A103391 -id:A103391 - cp's OEIS frontend

A331743 Lexicographically earliest infinite sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A323901(i) = A323901(j) for all i, j.

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

Offset: 0

Antti Karttunen, Feb 05 2020

nonn

Restricted growth sequence transform of the ordered pair [A002487(n), A002487(A163511(n))].
For all i, j:
  a(i) = a(j) => A331748(i) = A331748(j),
  a(i) = a(j) => A331749(i) = A331749(j).

Cf. A002487, A163511, A323901, A324400, A331742, A331744, A331748, A331749.
Differs from A331745 for the first time at n=77, where a(77) = 40, while A331745(77) = 24.
Differs from A103391(1+n) for the first time at n=191, where a(191) = 23, while A103391(192) = 97.

\\ 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; };
Aux331743(n) = [A002487(n), A323901(n)];
v331743 = rgs_transform(vector(1+up_to, n, Aux331743(n-1)));
A331743(n) = v331743[1+n];

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

A365718 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365717(i) = A365717(j) for all i, j >= 0, where A365717(n) = A348717(A356867(1+n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 17 2023

Restricted growth sequence transform of A365717.
For all i, j >= 0: a(i) = a(j) => A365720(i) = A365720(j).
In contrast to austere A103391, which is easily computed from n's binary expansion, the scatter plot here with its slender seaweed-like branchings suggests that this sequence is not just a simple derivation of base-3 expansion of n.

Antti Karttunen, Table of n, a(n) for n = 0..59049

Cf. A348717, A356867, A365720.

Cf. also A103391 (similar transformation applied to A005940) and A365715 (compare the scatter plot).

up_to = 59049; \\ = 3^10.
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; };
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
A356867list(up_to) = { my(v=vector(up_to),met=Map(),h=0,ak); for(i=1,#v,if(1==vecsum(digits(i,3)), v[i] = i; h = i, ak = v[i-h]; forprime(p=2,,if(3!=p && !mapisdefined(met,p*ak), v[i] = p*ak; break))); mapput(met,v[i],i)); (v); };
v365718 = rgs_transform(apply(A348717,A356867list(1+up_to)));
A365718(n) = v365718[1+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).

A351454 Lexicographically earliest infinite sequence such that a(i) = a(j) => A006530(i) = A006530(j), A329697(i) = A329697(j) and A331410(i) = A331410(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 11 2022

nonn

Restricted growth sequence transform of the triplet [A006530(n), A329697(n), A331410(n)], or equally, of the ordered pair [A006530(n), A335880(n)].

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

			a(99) = a(121) because 99 = 3^2 * 11 and 121 = 11^2, so they have equal largest prime factor (A006530), and they also agree on A329697(99) = A329697(121) = 4 and on A331410(99) = A331410(121) = 4, therefore they get equal value (which is 51) allotted to them by the restricted growth sequence transform. - _Antti Karttunen_, Feb 14 2022

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

Cf. A006530, A329697, A331410, A335880.
Cf. also A324400, A336936, A351453.
Differs from A351452 for the first time at n=49, where a(49) = 26, while A351452(49) = 19.
Differs from A351460 for the first time at n=121, where a(121) = 51, while A351460(121) = 62.
Differs from A103391(1+n) for the first time after n=1 at n=121, where a(121) = 51, while A103391(122) = 62.

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]);
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A331410(n) = if(!bitand(n,n-1),0,1+A331410(n+(n/vecmax(factor(n)[, 1]))));
Aux351454(n) = [A006530(n), A329697(n), A331410(n)];
v351454 = rgs_transform(vector(up_to, n, Aux351454(n)));
A351454(n) = v351454[n];

A103383 Primes in A103373.

Original entry on oeis.org

2, 3, 5, 7, 17, 31, 71, 157, 227, 293, 349, 419, 557, 1663, 2269, 2609, 3547, 3943, 15761, 17477, 37243, 70481, 105557, 23913779, 84394837, 7057254647, 3915885721591, 4641244746324673, 5266511621347511, 565552908731370799

Offset: 1

Jonathan Vos Post, Feb 03 2005

Intersection of A103373 with A000040.

			105557 is an element of this sequence because A103373(95) = 105557.

Cf. A000040, A103373, A103382-A103391, A103393.

k = 5; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 350], PrimeQ]]

Edited and extended by Ray Chandler and Robert G. Wilson v, Feb 06 2005

A110962 Fractalization of A025480, zero-based version of Kimberling's paraphrases sequence.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 1, 0, 3, 0, 0, 0, 4, 2, 2, 1, 5, 1, 1, 0, 6, 3, 3, 0, 7, 0, 0, 0, 8, 4, 4, 2, 9, 2, 2, 1, 10, 5, 5, 1, 11, 1, 1, 0, 12, 6, 6, 3, 13, 3, 3, 0, 14, 7, 7, 0, 15, 0, 0, 0, 16, 8, 8, 4, 17, 4, 4, 2, 18, 9, 9, 2, 19, 2, 2, 1, 20, 10, 10, 5, 21, 5, 5, 1, 22, 11, 11, 1, 23, 1, 1, 0, 24, 12, 12

Offset: 0

Alexandre Wajnberg, Sep 26 2005

Self-descriptive sequence: the terms at odd indices are the sequence itself, while the terms at even indices (the skeleton of this sequence) are the terms of A025480, which is a zero-based sequence of Kimberling's paraphrases sequence, A003602.

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

Cf. A000265, A003602, A025480, A089309, A103391, A110812, A110779, A110766.

One less than A110963 (note also the different starting offsets).

A025480(n) = (n>>valuation(n*2+2, 2));
A110962(n) = if(!(n%2), A025480(n/2), A110962((n-1)/2)); \\ Antti Karttunen, Apr 18 2022

a(n) = n++; n>>=valuation(n, 2); n>>valuation(2*n+2, 2); \\ Ruud H.G. van Tol, Jun 23 2024

For even n, a(n) = A025480(n/2), for odd n, a(n) = a((n-1)/2). - Antti Karttunen, Apr 18 2022
a(2n+1) = a(4n+3) = a(n).
a(2n) = a(4n+1) = a(4n+2) = A025480(n/2).
a(4n) = a(8n+1) = a(8n+2) = n.
a(n) = A110963(1+n) - 1.

Entry edited and more terms added by Antti Karttunen, Apr 18 2022

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).

A103384 Primes in A103374.

Original entry on oeis.org

2, 3, 5, 7, 17, 31, 71, 127, 157, 227, 293, 349, 419, 521, 1873, 3877, 4889, 6607, 12289, 19843, 31873, 42703, 80309, 137957, 884987, 976091, 9979037, 15614983, 29738231, 45024979, 812280583, 6882259301, 8479913153, 88930242859, 356874733421

Offset: 1

Jonathan Vos Post, Feb 03 2005

Intersection of A103374 with A000040.

			1873 is an element of this sequence because A103374(74) = 1873.

Cf. A000040, A103374, A103382-A103391, A103394.

k = 6; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 260], PrimeQ]]

Edited, corrected and extended by Ray Chandler and Robert G. Wilson v, Feb 06 2005

A103385 Primes in A103375.

Original entry on oeis.org

2, 3, 5, 7, 17, 31, 71, 127, 157, 227, 257, 293, 349, 419, 503, 1993, 7907, 26293, 29311, 34603, 67477, 3147311, 9159547, 973669469, 6797534657, 9627183689, 297222052181, 4530692779838851, 41748646469705167, 266359428042546661

Offset: 1

Jonathan Vos Post, Feb 03 2005

Intersection of A103375 with A000040.

			17 is an element of this sequence because A103375(36) = 17.

Cf. A000040, A103375, A103382-A103391, A103395.

k = 7; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 475], PrimeQ]]

Edited, corrected and extended by Ray Chandler and Robert G. Wilson v, Feb 06 2005

A103386 Primes in A103376.

Original entry on oeis.org

2, 3, 5, 7, 17, 31, 71, 127, 157, 227, 257, 293, 349, 419, 503, 65657, 152833, 172297, 11229341, 12584983, 26532901, 31220807, 1164893671, 1217349652999, 2346608054761, 8116583338373, 53091879496979, 9758833144565411

Offset: 1

Jonathan Vos Post, Feb 05 2005

Intersection of A103376 with A000040.

			7 is in this sequence because A103376(27) = 7, which is prime.

Cf. A000040, A103376, A103382-A103391, A103396.

k = 8; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 500], PrimeQ]]

Edited and extended by Ray Chandler, Feb 10 2005

cp's OEIS Frontend

A331743 Lexicographically earliest infinite sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A323901(i) = A323901(j) for all i, j.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A365718 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365717(i) = A365717(j) for all i, j >= 0, where A365717(n) = A348717(A356867(1+n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Scheme

Formula

A351454 Lexicographically earliest infinite sequence such that a(i) = a(j) => A006530(i) = A006530(j), A329697(i) = A329697(j) and A331410(i) = A331410(j) for all i, j >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A103383 Primes in A103373.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A110962 Fractalization of A025480, zero-based version of Kimberling's paraphrases sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Scheme

Formula

A103384 Primes in A103374.

Views

Author

Keywords

Comments

Examples