A304101 -id:A304101 - cp's OEIS frontend

A304102 a(n) = Product_{d|n, dA304101(d)-1).

1, 2, 2, 4, 2, 8, 2, 12, 4, 8, 2, 120, 2, 12, 8, 24, 2, 200, 2, 120, 12, 12, 2, 1680, 4, 8, 20, 180, 2, 2000, 2, 120, 12, 44, 12, 12600, 2, 44, 8, 1680, 2, 1200, 2, 180, 200, 20, 2, 42000, 6, 440, 44, 120, 2, 7800, 12, 3960, 44, 12, 2, 3234000, 2, 44, 120, 840, 8, 10200, 2, 264, 20, 3000, 2, 630000, 2, 20, 440, 1452, 18, 2000, 2, 109200, 260, 44, 2, 1386000

Offset: 1

Antti Karttunen, May 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4181

Cf. A304101, A304103 (restricted growth sequence transform of this sequence), A304104.

\\ Needs also code from A304101:
A304102(n) = { my(m=1); fordiv(n,d,if(dA304101(d)-1))); (m); };

a(n) = Product_{d|n, dA000040(A304101(d)-1).
a(n) = 2*A304104(n) / A000040(A304101(n)-1).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A293435(n).
A007949(a(n)) = A304095(n).

A304104 a(n) = Product_{d|n, d>1} prime(A304101(d)-1).

Original entry on oeis.org

1, 2, 2, 6, 2, 20, 3, 12, 10, 20, 3, 420, 2, 30, 20, 60, 11, 300, 11, 420, 12, 30, 5, 4200, 22, 20, 130, 990, 3, 11000, 11, 420, 102, 44, 30, 31500, 5, 242, 20, 10920, 11, 3000, 13, 1170, 1100, 190, 3, 231000, 33, 2420, 506, 420, 19, 66300, 12, 9900, 110, 30, 11, 8085000, 13, 242, 300, 5460, 52, 56100, 19, 660, 130, 19500, 13, 9135000, 11, 290, 4180, 2178, 99

Offset: 1

Antti Karttunen, May 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4181

Cf. A304101, A304102, A304105 (restricted growth sequence transform of this sequence).

\\ Needs also code from A304101:
A304104(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(A304101(d)-1))); (m); };

a(n) = Product_{d|n, d>1} A000040(A304101(d)-1).
a(n) = (1/2) * A304102(n) * A000040(A304101(n)-1).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A001511(a(n)) = A005086(n).
A007949(a(n)) = A304096(n).

A286622 Restricted growth sequence computed for filter-sequence A278222, related to 1-runs in the binary representation of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 11 2017

When filtering sequences (by equivalence class partitioning), this sequence can be used instead of A278222, because for all i, j it holds that: a(i) = a(j) <=> A278222(i) = A278222(j).
For example, for all i, j: a(i) = a(j) => A000120(i) = A000120(j), and for all i, j: a(i) = a(j) => A001316(i) = A001316(j).
The sequence allots a distinct value for each distinct multiset formed from the lengths of 1-runs in the binary representation of n. See the examples. - Antti Karttunen, Jun 04 2017

			For n = 0, there are no 1-runs, thus the multiset is empty [], and it is allotted the number 1, thus a(0) = 1.
For n = 1, in binary also "1", there is one 1-run of length 1, thus the multiset is [1], which has not been encountered before, and a new number is allotted for that, thus a(1) = 2.
For n = 2, in binary "10", there is one 1-run of length 1, thus the multiset is [1], which was already encountered at n=1, thus a(2) = a(1) = 2.
For n = 3, in binary "11", there is one 1-run of length 2, thus the multiset is [2], which has not been encountered before, and a new number is allotted for that, thus a(3) = 3.
For n = 4, in binary "100", there is one 1-run of length 1, thus the multiset is [1], which was already encountered at n=1 for the first time, thus a(4) = a(1) = 2.
For n = 5, in binary "101", there are two 1-runs, both of length 1, thus the multiset is [1,1], which has not been encountered before, and a new number is allotted for that, thus a(5) = 4.

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

Cf. A278222, A000120, A001316.
Cf. A286552 (ordinal transform).
Cf. also A101296, A286581, A286589, A286597, A286599, A286600, A286602, A286603, A286605, A286610, A286619, A286621, A286626, A286378, A304101 for similarly constructed or related sequences.
Cf. also A305793, A305795.

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 }; \\ 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));
v286622 = rgs_transform(vector(1+65537, n, A278222(n-1)));
A286622(n) = v286622[1+n];

Example section added by Antti Karttunen, Jun 04 2017

A048679 Compressed fibbinary numbers (A003714), with rewrite 0->0, 01->1 applied to their binary expansion.

Original entry on oeis.org

0, 1, 2, 4, 3, 8, 5, 6, 16, 9, 10, 12, 7, 32, 17, 18, 20, 11, 24, 13, 14, 64, 33, 34, 36, 19, 40, 21, 22, 48, 25, 26, 28, 15, 128, 65, 66, 68, 35, 72, 37, 38, 80, 41, 42, 44, 23, 96, 49, 50, 52, 27, 56, 29, 30, 256, 129, 130, 132, 67, 136, 69, 70, 144, 73, 74, 76, 39, 160, 81

Offset: 0

Antti Karttunen

Permutation of the nonnegative integers (A001477); inverse permutation of A048680 i.e. A048679[ A048680[ n ] ] = n for all n.

Alois P. Heinz, Table of n, a(n) for n = 0..17710 (terms n = 10001..10945 from Antti Karttunen)
Index entries for sequences that are permutations of the natural numbers

Cf. A000045, A003714, A005203, A048678, A048680, A072650, A087808, A106151, A200714, A227351, A232559, A277006, A304100, A304101.

a(n) = rewrite_0to0_x1to1(fibbinary(j)) (where fibbinary(j) = A003714[ n ])
rewrite_0to0_x1to1 := proc(n) option remember; if(0 = n) then RETURN(n); else RETURN((2 * rewrite_0to0_x1to1(floor(n/(2^(1+(n mod 2)))))) + (n mod 2)); fi; end;
fastfib := n -> round((((sqrt(5)+1)/2)^n)/sqrt(5)); fibinv_appr := n -> floor(log[ (sqrt(5)+1)/2 ](sqrt(5)*n)); fibinv := n -> (fibinv_appr(n) + floor(n/fastfib(1+fibinv_appr(n)))); fibbinary := proc(n) option remember; if(n <= 2) then RETURN(n); else RETURN((2^(fibinv(n)-2))+fibbinary_seq(n-fastfib(fibinv(n)))); fi; end;
# second Maple program:
b:= proc(n) is(n=0) end:
a:= proc(n) option remember; local h; h:= iquo(a(n-1), 2)+1;
      while b(h) do h:= h*2 od; b(h):=true; h
    end: a(0):=0:
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 22 2014

b[n_] := n==0; a[n_] := a[n] = Module[{h}, h = Quotient[a[n-1], 2] + 1; While[b[h], h = h*2]; b[h] = True; h]; a[0]=0; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 27 2016, after Alois P. Heinz *)

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A007814(n) = valuation(n,2);
A000265(n) = (n/2^valuation(n, 2));
A106151(n) = if(n<=1,n,if(n%2,1+(2*A106151((n-1)/2)),(2^(A007814(n)-1))*A106151(A000265(n))));
A048679(n) = if(!n,n,A106151(2*A003714(n))); \\ Antti Karttunen, May 13 2018, after Reinhard Zumkeller's May 09 2005 formula.

from itertools import count, islice
def A048679_gen(): # generator of terms
    return map(lambda n: int(bin(n)[2:].replace('01','1'),2),filter(lambda n:not (n<<1)&n,count(0)))
A048679_list = list(islice(A048679_gen(),20)) # Chai Wah Wu, Mar 18 2024

def A048679(n):
    tlist, s = [1,2], 0
    while tlist[-1]+tlist[-2] <= n: tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        if d <= n:
            s += 1
            n -= d
        else:
            s <<= 1
    return s # Chai Wah Wu, Apr 24 2025

a(n) = A106151(2*A003714(n)) for n > 0. - Reinhard Zumkeller, May 09 2005
a(n+1) = min{([a(n)/2]+1)*2^k} such that it is not yet in the sequence. - Gerard Orriols, Jun 07 2014
a(n) = A072650(A003714(n)) = A003188(A227351(n)). - Antti Karttunen, May 13 2018

A304105 Restricted growth sequence transform of A304104, a filter sequence related to how the divisors of n are expressed in Fibonacci number system.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 5, 6, 7, 4, 5, 8, 2, 9, 4, 10, 11, 12, 11, 8, 6, 9, 13, 14, 15, 4, 16, 17, 5, 18, 11, 8, 19, 20, 9, 21, 13, 22, 4, 23, 11, 24, 25, 26, 27, 28, 5, 29, 30, 31, 32, 8, 33, 34, 6, 35, 36, 9, 11, 37, 25, 22, 12, 38, 39, 40, 33, 41, 16, 42, 25, 43, 11, 44, 45, 46, 47, 18, 11, 48, 49, 50, 51, 52, 53, 54, 19, 55, 2, 56, 9, 57, 22, 9, 58, 59, 13, 60

Offset: 1

Antti Karttunen, May 13 2018

nonn

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

Cf. A005086, A304096, A300837, A304101, A304103, A304104.

\\ Needs also code from A304101:
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])); }
A304104(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(A304101(d)-1))); (m); };
write_to_bfile(1,rgs_transform(vector(up_to,n,A304104(n))),"b304105.txt");

For all i, j: a(i) = a(j) => b(i) = b(j), where b can be any of {A000005, A005086, A304096 or A300837} for example.

A304103 Restricted growth sequence transform of A304102, a filter sequence related to the proper divisors of n expressed in Fibonacci number system.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 13 2018

nonn

For all i, j: a(i) = a(j) => b(i) = b(j), where b can be any of {A000005, A293435, A304095 or A300836} for example.

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

Cf. A293435, A304095, A300836, A304102.
Cf. also A300835, A304105, A305800.
Cf. A305793 (analogous filter for base 2).

\\ Needs also code from A304101.
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])); }
A304102(n) = { my(m=1); fordiv(n,d,if(dA304101(d)-1))); (m); };
write_to_bfile(1,rgs_transform(vector(up_to,n,A304102(n))),"b304103.txt");

A304100 a(n) = A003602(A048679(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 1, 5, 3, 2, 4, 1, 9, 5, 3, 6, 2, 7, 4, 1, 17, 9, 5, 10, 3, 11, 6, 2, 13, 7, 4, 8, 1, 33, 17, 9, 18, 5, 19, 10, 3, 21, 11, 6, 12, 2, 25, 13, 7, 14, 4, 15, 8, 1, 65, 33, 17, 34, 9, 35, 18, 5, 37, 19, 10, 20, 3, 41, 21, 11, 22, 6, 23, 12, 2, 49, 25, 13, 26, 7, 27, 14, 4, 29, 15, 8, 16, 1, 129, 65, 33, 66, 17, 67, 34, 9, 69, 35, 18, 36, 5, 73

Offset: 1

Antti Karttunen, May 13 2018

nonn

Positions of ones is given by the positive Fibonacci numbers: 1, 2, 3, 5, 8, 13, 21, ..., that is, A000045(n) from n >= 2 onward.
Positions of 2's is given by Lucas numbers larger than 3: 4, 7, 11, 18, ..., that is, A000032(n) from n >= 3 onward.
The restricted growth sequence transform of this sequence (almost certainly) is A003603.

Antti Karttunen, Table of n, a(n) for n = 1..17710

Cf. A000032, A000045, A003602, A003603, A035513, A048679, A304101.

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A007814(n) = valuation(n,2);
A000265(n) = (n/2^valuation(n, 2));
A106151(n) = if(n<=1,n,if(n%2,1+(2*A106151((n-1)/2)),(2^(A007814(n)-1))*A106151(A000265(n))));
A048679(n) = if(!n,n,A106151(2*A003714(n)));
A003602(n) = (1+A000265(n))/2;
A304100(n) = A003602(A048679(n));

a(n) = A003602(A048679(n)).

For all i, j: a(i) = a(j) => A304101(i) = A304101(j).

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];

A304738 Restricted growth sequence transform of A278222(A048673(n)).

Original entry on oeis.org

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

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 A048673(n). Compare to the scatter plot of A286622.

Cf. A003961, A048673, A278222, A286622.

Cf. also A286243, A304101, A304737.

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));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
v304738 = rgs_transform(vector(65539,n,A278222(A048673(n))));
A304738(n) = v304738[n];

A305820 Filter sequence for a(Fibonacci numbers > 1) = constant sequences.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 5, 2, 6, 7, 8, 9, 2, 10, 11, 12, 13, 14, 15, 16, 2, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 2, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 2, 82, 83, 84, 85, 86

Offset: 0

Antti Karttunen, Jun 16 2018

nonn

For all i, j:

a(i) = a(j) => A003603(i) = A003603(j) => A304101(i) = A304101(j) => A007895(i) = A007895(j).

Cf. A000045, A003603, A007895, A010056, A072649, A304101.

A010056(n) = { my(k=n^2); k+=(k+1)<<2; (issquare(k) || (n>0 && issquare(k-8))) }; \\ From A010056
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A305820(n) = if(n<=1, n, if(1==A010056(n),2,2+n-A072649(n)));

a(0)= 0, a(1) = 1, for n > 1, a(n) = 2 if n is a Fibonacci number > 1, otherwise a(n) = 2+n-A072649(n) = running count from 3 onward for non-Fibonacci numbers.

cp's OEIS Frontend

A304102 a(n) = Product_{d|n, dA304101(d)-1).

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A304104 a(n) = Product_{d|n, d>1} prime(A304101(d)-1).

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A286622 Restricted growth sequence computed for filter-sequence A278222, related to 1-runs in the binary representation of n.

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A048679 Compressed fibbinary numbers (A003714), with rewrite 0->0, 01->1 applied to their binary expansion.

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A304105 Restricted growth sequence transform of A304104, a filter sequence related to how the divisors of n are expressed in Fibonacci number system.

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A304103 Restricted growth sequence transform of A304102, a filter sequence related to the proper divisors of n expressed in Fibonacci number system.

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A304100 a(n) = A003602(A048679(n)).

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

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