A305795 -id:A305795 - cp's OEIS frontend

A305801 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n is an odd prime, with f(n) = n for all other n.

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

Offset: 1

Antti Karttunen, Jun 14 2018

nonn

The original name was: "Filter sequence for a(odd prime) = constant sequences", which stemmed from the fact that for all i, j, a(i) = a(j) => b(i) = b(j) for any sequence b that obtains a constant value for all odd primes A065091.
For example, we have for all i, j:
  a(i) = a(j) => A305800(i) = A305800(j),
  a(i) = a(j) => A007814(i) = A007814(j),
  a(i) = a(j) => A305891(i) = A305891(j) => A291761(i) = A291761(j).
There are several filter sequences "above" this one (meaning that they have finer equivalence class partitioning), for example, we have, for all i, j:
[where odd primes are further distinguished by]
  A305900(i) = A305900(j) => a(i) = a(j),   [whether p = 3 or > 3]
  A319350(i) = A319350(j) => a(i) = a(j),   [A007733(p)]
  A319704(i) = A319704(j) => a(i) = a(j),   [p mod 4]
  A319705(i) = A319705(j) => a(i) = a(j),   [A286622(p)]
  A331304(i) = A331304(j) => a(i) = a(j),   [parity of A000720(p)]
  A336855(i) = A336855(j) => a(i) = a(j).   [distance to the next larger prime]

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000720, A065091, A305800, A305900.
Cf. A305900, A319350, A319704, A319705, A331304, A336855 (sequences with finer equivalence class partitioning).
Cf. A305800, A305890, A305891, A305896, A318500, A318888, A319346, A319347, A319349, A319701, A322591, A322809, A322810, A323078, A323367, A323082, A323369, A323370, A323371, A323374, A323400, A324401, A326199, A326201, A326203, A326203, A328470, A329608, A331174, A331730, A331301 (sequences with coarser equivalence class partitioning).
Cf. also A003602, A103391, A295300, A305795, A324400, A331300, A336460 (for similar constructions or similarly useful sequences).

Array[If[# <= 2, #, If[PrimeQ[#], 3, 2 + # - PrimePi[#]]] &, 105] (* Michael De Vlieger, Oct 18 2021 *)

A305801(n) = if(n<=2,n,if(isprime(n),3,2+n-primepi(n)));

a(1) = 1, a(2) = 2; for n > 2, a(n) = 3 for odd primes, and a(n) = 2+n-A000720(n) for composite n.

For n > 2, a(n) = 1 + A305800(n).

Name changed and Comment section rewritten by Antti Karttunen, Oct 17 2021

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

A305793 Restricted growth sequence transform of A305792, a filter sequence constructed from binary expansions of the proper divisors of n.

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, 15, 10, 18, 2, 19, 2, 20, 21, 7, 22, 23, 2, 15, 21, 24, 2, 25, 2, 26, 27, 28, 2, 29, 30, 31, 10, 26, 2, 32, 33, 34, 21, 28, 2, 35, 2, 36, 37, 38, 33, 39, 2, 13, 40, 41, 2, 42, 2, 43, 44, 26, 45, 46, 2, 47, 48, 43, 2, 49, 50, 51, 40, 52, 2, 53, 45, 54, 55, 56, 33, 57, 2, 58, 59

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A278222, A286622, A305792, A305795.

Cf. also A293215, A300833, A304103, A305813.

\\ Needs also code from 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; };
A305792(n) = { my(m=1); fordiv(n,d,if(dA286622(d)-1))); (m); };
v305793 = rgs_transform(vector(up_to, n, A305792(n)));
A305793(n) = v305793[n];

For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j).
  a(i) = a(j) => A292257(i) = A292257(j).
  a(i) = a(j) => A305426(i) = A305426(j).
  a(i) = a(j) => A305435(i) = A305435(j).

A305815 Restricted growth sequence transform of A305814, a filter sequence constructed from the GF(2)[X]-factorization signatures of the divisors of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A278233, A305788, A305813, A305814.

Cf. also A305795.

\\ Needs also code from A305788:
up_to = 65537;
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; };
A305814(n) = { my(m=1); fordiv(n, d, if(d>1, m *= prime(A305788(d)-1))); (m); };
v305815 = rgs_transform(vector(up_to, n, A305814(n)));
A305815(n) = v305815[n];

For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j).
  a(i) = a(j) => A294883(i) = A294883(j).
  a(i) = a(j) => A294884(i) = A294884(j).

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];

A305794 a(n) = Product_{d|n, d>1} prime(A286622(d)-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 18, 7, 8, 15, 50, 11, 108, 11, 98, 195, 16, 5, 450, 11, 500, 357, 242, 19, 648, 55, 242, 345, 1372, 19, 76050, 29, 32, 165, 50, 385, 13500, 17, 242, 627, 5000, 17, 254898, 31, 5324, 30225, 722, 37, 3888, 77, 6050, 345, 5324, 31, 238050, 2255, 19208, 627, 722, 41, 29659500, 37, 1682, 76755, 64, 275, 54450, 11, 500, 969, 296450, 19, 405000, 17

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A008578, A278222, A286622, A305792, A305795 (rgs-transform).

Cf. also A304104.

A305794(n) = { my(m=1); fordiv(n, d, if(d>1, m *= prime(A286622(d)-1))); (m); }; \\ Needs also code from A286622.

a(n) = Product_{d|n, d>1} A008578(A286622(d)).

For all k >= 0, a(2^k) = 2^k.

A319705 Filter sequence which for primes p records a distinct value for each distinct multiset formed from the lengths of 1-runs in its binary representation [A286622(p)], and for all other numbers assigns a unique number.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 26 2018

nonn

Restricted growth sequence transform of function f defined as f(n) = A278222(n) when n is a prime, otherwise -n.
After its initial term 3, Fermat primes (A019434) gives the positions of 5 in this sequence, while the Mersenne primes (A000668) are each assigned to their own singleton equivalence class.
For all i, j:
  a(i) = a(j) => A305900(i) = A305900(j),
  a(i) = a(j) => A286622(i) = A286622(j),
  a(i) = a(j) => A305795(i) = A305795(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000668, A019434, A278222, A286163, A286622, A305795, A305900.

Cf. also A319704, A319706.

up_to = 100000;
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));
A319705aux(n) = if(isprime(n),A278222(n),-n);
v319705 = rgs_transform(vector(up_to,n,A319705aux(n)));
A319705(n) = v319705[n];

cp's OEIS Frontend

A305801 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n is an odd prime, with f(n) = n for all other n.

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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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A305793 Restricted growth sequence transform of A305792, a filter sequence constructed from binary expansions of the proper divisors of n.

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A305815 Restricted growth sequence transform of A305814, a filter sequence constructed from the GF(2)[X]-factorization signatures of the divisors of n.

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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.

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A305794 a(n) = Product_{d|n, d>1} prime(A286622(d)-1).

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A319705 Filter sequence which for primes p records a distinct value for each distinct multiset formed from the lengths of 1-runs in its binary representation [A286622(p)], and for all other numbers assigns a unique number.

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