A286378 -id:A286378 - 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");

A323889 Lexicographically earliest positive sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A278222(i) = A278222(j), for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 09 2019

Restricted growth sequence transform of the ordered pair [A002487(n), A278222(n)].

Cf. A002487, A278222, A286622, A324400.

Cf. also A103391, A278243, A286378, A318311, A323892, A323897 and A324533 for a "deformed variant".

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
A278222(n) = A046523(A005940(1+n));
Aux323889(n) = [A002487(n), A278222(n)];
v323889 = rgs_transform(vector(1+up_to,n,Aux323889(n-1)));
A323889(n) = v323889[1+n];

a(2^n) = 2 for all n >= 0.

A317943 Filter sequence constructed from the coefficients of the Stern polynomials B(d,t) collected for each proper divisor d of n; Restricted growth sequence transform of A317942.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 57, 58, 2, 59, 60, 61, 2, 62, 63, 64, 65, 66, 2, 67, 68, 69, 70, 71, 72, 73, 2, 74, 75, 76, 2, 77, 2, 78, 79, 80, 2, 81, 2, 82, 83, 84, 2, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 86

Offset: 1

Antti Karttunen, Aug 12 2018

nonn

For all i, j: a(i) = a(j) => A317837(i) = A317837(j).

			Proper divisors of 115 are 1, 5 and 23 and proper divisors of 125 are 1, 5 and 25. The divisors 1 and 5 occur in both, while for the Stern polynomials B(23,t) and B(25,t) (see A125184) the nonzero coefficients are {1, 2, 3, 1} and {1, 3, 2, 1}, that is, they are equal as multisets, thus A286378(23) = A286378(25). From this follows that a(115) = a(125).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A125184, A286378, A317837, A317942, A317945.
Cf. also A293217, A305793.
Differs from A305800 and A296073 for the first time at n=125, where a(125) = 86.

\\ Needs also code from A286378:
up_to = 65537;
A317942(n) = { my(m=1); fordiv(n,d,if(dA286378(d)-1))); (m); };
v317943 = rgs_transform(vector(up_to, n, A317942(n)));
A317943(n) = v317943[n];

A317945 Filter sequence constructed from the coefficients of the Stern polynomials B(d,t) collected for each divisor d of n. Restricted growth sequence transform of A317944.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 12 2018

nonn

Differs from A000027(n) = n (positive integers) from a(193) = 191 on.
For all i, j: a(i) = a(j) => A317838(i) = A317838(j).
There are certain prime pairs p, q for which the Stern polynomials B(p,t) and B(q,t) (see table A125184) have equal multisets of nonzero coefficients. For example, for primes 191 and 193 these coefficients are {1, 2, 2, 2, 2, 3, 1} and {1, 2, 2, 2, 3, 2, 1} (from which follows that A278243(191) = A278243(193), A286378(191) = A286378(193) and thus => a(191) = a(193) => A002487(191) = A002487(193) as well). Other such prime pairs currently known are {419, 461}, {2083, 2143} and {11777, 12799}. Whenever a(p) = a(q) for such a prime pair, then also a(2^k * p) = a(2^k * q) for all k >= 0. It would be nice to know whether there could exist any other cases of a(i) = a(j), i != j, but for example both i and j being odd semiprimes?

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A002487, A125184, A260443, A286378, A317943, A317944.

Cf. also A305795, A317838.

\\ Needs also code from A286378.
up_to = 65537;
A317944(n) = { my(m=1); fordiv(n,d, if(d>1, m *= prime(A286378(d)-1))); (m); };
v317945 = rgs_transform(vector(up_to, n, A317944(n)));
A317945(n) = v317945[n];

A286377 a(n) = A278243(n^2).

Original entry on oeis.org

1, 2, 2, 60, 2, 2520, 60, 138600, 2, 87318000, 2520, 189189000, 60, 792148896000000, 138600, 70756686000, 2, 2288271225240000, 87318000, 944154902157667200000000, 2520, 20388496616888400000000, 189189000, 127170673342713000000, 60, 701323506627727183200000000, 792148896000000, 21149759041410320377056000000000000000, 138600

Offset: 0

Antti Karttunen, May 09 2017

nonn

Observation: the restricted growth sequence computed for this sequence seems to give A103391 (apart from the fact that the latter uses starting offset 1 instead of 0. Checked up to n=2048). If this holds, then A103391 works as a more practical filtering sequence (than this sequence, with its huge terms) matching for example to sequences like A286387. Compare also to A286378.

Antti Karttunen, Table of n, a(n) for n = 0..256
Index entries for sequences related to Stern's sequences

Cf. A000290, A103391, A278243, A286374, A286378, A286387.

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
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2))));
A278243(n) = A046523(A260443(n));
A286377(n) = A278243(n*n);
for(n=0, 256, write("b286377.txt", n, " ", A286377(n)));

Scheme
```
(define (A286377 n) (A278243 (* n n)))
```

a(n) = A278243(A000290(n)) = A278243(n^2).

A286387 a(n) = A002487(n^2).

Original entry on oeis.org

0, 1, 1, 4, 1, 7, 4, 9, 1, 14, 7, 13, 4, 29, 9, 16, 1, 23, 14, 43, 7, 36, 13, 29, 4, 43, 29, 64, 9, 67, 16, 25, 1, 34, 23, 89, 14, 115, 43, 46, 7, 85, 36, 79, 13, 46, 29, 79, 4, 97, 43, 142, 29, 89, 64, 91, 9, 136, 67, 157, 16, 121, 25, 36, 1, 47, 34, 151, 23, 236, 89, 157, 14, 211, 115, 104, 43, 225, 46, 109, 7, 162, 85, 235, 36, 139, 79, 174, 13, 101, 46

Offset: 0

Antti Karttunen, May 09 2017

nonn

Conjecture: For all i >= 0, j >= 0: A103391(1+i) = A103391(1+j) => a(i) = a(j). This would be an implication of observation made at A286377, which has been checked up to n=2048. See also A286378.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to Stern's sequences

Cf. A000290, A002487, A103391, A286377, A286378.

from functools import reduce
def A286387(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n**2)[-1:2:-1],(1,0))) if n else 0 # Chai Wah Wu, May 18 2023

Scheme
```
(define (A286387 n) (A002487 (* n n)))
```

a(n) = A002487(A000290(n)) = A002487(n^2).

A331742 Lexicographically earliest infinite sequence such that a(i) = a(j) => A323901(i) = A323901(j) for all i, j.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 05 2020

Restricted growth sequence transform of function A323901(n) = A002487(A163511(n)).

Cf. A002487, A163511, A323901.

Cf. also A286378, A305899, A331743, A331744, A331745.

\\ Needs also code from A323901.
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; };
v331742 = rgs_transform(vector(1+up_to, n, A323901(n-1)));
A331742(n) = v331742[1+n];

a(2^n) = 1 for all n >= 0.

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

A323889 Lexicographically earliest positive sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A278222(i) = A278222(j), for all i, j >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A317943 Filter sequence constructed from the coefficients of the Stern polynomials B(d,t) collected for each proper divisor d of n; Restricted growth sequence transform of A317942.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A317945 Filter sequence constructed from the coefficients of the Stern polynomials B(d,t) collected for each divisor d of n. Restricted growth sequence transform of A317944.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A286377 a(n) = A278243(n^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Scheme

Formula

A286387 a(n) = A002487(n^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Scheme

Formula

A331742 Lexicographically earliest infinite sequence such that a(i) = a(j) => A323901(i) = A323901(j) for all i, j.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula