A278222 -id:A278222 - cp's OEIS frontend

A351452 Lexicographically earliest infinite sequence such that a(i) = a(j) => A006530(i) = A006530(j) and A278222(i) = A278222(j) for all i, j >= 1.

1, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 14, 8, 15, 5, 16, 9, 17, 2, 18, 10, 19, 6, 20, 11, 21, 4, 22, 12, 23, 7, 24, 13, 25, 3, 19, 14, 26, 8, 27, 15, 28, 5, 29, 16, 30, 9, 31, 17, 32, 2, 33, 18, 34, 10, 35, 19, 36, 6, 37, 20, 24, 11, 38, 21, 39, 4, 40, 22, 41, 12, 42, 23, 43, 7, 44, 24, 45, 13, 46, 25, 47, 3, 48, 19, 49, 14, 50, 26, 51, 8, 52

Offset: 1

Antti Karttunen, Feb 11 2022

nonn

Restricted growth sequence transform of the ordered pair [A006530(n), A278222(n)].

For all i, j >= 1: A324400(i) = A324400(j) => a(i) = a(j).

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

Cf. A006530, A278222, A286622.
Cf. also A324400, A336935, A351453, A351454.
Differs from A351454 and A351460 for the first time at n=49, where a(49) = 19, while A351454(49) = A351460(49) = 26.

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; };
A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
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
A278222(n) = A046523(A005940(1+n));
Aux351452(n) = [A006530(n), A278222(n)];
v351452 = rgs_transform(vector(up_to, n, Aux351452(n)));
A351452(n) = v351452[n];

A351578 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(f(i)) = A007814(f(j)) and A278222(f(i)) = A278222(f(j)), for all i, j >= 1, where f(k) = A109812(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 6, 7, 10, 6, 11, 12, 13, 14, 6, 15, 16, 15, 16, 17, 18, 7, 16, 19, 20, 21, 22, 18, 7, 16, 22, 6, 22, 18, 17, 23, 24, 25, 15, 16, 26, 16, 27, 28, 23, 29, 30, 23, 22, 6, 31, 16, 7, 32, 33, 15, 33, 18, 22, 18, 27, 33, 16, 22, 34, 23, 17, 25, 27, 16, 35, 36, 37, 38, 32, 28, 32, 39, 18, 40, 16

Offset: 1

Antti Karttunen, Apr 07 2022

Restricted growth sequence transform of the ordered pair [A007814(A109812(n)), A046523(A005940(1+A109812(n)))].
The sequence allots a new distinct number for each newly encountered combination of the 2-adic valuation of A109812 (A351964), and the multiset of the lengths of 1-runs in the odd part of A109812 (A351965). See the examples.
For all i, j: a(i) = a(j) => A352889(i) = A352889(j).

			   n   A109812(n)  [base-2]   A351964(n)           Lengths of       a(n)
                              (# of trailing 0's)  1-runs       (allotted #)
-----+----------------------------------------------------------------------
   1 :          1       [1],  0                    [1]               1
   2 :          2      [10],  1                    [1]               2
   3 :          4     [100],  2                    [1]               3
   4 :          3      [11],  0                    [2]               4
   5 :          8    [1000],  3                    [1]               5
   6 :          5     [101],  0                    [1,1]             6
   7 :         10    [1010],  1                    [1,1]             7
   8 :         16   [10000],  4                    [1]               8
   9 :          6     [110],  1                    [2]               9
  10 :          9    [1001],  0                    [1,1]             6
  11 :         18   [10010],  1                    [1,1]             7
Because the combinations of the multiset of 1-runs in the binary expansion of A109812(n) and the number of trailing zeros in it (A351964) are unique for n = 1 .. 9, a unique increasing number (starting from 1) is allotted for each, and a(n) = n for n <= 9. On the other hand, at n=10, the binary expansion is [1001], for which these two measures are equal to that of binary expansion [101] found first time at n=6, therefore the rgs-transform allots for 10 the same number as for 6, and a(10) = a(6) = 6. At n=11, the binary expansion is [10010], where these two measures coincide with that of [1010] found first time at n=7, therefore a(10) = a(7) = 7.

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

Cf. A007814, A109812, A278222 (A286622), A351963, A351964, A351965, A352888, A352889.

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); };
A007814(n) = valuation(n,2);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
v351578 = rgs_transform(vector(up_to, n, [A007814(A109812(n)), A046523(A005940(1+A109812(n)))]));
A351578(n) = v351578[n];

A286379 Compound filter ("discard the smallest prime factor" & "signature for 1-runs in base-2"): a(n) = P(A032742(n), A278222(n)), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = 1.

Original entry on oeis.org

1, 2, 7, 5, 16, 18, 29, 14, 31, 50, 67, 42, 67, 98, 195, 44, 16, 100, 67, 115, 637, 242, 277, 117, 125, 289, 955, 224, 277, 450, 497, 152, 131, 248, 160, 271, 436, 454, 643, 320, 436, 1246, 1771, 550, 2716, 1058, 1129, 375, 160, 655, 1343, 692, 1771, 1918, 3332, 623, 880, 1355, 2557, 1020, 1129, 1922, 3507, 560, 166, 736, 67, 775, 1349, 1070, 277, 856, 436

Offset: 1

Antti Karttunen, May 13 2017

nonn

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

Cf. A032742, A278222, A285729.

A032742(n) = if(1==n,n,n/vecmin(factor(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));
A286379(n) = if(1==n,n,(1/2)*(2 + ((A032742(n)+A278222(n))^2) - A032742(n) - 3*A278222(n)));
for(n=1, 16384, write("b286379.txt", n, " ", A286379(n)));

from sympy import factorint, divisors
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 a(n): return 1 if n==1 else T(divisors(n)[-2], a278222(n)) # Indranil Ghosh, May 13 2017

(define (A286379 n) (if (= 1 n) n (* (/ 1 2) (+ (expt (+ (A032742 n) (A278222 n)) 2) (- (A032742 n)) (- (* 3 (A278222 n))) 2))))

a(1) = 1, for n > 1, a(n) = (1/2)*(2 + ((A032742(n)+A278222(n))^2) - A032742(n) - 3*A278222(n)).

A286536 a(n) = A278222(A276445(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 17 2017

nonn

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

Cf. A267111, A276445, A278219, A278222, A286538.

Cf. A286537 (rgs-version of this sequence).

(define (A286536 n) (A278219 (A267111 n)))
(define (A286536 n) (A278222 (A276445 n)))

a(n) = A278222(A276445(n)).

a(n) = A278219(A267111(n)).

A286596 a(n) = A278222(A153141(n)).

Original entry on oeis.org

1, 2, 4, 2, 8, 4, 2, 6, 16, 8, 4, 12, 2, 6, 6, 12, 32, 16, 8, 24, 4, 12, 12, 36, 2, 6, 6, 12, 6, 30, 12, 24, 64, 32, 16, 48, 8, 24, 24, 72, 4, 12, 12, 36, 12, 60, 36, 72, 2, 6, 6, 12, 6, 30, 12, 24, 6, 30, 30, 60, 12, 60, 24, 48, 128, 64, 32, 96, 16, 48, 48, 144, 8, 24, 24, 72, 24, 120, 72, 216, 4, 12, 12, 36, 12, 60, 36, 72, 12

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. A153141, A278222, A286597 (rgs-version of this sequence), A286598.

(define (A286596 n) (A278222 (A153141 n)))

a(n) = A278222(A153141(n)).

A286598 a(n) = A278222(A153142(n)).

Original entry on oeis.org

1, 2, 4, 2, 4, 8, 6, 2, 4, 12, 8, 16, 6, 12, 6, 2, 4, 12, 12, 36, 8, 24, 16, 32, 6, 30, 12, 24, 6, 12, 6, 2, 4, 12, 12, 36, 12, 60, 36, 72, 8, 24, 24, 72, 16, 48, 32, 64, 6, 30, 30, 60, 12, 60, 24, 48, 6, 30, 12, 24, 6, 12, 6, 2, 4, 12, 12, 36, 12, 60, 36, 72, 12, 60, 60, 180, 36, 180, 72, 144, 8, 24, 24, 72, 24, 120, 72, 216, 16, 48, 48

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. A153142, A278222, A286596, A286599 (rgs-version of this sequence).

(define (A286598 n) (A278222 (A153142 n)))

a(n) = A278222(A153142(n)).

A302795 Restricted growth sequence transform of A278222(A302793(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 26 2018

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A278222, A302793.

Cf. also A286602, A286622 (compare the scatter-plots).

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,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])); }
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ After code in A005940
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
A302793(n) = if(!n,n,A193231(1+A193231(n-1)));
write_to_bfile(0,rgs_transform(vector(65538,n,A278222(A302793(n-1)))),"b302795.txt");

A304097 Restricted growth sequence transform of A278222(A302853(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 17 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..60822

Cf. A278222, A302791, A282291, A302853, A304098.

Compare also the scatter-plot to that of A304535.

\\ Needs also code from A282291 and A302853:
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
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; };
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
A278222(n) = A046523(A005940(1+n));
write_to_bfile(0,rgs_transform(vector(60823,n,A278222(A302853(n-1)))),"b304097.txt");

A304737 Restricted growth sequence transform of A278222(A064413(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 18 2018

nonn

Sequence allots a distinct value for each distinct multiset formed from the lengths of 1-runs in the binary representation of A064413(n), the n-th term of EKG-sequence. Compare to the scatter plot of A286622.

Cf. A064301, A064413, A278222, A286622.

Cf. also A304738.

\\ Needs also code for A064413.
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; };
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
A278222(n) = A046523(A005940(1+n));
v304737 = rgs_transform(vector(65539,n,A278222(A064413(n))));
A304737(n) = v304737[n];

A318831 Restricted growth sequence transform of A278222(A000010(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 1, 1, 1, 2, 3, 1, 2, 3, 4, 1, 3, 2, 3, 2, 5, 1, 6, 1, 3, 1, 2, 2, 3, 3, 2, 1, 3, 2, 7, 3, 2, 4, 8, 1, 7, 3, 1, 2, 4, 3, 3, 2, 3, 5, 8, 1, 6, 6, 3, 1, 2, 3, 3, 1, 4, 2, 4, 2, 3, 3, 3, 3, 6, 2, 8, 1, 9, 3, 7, 2, 1, 7, 5, 3, 4, 2, 3, 4, 6, 8, 3, 1, 2, 7, 6, 3, 4, 1, 9, 2, 2

Offset: 1

Antti Karttunen, Sep 04 2018

nonn

Sequence allots a distinct value for each distinct multiset formed from the lengths of 1-runs in the binary expansion of A000010(n).

For all i, j: a(i) = a(j) => A295660(i) = A295660(j).

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

Cf. also A000010, A278222, A295660, A318835.

Compare also with the scatterplots of A286622, A304101 and A318832.

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; };
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
A278222(n) = A046523(A005940(1+n));
v318831 = rgs_transform(vector(up_to,n,A278222(eulerphi(n))));
A318831(n) = v318831[n];

cp's OEIS Frontend

A351452 Lexicographically earliest infinite sequence such that a(i) = a(j) => A006530(i) = A006530(j) and A278222(i) = A278222(j) for all i, j >= 1.

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A351578 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(f(i)) = A007814(f(j)) and A278222(f(i)) = A278222(f(j)), for all i, j >= 1, where f(k) = A109812(k).

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A286379 Compound filter ("discard the smallest prime factor" & "signature for 1-runs in base-2"): a(n) = P(A032742(n), A278222(n)), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = 1.

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Python

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A286536 a(n) = A278222(A276445(n)).

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A286596 a(n) = A278222(A153141(n)).

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A286598 a(n) = A278222(A153142(n)).

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A302795 Restricted growth sequence transform of A278222(A302793(n)).

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A304097 Restricted growth sequence transform of A278222(A302853(n)).

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A304737 Restricted growth sequence transform of A278222(A064413(n)).

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