A278222 -id:A278222 - cp's OEIS frontend

A351963 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A109812(i)) = A278222(A109812(j)), for all i, j >= 1.

1, 1, 1, 2, 1, 3, 3, 1, 2, 3, 3, 2, 3, 4, 1, 4, 2, 3, 3, 5, 3, 5, 3, 6, 3, 5, 2, 7, 1, 5, 6, 3, 5, 5, 3, 5, 6, 3, 8, 2, 9, 3, 5, 4, 5, 5, 3, 8, 5, 1, 8, 5, 3, 7, 5, 3, 10, 6, 3, 6, 6, 5, 6, 5, 6, 5, 5, 4, 8, 3, 9, 5, 5, 7, 2, 11, 3, 10, 3, 10, 6, 6, 9, 5, 8, 6, 5, 8, 5, 10, 6, 12, 6, 10, 6, 5, 10, 4, 6, 8, 5, 13

Offset: 1

Antti Karttunen, Apr 06 2022

Restricted growth sequence transform of A278222(A109812(n)), or equally of, A278222(A351965(n)).

For all i, j: A351578(i) = A351578(j) => a(i) = a(j) => A352884(i) = A352884(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to binary expansion of n

Cf. A005940, A046523, A109812, A278222, A286622, A351965, A352884.

Cf. also A351578, A352888.

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; };
v109812 = readvec("b109812_to10e5.txt"); \\ Prepared from b-file data with gawk ' { print $2 } '
up_to = #v109812;
A109812(n) = v109812[n];
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
v351963 = rgs_transform(vector(up_to, n, A046523(A005940(1+A109812(n)))));
A351963(n) = v351963[n];

A278262 a(n) = A278222(A179016(n)).

Original entry on oeis.org

2, 4, 2, 8, 2, 12, 16, 2, 12, 24, 12, 32, 2, 12, 24, 30, 24, 12, 60, 24, 64, 2, 12, 24, 30, 24, 30, 210, 60, 48, 12, 60, 12, 180, 8, 24, 16, 128, 2, 12, 24, 30, 24, 30, 210, 60, 48, 30, 210, 30, 420, 12, 60, 24, 96, 12, 60, 12, 180, 60, 360, 360, 24, 216, 72, 16, 432, 256, 2, 12, 24, 30, 24, 30, 210, 60, 48, 30, 210, 30, 420, 12, 60, 24, 96, 30, 210, 30, 420

Offset: 1

Antti Karttunen, Nov 22 2016

Antti Karttunen, Table of n, a(n) for n = 1..16405
Index entries for sequences related to binary expansion of n

Cf. A179016, A278222.

Cf. also A213712, A278232.

(define (A278262 n) (A278222 (A179016 n)))

a(n) = A278222(A179016(n)).

A286242 Compound filter: a(n) = P(A278222(n), A278219(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 12, 14, 12, 84, 40, 44, 12, 142, 216, 183, 40, 265, 86, 152, 12, 142, 826, 265, 216, 1860, 607, 489, 40, 832, 607, 1117, 86, 619, 226, 560, 12, 142, 826, 265, 826, 5080, 2497, 619, 216, 2956, 4308, 4155, 607, 8575, 1105, 1533, 40, 832, 2497, 2116, 607, 5731, 4501, 3475, 86, 1402, 1105, 3475, 226, 1759, 698, 2144, 12, 142, 826, 265, 826, 5080, 2497, 619, 826

Offset: 0

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
MathWorld, Pairing Function

Cf. A000027, A003188, A278219, A278222, A286240, A286241, A286242.

A003188(n) = bitxor(n, n>>1);
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));
A278219(n) = A278222(A003188(n));
A286242(n) = (2 + ((A278222(n)+A278219(n))^2) - A278222(n) - 3*A278219(n))/2;
for(n=0, 16383, write("b286242.txt", n, " ", A286242(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 a003188(n): return n^(n>>1)
def a243353(n): return a005940(1 + a003188(n))
def a278219(n): return a046523(a243353(n))
def a278222(n): return a046523(a005940(n + 1))
def a(n): return T(a278222(n), a278219(n)) # Indranil Ghosh, May 07 2017

(define (A286242 n) (* (/ 1 2) (+ (expt (+ (A278222 n) (A278219 n)) 2) (- (A278222 n)) (- (* 3 (A278219 n))) 2)))

a(n) = (1/2)*(2 + ((A278222(n)+A278219(n))^2) - A278222(n) - 3*A278219(n)).

a(n) = (1/2)*(2 + ((A278222(n)+A278222(A003188(n)))^2) - A278222(n) - 3*A278222(A003188(n))).

A286374 a(n) = A278222(n^2).

Original entry on oeis.org

1, 2, 2, 6, 2, 12, 6, 12, 2, 30, 12, 48, 6, 210, 12, 24, 2, 30, 30, 420, 12, 360, 48, 30, 6, 120, 210, 1260, 12, 420, 24, 48, 2, 30, 30, 420, 30, 4620, 420, 480, 12, 420, 360, 1080, 48, 960, 30, 210, 6, 420, 120, 2310, 210, 3360, 1260, 1680, 12, 1260, 420, 6300, 24, 840, 48, 96, 2, 30, 30, 420, 30, 4620, 420, 2520, 30, 4620, 4620, 6720, 420, 9240, 480, 180

Offset: 0

Antti Karttunen, May 09 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A000290, A278222, A286243, A286375, A286376, A286377.

Cf. A159918 (one of the matched sequences).

from sympy import prime, factorint
import math
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 a(n): return a278222(n**2) # Indranil Ghosh, May 09 2017

Scheme
```
(define (A286374 n) (A278222 (* n n)))
```

a(n) = A278222(A000290(n)) = A278222(n^2).

A286375 a(n) = A278222(n^3).

Original entry on oeis.org

1, 2, 2, 36, 2, 96, 36, 840, 2, 1260, 96, 1260, 36, 2310, 840, 5040, 2, 1260, 1260, 69300, 96, 4620, 1260, 4320, 36, 1680, 2310, 12600, 840, 36960, 5040, 30240, 2, 1260, 1260, 831600, 1260, 415800, 69300, 2646000, 96, 27720, 4620, 900900, 1260, 221760, 4320, 277200, 36, 83160, 1680, 13860, 2310, 180180, 12600, 166320, 840, 15315300, 36960, 13860, 5040

Offset: 0

Antti Karttunen, May 09 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A000578, A278222, A286243, A286375, A286376.

Cf. A192085 (one of the matched sequences).

(define (A286375 n) (A278222 (* n n n)))

a(n) = A278222(A000578(n)) = A278222(n^3).

A286376 a(n) = A278222(A277699(n)).

Original entry on oeis.org

2, 2, 6, 2, 24, 6, 12, 2, 60, 24, 180, 6, 120, 12, 24, 2, 60, 60, 420, 24, 180, 180, 60, 6, 192, 120, 384, 12, 240, 24, 48, 2, 60, 60, 420, 60, 720, 420, 360, 24, 420, 180, 1260, 180, 4620, 60, 420, 6, 720, 192, 5040, 120, 5040, 384, 1680, 12, 1440, 240, 4320, 24, 480, 48, 96, 2, 60, 60, 420, 60, 2520, 420, 1260, 60, 420, 720, 1260, 420, 1260, 360, 180, 24

Offset: 1

Antti Karttunen, May 09 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A277699, A278222, A278239, A286243, A286374, A286375.

(define (A286376 n) (A278222 (A277699 n)))

a(n) = A278222(A277699(n)).

A286464 Compound filter (3-adic valuation & signature for base-2 1-runs): a(n) = P(A051064(n), A278222(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 2, 12, 2, 16, 12, 29, 2, 31, 16, 67, 12, 67, 29, 138, 2, 16, 31, 67, 16, 467, 67, 277, 12, 67, 67, 745, 29, 277, 138, 497, 2, 23, 16, 67, 31, 436, 67, 302, 16, 436, 467, 1771, 67, 1894, 277, 1129, 12, 67, 67, 668, 67, 1771, 745, 2557, 29, 302, 277, 2557, 138, 1129, 497, 2148, 2, 16, 23, 67, 16, 467, 67, 277, 31, 436, 436, 1832, 67, 1771, 302, 1129, 16, 566

Offset: 1

Antti Karttunen, May 10 2017

nonn

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

Cf. A000027, A051064, A278222, A286462, A286463.

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
A051064(n) = if(n<1, 0, 1+valuation(n, 3));
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));
A286464(n) = (1/2)*(2 + ((A051064(n)+A278222(n))^2) - A051064(n) - 3*A278222(n))
for(n=1, 10000, write("b286464.txt", n, " ", A286464(n)));

(define (A286464 n) (* (/ 1 2) (+ (expt (+ (A051064 n) (A278222 n)) 2) (- (A051064 n)) (- (* 3 (A278222 n))) 2)))

a(n) = (1/2)*(2 + ((A051064(n)+A278222(n))^2) - A051064(n) - 3*A278222(n)).

A286538 a(n) = A278222(A267111(n)).

Original entry on oeis.org

2, 2, 4, 2, 6, 8, 4, 2, 6, 12, 16, 6, 12, 8, 4, 2, 6, 12, 24, 32, 6, 30, 36, 12, 24, 16, 6, 12, 12, 8, 4, 2, 6, 12, 24, 48, 64, 6, 30, 60, 72, 12, 60, 72, 24, 48, 32, 6, 30, 36, 30, 60, 36, 12, 24, 24, 16, 6, 12, 12, 12, 8, 4, 2, 6, 12, 24, 48, 96, 128, 6, 30, 60, 120, 144, 12, 60, 180, 216, 24, 120, 144, 48, 96, 64, 6

Offset: 1

Antti Karttunen, May 17 2017

nonn

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

Cf. A267111, A278222, A286536, A286539 (rgs-version of this sequence).

(define (A286538 n) (A278222 (A267111 n)))

a(n) = A278222(A267111(n)).

A286552 Ordinal transform of A286622, or equally, of A278222.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 21 2017

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for sequences related to binary expansion of n

Cf. A263017, A278222, A286622, A286554.

A286601 a(n) = A278222(A193231(n)).

Original entry on oeis.org

1, 2, 4, 2, 6, 2, 4, 8, 16, 8, 4, 12, 6, 12, 6, 2, 6, 2, 6, 12, 6, 30, 24, 12, 16, 32, 24, 8, 36, 12, 4, 12, 36, 12, 4, 12, 36, 72, 60, 12, 16, 48, 64, 32, 24, 8, 24, 72, 6, 12, 6, 2, 24, 12, 6, 30, 60, 12, 24, 48, 6, 30, 60, 30, 210, 30, 60, 120, 6, 30, 60, 30, 60, 180, 60, 12, 96, 48, 24, 120, 6, 30, 24, 12, 6, 2, 6, 12, 60, 30, 6, 30

Offset: 0

Antti Karttunen, Jun 04 2017

Antti Karttunen, Table of n, a(n) for n = 0..8191
Indranil Ghosh, Python program to generate the sequence
Index entries for sequences related to binary expansion of n

Cf. A193231, A234022, A278222, A278231, A278233, A286602 (rgs-version of this sequence).

(define (A286601 n) (A278222 (A193231 n)))

a(n) = A278222(A193231(n)).

cp's OEIS Frontend

A351963 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A109812(i)) = A278222(A109812(j)), for all i, j >= 1.

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A278262 a(n) = A278222(A179016(n)).

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A286242 Compound filter: a(n) = P(A278222(n), A278219(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286374 a(n) = A278222(n^2).

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A286375 a(n) = A278222(n^3).

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A286376 a(n) = A278222(A277699(n)).

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A286464 Compound filter (3-adic valuation & signature for base-2 1-runs): a(n) = P(A051064(n), A278222(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286538 a(n) = A278222(A267111(n)).

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A286552 Ordinal transform of A286622, or equally, of A278222.

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