A286251 -id:A286251 - cp's OEIS frontend

A286161 Compound filter: a(n) = T(A001511(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

1, 5, 2, 18, 2, 23, 2, 59, 7, 23, 2, 94, 2, 23, 16, 195, 2, 80, 2, 94, 16, 23, 2, 355, 7, 23, 29, 94, 2, 467, 2, 672, 16, 23, 16, 706, 2, 23, 16, 355, 2, 467, 2, 94, 67, 23, 2, 1331, 7, 80, 16, 94, 2, 302, 16, 355, 16, 23, 2, 1894, 2, 23, 67, 2422, 16, 467, 2, 94, 16, 467, 2, 2779, 2, 23, 67, 94, 16, 467, 2, 1331, 121, 23, 2, 1894, 16, 23, 16, 355, 2, 1832

Offset: 1

Antti Karttunen, May 04 2017

nonn

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

Cf. A000027, A001511, A046523, A286160, A286162, A286163, A286164, A286251.

A001511(n) = (1+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
A286161(n) = (2 + ((A001511(n)+A046523(n))^2) - A001511(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286161.txt", n, " ", A286161(n)));

from sympy import 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 a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), a046523(n)) # Indranil Ghosh, May 06 2017

(define (A286161 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A046523 n)) 2) (- (A001511 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A046523(n))^2) - A001511(n) - 3*A046523(n)).

A291761 Restricted growth sequence transform of ((-1)^n)*A046523(n); filter combining the parity and the prime signature of n.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 5, 3, 8, 3, 5, 9, 10, 3, 8, 3, 8, 9, 5, 3, 11, 7, 5, 12, 8, 3, 13, 3, 14, 9, 5, 9, 15, 3, 5, 9, 11, 3, 13, 3, 8, 16, 5, 3, 17, 7, 8, 9, 8, 3, 11, 9, 11, 9, 5, 3, 18, 3, 5, 16, 19, 9, 13, 3, 8, 9, 13, 3, 20, 3, 5, 16, 8, 9, 13, 3, 17, 21, 5, 3, 18, 9, 5, 9, 11, 3, 18, 9, 8, 9, 5, 9, 22, 3, 8, 16, 15, 3, 13, 3, 11, 23, 5, 3, 20, 3, 13

Offset: 1

Antti Karttunen, Sep 11 2017

nonn

Equally, restricted growth sequence transform of sequence b defined as b(1) = 1, and for n > 1, b(n) = A046523(n) + A000035(n), which starts as 1, 2, 3, 4, 3, 6, 3, 8, 5, 6, 3, 12, 3, 6, 7, 16, 3, 12, 3, 12, ...

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

Cf. A291767, A291768 (bisections), A147516.
Cf. A046523, A101296, A286161, A286251, A286367, A291762 (related or similar filtering sequences).
Cf. A065091 (positions of 3's), A100484 (of 4 and 5's), A001248 (of 4 and 7's), A046388 (of 9's), A030078 (of 6 and 12's).
Cf. A098108 (one of the matching sequences).

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])); }
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
write_to_bfile(1,rgs_transform(vector(65537,n,((-1)^n)*A046523(n))),"b291761.txt");
\\ Or alternatively:
f(n) = if(1==n,n,A046523(n)+(n%2));
write_to_bfile(1,rgs_transform(vector(16385,n,f(n))),"b291761.txt");

A286252 Compound filter: a(n) = P(A001511(1+n), A278222(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 2, 18, 2, 23, 7, 59, 2, 23, 16, 94, 7, 80, 29, 195, 2, 23, 16, 94, 16, 467, 67, 355, 7, 80, 67, 706, 29, 302, 121, 672, 2, 23, 16, 94, 16, 467, 67, 355, 16, 467, 436, 1894, 67, 1832, 277, 1331, 7, 80, 67, 706, 67, 1832, 631, 2779, 29, 302, 277, 2704, 121, 1178, 497, 2422, 2, 23, 16, 94, 16, 467, 67, 355, 16, 467, 436, 1894, 67, 1832, 277, 1331, 16, 467, 436

Offset: 0

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
MathWorld, Pairing Function

Cf. A000027, A001511, A278222, A286162, A286251, A286253, A286254.

A001511(n) = (1+valuation(n,2));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
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
A278222(n) = A046523(A005940(1+n));
A286252(n) = (2 + ((A001511(1+n)+A278222(n))^2) - A001511(1+n) - 3*A278222(n))/2;
for(n=0, 16384, write("b286252.txt", n, " ", A286252(n)));

from sympy import prime, factorint
import math
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
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 a278222(n): return a046523(a005940(n + 1))
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n + 1), a278222(n)) # Indranil Ghosh, May 07 2017

(define (A286252 n) (* (/ 1 2) (+ (expt (+ (A001511 (+ 1 n)) (A278222 n)) 2) (- (A001511 (+ 1 n))) (- (* 3 (A278222 n))) 2)))

a(n) = (1/2)*(2 + ((A001511(1+n)+A278222(n))^2) - A001511(1+n) - 3*A278222(n)).

A286253 Compound filter: a(n) = P(A055396(n), A001511(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 1, 8, 1, 9, 1, 25, 1, 5, 1, 26, 1, 27, 1, 17, 1, 35, 1, 53, 1, 5, 1, 75, 1, 9, 1, 8, 1, 65, 1, 131, 1, 5, 1, 13, 1, 90, 1, 12, 1, 104, 1, 134, 1, 5, 1, 186, 1, 14, 1, 8, 1, 152, 1, 18, 1, 5, 1, 188, 1, 189, 1, 30, 1, 9, 1, 229, 1, 5, 1, 273, 1, 252, 1, 8, 1, 14, 1, 347, 1, 5, 1, 323, 1, 9, 1, 12, 1, 324, 1, 19, 1, 5, 1, 31, 1, 350, 1, 8, 1, 377, 1, 462, 1, 5

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, A055396, A286164, A286251, A286252, A286254.

A001511(n) = (1+valuation(n,2));
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])); \\ This function from Charles R Greathouse IV, Apr 23 2015
A286253(n) = (2 + ((A055396(n)+A001511(1+n))^2) - A055396(n) - 3*A001511(1+n))/2;
for(n=1, 10000, write("b286253.txt", n, " ", A286253(n)));

from sympy import primepi, isprime, primefactors
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a055396(n), a001511(n + 1)) # Indranil Ghosh, May 07 2017

(define (A286253 n) (* (/ 1 2) (+ (expt (+ (A055396 n) (A001511 (+ 1 n))) 2) (- (A055396 n)) (- (* 3 (A001511 (+ 1 n)))) 2)))

a(n) = (1/2)*(2 + ((A055396(n)+A001511(1+n))^2) - A055396(n) - 3*A001511(1+n)).

A286254 Compound filter: a(n) = P(A001511(n), A055396(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 1, 13, 1, 12, 1, 14, 1, 17, 1, 31, 1, 5, 1, 60, 1, 38, 1, 9, 1, 47, 1, 19, 1, 5, 1, 69, 1, 68, 1, 27, 1, 8, 1, 94, 1, 5, 1, 124, 1, 107, 1, 9, 1, 122, 1, 33, 1, 5, 1, 156, 1, 8, 1, 14, 1, 155, 1, 193, 1, 5, 1, 43, 1, 192, 1, 9, 1, 212, 1, 280, 1, 5, 1, 18, 1, 255, 1, 20, 1, 278, 1, 13, 1, 5, 1, 355, 1, 12, 1, 9, 1, 8, 1, 441, 1, 5, 1, 381, 1, 380, 1, 14

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, A055396, A253790, A286161, A286251, A286252, A286253.

A001511(n) = (1+valuation(n,2));
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])); \\ This function from Charles R Greathouse IV, Apr 23 2015
A286254(n) = (2 + ((A001511(n)+A055396(1+n))^2) - A001511(n) - 3*A055396(1+n))/2;
for(n=1, 10000, write("b286254.txt", n, " ", A286254(n)));

from sympy import primepi, isprime, primefactors
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), a055396(n + 1)) # Indranil Ghosh, May 07 2017

(define (A286254 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A055396 (+ 1 n))) 2) (- (A001511 n)) (- (* 3 (A055396 (+ 1 n)))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A055396(1+n))^2) - A001511(n) - 3*A055396(1+n)).

A286362 Compound filter: a(n) = P(A089309(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 7, 2, 23, 9, 29, 7, 16, 5, 80, 2, 31, 40, 121, 2, 67, 5, 67, 16, 23, 9, 302, 7, 16, 38, 94, 2, 532, 20, 497, 16, 16, 23, 631, 2, 23, 31, 277, 2, 436, 5, 80, 67, 31, 14, 1178, 7, 67, 23, 67, 2, 302, 31, 328, 16, 16, 5, 1957, 2, 50, 142, 2017, 16, 436, 5, 67, 16, 467, 9, 2557, 2, 16, 80, 80, 16, 499, 14, 1129, 121, 16, 5, 1771, 16, 23, 31, 302, 2, 1771

Offset: 1

Antti Karttunen, May 08 2017

nonn

For all odd i, odd j: a(i) = a(j) <=> A286251(i) = A286251(j).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A046523, A089309, A286161, A286251, A286365.

(define (A286362 n) (* (/ 1 2) (+ (expt (+ (A089309 n) (A046523 n)) 2) (- (A089309 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A089309(n)+A046523(n))^2) - A089309(n) - 3*A046523(n)).

cp's OEIS Frontend

A286161 Compound filter: a(n) = T(A001511(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A291761 Restricted growth sequence transform of ((-1)^n)*A046523(n); filter combining the parity and the prime signature of n.

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A286252 Compound filter: a(n) = P(A001511(1+n), A278222(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286253 Compound filter: a(n) = P(A055396(n), A001511(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286254 Compound filter: a(n) = P(A001511(n), A055396(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

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PARI

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Formula

A286362 Compound filter: a(n) = P(A089309(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

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