A286619 -id:A286619 - cp's OEIS frontend

A286605 Restricted growth sequence computed for number of divisors, d(n) (A000005).

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

Offset: 1

Antti Karttunen, May 11 2017

nonn

For all i, j: A101296(i) = A101296(j) => a(i) = a(j).

For all i, j: a(i) = a(j) <=> A000005(i) = A000005(j).

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

Cf. A000005, A007416 (positions of records, and also the first occurrence of each n).

Cf. also A101296, A286603, A286610, A286619, A286621, A286622, A286626, A286378.

With[{nn = 119}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Array[DivisorSigma[0, #] &, nn]] (* Michael De Vlieger, May 12 2017, Version 10 *)

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])); }
A000005(n) = numdiv(n);
write_to_bfile(1,rgs_transform(vector(10000,n,A000005(n))),"b286605.txt");

A286617 Restricted growth sequence of A278217 (prime-signature of A075159(1+n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 17 2017

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A075159, A278217, A286539, A286619, A286622.

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])); }
A005811(n) = hammingweight(bitxor(n, n>>1));  \\ This function from Gheorghe Coserea, Sep 03 2015
A286468(n) = { my(p=((n+1)%2), i=0, m=1); while(n>0, if(((n%2)==p), m *= prime(i), p = (n%2); i = i+1); n = n\2); m };
A075157(n) = if(!n,n,(prime(A005811(n))*A286468(n))-1);
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
A278217(n) = A046523(1+A075157(n));
write_to_bfile(0,rgs_transform(vector(65538,n,A278217(n-1))),"b286617.txt");

A286539 Restricted growth sequence of A286538.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 17 2017

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

Cf. A267111, A286537, A286538, A286617, A286619, A278222.

A324390 Lexicographically earliest positive sequence such that a(i) = a(j) => A278219(i) = A278219(j) and A324386(i) = A324386(j), for all i, j >= 0.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 5, 2, 6, 7, 8, 4, 5, 9, 10, 2, 3, 7, 11, 7, 8, 12, 13, 4, 14, 15, 16, 9, 17, 18, 14, 2, 6, 7, 19, 20, 11, 21, 22, 7, 23, 24, 25, 26, 27, 28, 27, 4, 14, 29, 30, 15, 31, 32, 33, 9, 34, 35, 36, 9, 37, 38, 37, 2, 39, 40, 11, 7, 41, 42, 43, 40, 41, 44, 45, 46, 47, 48, 47, 7, 49, 50, 51, 24, 52, 53, 54, 55, 56, 57, 58, 42, 59, 60, 56, 4, 61, 62, 63

Offset: 0

Antti Karttunen, Feb 27 2019

nonn

Restricted growth sequence transform of the ordered pair [A278219(n), A324386(n)].

Cf. A278219, A324386.

Cf. also A286619, A324343, A324344, A324380 (compare the scatter-plots).

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 }; \\ From 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));
A003188(n) = bitxor(n, n>>1);
A278219(n) = A278222(A003188(n));
Aux324390(n) = [A278219(n), A324386(n)]; \\ See code for A324386 in that entry.
v324390 = rgs_transform(vector(1+up_to,n,Aux324390(n-1)));
A324390(n) = v324390[1+n];

a(A000225(n)) = 2 for all n >= 1.

A286581 Restricted growth sequence transform of A286580.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 03 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65536

Cf. A233275, A286580, A286537, A286539, A286617, A286619, A286622.

A352888 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278219(A109812(i)) = A278219(A109812(j)), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 07 2022

Restricted growth sequence transform of A278219(A109812(n)).

For all i, j: a(i) = a(j) => A352889(i) = A352889(j).

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

Cf. A109812, A278219, A352889.

Cf. also A286619, A351578, A351963.

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;
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
A003188(n) = bitxor(n, n>>1);
A278219(n) = A046523(A005940(1+A003188(n)));
A109812(n) = v109812[n];
v352888 = rgs_transform(vector(up_to, n, A278219(A109812(n))));
A352888(n) = v352888[n];

A366262 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366261(i) = A366261(j) for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 05 2023

nonn

Restricted growth sequence transform of A366261.

For all i, j >= 0: a(i) = a(j) => A366254(i) = A366254(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A366254, A366260, A366261.

Cf. also A286602, A286619, A286622.

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]); };
A209229(n) = (n && !bitand(n,n-1));
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); };
A303767(n) = if(!n,n,if(A209229(n),n+A303767(n-1),A053644(n)+A303767(n-A053644(n)-1)));
A366260(n) = A005940(1+A303767(n));
A366261(n) = A046523(A366260(n));
v366262 = rgs_transform(vector(1+up_to,n,A366261(n-1)));
A366262(n) = v366262[1+n];

A304745 Restricted growth sequence transform of A046523(A207901(n)).

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 7, 4, 5, 8, 9, 3, 10, 3, 2, 3, 10, 11, 10, 12, 13, 8, 9, 4, 5, 8, 9, 3, 10, 3, 2, 4, 9, 8, 5, 14, 15, 14, 16, 17, 18, 19, 18, 8, 13, 12, 9, 4, 9, 8, 5, 14, 15, 14, 16, 20, 21, 22, 21, 6, 5, 4, 7, 4, 9, 8, 5, 14, 15, 14, 16, 17, 18, 19, 18, 8, 13, 12, 9, 12, 23, 24, 13, 25, 26, 25, 27, 17, 18, 19, 18, 8, 13, 12, 9, 3, 10, 11, 10, 12, 13, 8

Offset: 0

Antti Karttunen, May 27 2018

nonn

For all i, j: a(i) = a(j) => A005811(i) = A005811(j).

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

Cf. A046523, A207901.

Cf. also A286619 (A278219), A304744.

up_to_e = 17; \\ Good for computing up to n = (2^up_to_e)-1
v050376 = vector(up_to_e);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A003188(n) = bitxor(n, n>>1);
A207901(n) = A052330(A003188(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
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; };
v304745 = rgs_transform(vector(65538,n,A046523(A207901(n-1))));
A304745(n) = v304745[1+n];

A324533 Lexicographically earliest positive sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A278219(i) = A278219(j), for all i, j >= 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 6, 7, 3, 8, 9, 10, 6, 11, 12, 13, 3, 11, 14, 15, 9, 16, 17, 18, 6, 19, 17, 20, 12, 15, 21, 22, 3, 23, 24, 25, 14, 26, 27, 28, 9, 29, 30, 31, 17, 32, 33, 34, 6, 35, 27, 36, 17, 37, 38, 39, 12, 40, 33, 39, 21, 25, 41, 42, 3, 15, 43, 39, 24, 44, 45, 46, 14, 47, 48, 49, 27, 50, 51, 46, 9, 52, 48, 53, 30, 54, 55, 56, 17, 57, 58, 59, 33, 60, 61, 62, 6

Offset: 0

Antti Karttunen, Mar 05 2019

nonn

Restricted growth sequence transform of the ordered pair [A002487(n), A278219(n)].

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

Cf. A002487, A278219, A286619, A324400.

Cf. also A323889 (compare the scatterplots).

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; };
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
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
A003188(n) = bitxor(n, n>>1);
A278219(n) = A046523(A005940(1+A003188(n)));
Aux324533(n) = [A002487(n), A278219(n)];
v324533 = rgs_transform(vector(1+up_to,n,Aux324533(n-1)));
A324533(n) = v324533[1+n];

For n >= 1, a(2^n) = 3.

cp's OEIS Frontend

A286605 Restricted growth sequence computed for number of divisors, d(n) (A000005).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A286617 Restricted growth sequence of A278217 (prime-signature of A075159(1+n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A286539 Restricted growth sequence of A286538.

Views

Author

Keywords

Links

Crossrefs

A324390 Lexicographically earliest positive sequence such that a(i) = a(j) => A278219(i) = A278219(j) and A324386(i) = A324386(j), for all i, j >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A286581 Restricted growth sequence transform of A286580.

Views

Author

Keywords

Links

Crossrefs

A352888 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278219(A109812(i)) = A278219(A109812(j)), for all i, j >= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A366262 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366261(i) = A366261(j) for all i, j >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A304745 Restricted growth sequence transform of A046523(A207901(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A324533 Lexicographically earliest positive sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A278219(i) = A278219(j), for all i, j >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula