 5368, 2932, 5898, 1487, 10177, 2932, 10177, 3576, 4471, 1832, 25711, 1955, 5056, 6567, 18019, 3922

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, A286359, A286460.

Cf. A007503, A065608 (sequences matching to this filter), also A000203, A046523, A161942, A286034, A286357.

A000203(n) = sigma(n);
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
A286360(n) = (1/2)*(2 + ((A046523(n)+A000203(n))^2) - A046523(n) - 3*A000203(n));
for(n=1, 10000, write("b286360.txt", n, " ", A286360(n)));

from sympy import factorint, divisor_sigma as D
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 a(n): return T(a046523(n), D(n)) # Indranil Ghosh, May 12 2017

(define (A286360 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A000203 n)) 2) (- (A046523 n)) (- (* 3 (A000203 n))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A000203(n))^2) - A046523(n) - 3*A000203(n)).

A296089 Filter combining the sum of divisors (A000203) and 2-adic valuation of n (A007814); restricted growth sequence transform of A286460.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 15, 23, 24, 25, 26, 27, 28, 29, 21, 30, 31, 32, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 29, 31, 43, 44, 45, 46, 47, 48, 49, 46, 50, 51, 52, 53, 54, 55, 39, 56, 57, 58, 59, 60, 61, 62, 59, 46, 63, 64, 65, 66, 67, 62, 68, 51, 69, 70, 71, 58, 72, 73, 74, 75, 76, 77, 78, 79, 54, 80, 59, 75

Offset: 1

Antti Karttunen, Dec 07 2017

nonn

This seems to be also the restricted growth sequence transform of A286359.
For all i, j:
  a(i) = a(j) => A296088(i) = A296088(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A007814, A286359, A286460, A296088, A296090.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
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));
write_to_bfile(1,rgs_transform(vector(up_to,n,A286460(n))),"b296089.txt");

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)).

cp's OEIS Frontend

A286360 Compound filter (prime signature & sum of the divisors): a(n) = P(A046523(n), A000203(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A296089 Filter combining the sum of divisors (A000203) and 2-adic valuation of n (A007814); restricted growth sequence transform of A286460.

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PARI

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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Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Scheme

Formula