A305891 -id:A305891 - cp's OEIS frontend

A322022 Lexicographically earliest such sequence a that a(i) = a(j) => A305891(i) = A305891(j) and A319697(i) = A319697(j), for all i, j.

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

Offset: 1

Antti Karttunen, Nov 29 2018

nonn

Restricted growth sequence transform of the ordered pair [A305891(n), A319697(n)], or equally, of the triple [A007814(n), A046523(n), A319697(n)].

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

Cf. A007814, A046523, A097272, A305891, A319697, A319698, A322021.

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; };
A007814(n) = valuation(n,2);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A319697(n) = sumdiv(n, d, (!(d%2))*issquarefree(d)*d);
v322022 = rgs_transform(vector(up_to, n, [A007814(n), A046523(n), A319697(n)]));
A322022(n) = v322022[n];

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.

Original entry on oeis.org

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

A157226 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the sides of the unit cell of the parent lattice of index n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 4, 1, 3, 2, 2, 1, 3, 1, 4, 2, 3, 1, 4, 1, 3, 1, 4, 1, 6, 1, 2, 2, 3, 2, 4, 1, 3, 2, 4, 1, 6, 1, 4, 2, 3, 1, 4, 1, 3, 2, 4, 1, 3, 2, 4, 2, 3, 1, 8, 1, 3, 2, 2, 2, 6, 1, 4, 2, 6, 1, 4, 1, 3, 2, 4, 2, 6, 1, 4, 1, 3, 1, 8, 2, 3, 2

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

Andrey Zabolotskiy's new formula confirms that a(n) indeed is a function of A305891(n). - Antti Karttunen, Oct 01 2018

Andrey Zabolotskiy, Table of n, a(n) for n = 1..5000
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 5.]

Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157230, A157231, A304182, A007875, A029744.

A007875(n) = eulerphi(2^omega(n));
A157226(n) = if(n<=2,n-1,(A007875(n) + if(!(n%2),A007875(n/2)))); \\ Antti Karttunen, Oct 01 2018

From Andrey Zabolotskiy, Sep 30 2018: (Start)
Let b(n) = A007875(n) for n>1, b(1) = 0. Then
a(n) = b(n) for odd n,
a(n) = b(n) + b(n/2) for even n.
Thus the sorted list of all terms (except for a(1)=0) is A029744. (End)

New name and more terms from Andrey Zabolotskiy, May 09 2018

A305893 Filter sequence combining 3-adic valuation (A007949) and the prime signature (A046523) of n.

Original entry on oeis.org

1, 2, 3, 4, 2, 5, 2, 6, 7, 8, 2, 9, 2, 8, 5, 10, 2, 11, 2, 12, 5, 8, 2, 13, 4, 8, 14, 12, 2, 15, 2, 16, 5, 8, 8, 17, 2, 8, 5, 18, 2, 15, 2, 12, 11, 8, 2, 19, 4, 12, 5, 12, 2, 20, 8, 18, 5, 8, 2, 21, 2, 8, 11, 22, 8, 15, 2, 12, 5, 23, 2, 24, 2, 8, 9, 12, 8, 15, 2, 25, 26, 8, 2, 21, 8, 8, 5, 18, 2, 27, 8, 12, 5, 8, 8, 28, 2, 12, 11, 29, 2, 15, 2, 18, 15

Offset: 1

Antti Karttunen, Jun 14 2018

nonn

Restricted growth sequence transform of A286463, of the ordered pair [A007949(n), A046523(n)].

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

Cf. A007949, A046523, A286463, A305900.

Cf. also A305891.

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; };
A007949(n) = valuation(n,3);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux305893(n) = [A007949(n), A046523(n)];
v305893 = rgs_transform(vector(up_to,n,Aux305893(n)));
A305893(n) = v305893[n];

A318500 Filter sequence combining A305897 and the parity of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 24 2018

nonn

Restricted growth sequence transform of ordered pair [A000035(n), A246277(n)], or equally, of ordered pair [A007814(n), A246277(n)].
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j) => A305891(i) = A305891(j) => A291761(i) = A291761(j).

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

Cf. A246277, A291761, A305801, A305891, A305897.

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; };
A246277(n) = if(1==n, 0, 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)/2);
v318500 = rgs_transform(vector(up_to,n,[(n%2),A246277(n)]));
A318500(n) = v318500[n];

A318888 Filter sequence combining the 2-adic valuation of n (A007814) with the differences between odd primes in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 24 2018

nonn

Restricted growth sequence transform of an ordered pair [A007814(n), A318885(A000265(n))].

For all i, j: A305801(i) = A305801(j) => a(i) = a(j) => A305891(i) = A305891(j).

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

Cf. A305801, A305891, A318500, A318885, A318887.

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; };
A000265(n) = (n/2^valuation(n, 2));
A007814(n) = valuation(n,2);
A318885(n) = if(1==n,n,my(f=factor(n),m=2^f[1,2],i=1); for(k=2,#f~,i += (f[k,1]-f[k-1,1]); m *= prime(i)^f[k,2]); (m));
v318888 = rgs_transform(vector(up_to,n,[A007814(n), A318885(A000265(n))]));
A318888(n) = v318888[n];

A319347 Filter sequence combining A000035(n) (parity of n), A003557(n), and A046523(n) (prime signature of n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 24 2018

nonn

Restricted growth sequence transform of triple [A000035(n), A003557(n), A046523(n)], or equally, of triple [A007814(n), A003557(n), A046523(n)], or equally, of ordered pair [A000035(n), A291757(n)].

For all i, j: A305801(i) = A305801(j) => a(i) = a(j) => A305891(i) = A305891(j).

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

Cf. A000035, A003557, A007814, A046523, A291757, A305801, A305891.

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; };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p=0); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v319347 = rgs_transform(vector(up_to,n,[A003557(n),(n%2),A046523(n)]));
A319347(n) = v319347[n];

A365392 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = [A336158(n), A364255(n), A365425(n)].

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 04 2023

Restricted growth sequence transform of triplet [A336158(n), A364255(n), A365425(n)].
For all i, j >= 1:
  a(i) = a(j) => A286531(i) = A286531(j),
  a(i) = a(j) => A305891(i) = A305891(j),
  a(i) = a(j) => A365391(i) = A365391(j),
  a(i) = a(j) => A365421(i) = A365421(j).

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

Cf. A000265, A046523, A163511, A286531, A305891, A336158, A364255, A365425, A365391, A365421.

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; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A364255(n) = gcd(n, A163511(n));
A365392aux(n) = [A364255(n), A046523(A000265(n)), A046523(A000265(A163511(n)))];
v365392 = rgs_transform(vector(up_to,n,A365392aux(n)));
A365392(n) = v365392[n];

A358230 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j), A007949(i) = A007949(j) and A046523(i) = A046523(j), for all i, j, where A007814 and A007949 give the 2-adic and 3-adic valuation, and A046523 gives the prime signature of its argument.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 01 2022

nonn

Restricted growth sequence transform of the triple [A007814(n), A007949(n), A046523(n)].
For all i, j:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A305891(i) = A305891(j),
  a(i) = a(j) => A305893(i) = A305893(j),
  a(i) = a(j) => A322026(i) = A322026(j) => A072078(i) = A072078(j),
  a(i) = a(j) => A065333(i) = A065333(j).

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

Cf. A007814, A007949, A046523, A065333, A072078, A305891, A305893, A322026.

Cf. also A358747.

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; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
v358230 = rgs_transform(vector(up_to, n, [A007814(n), A007949(n), A046523(n)]));
A358230(n) = v358230[n];

A379002 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A112765(i) = A112765(j), for all i, j, where A046523 gives the least representative of the prime signature of n and A112765 gives the 5-adic valuation of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 15 2024

nonn

Restricted growth sequence transform of ordered pair [A046523(n), A112765(n)].
For all i, j:
  A379001(i) = A379001(j) => a(i) = a(j).

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

Cf. A046523, A112765, A379001.

Cf. also A305891, A305893.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v379002 = rgs_transform(vector(up_to, n, [A046523(n), valuation(n,5)]));
A379002(n) = v379002[n];

cp's OEIS Frontend

A322022 Lexicographically earliest such sequence a that a(i) = a(j) => A305891(i) = A305891(j) and A319697(i) = A319697(j), for all i, j.

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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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A157226 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the sides of the unit cell of the parent lattice of index n.

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A305893 Filter sequence combining 3-adic valuation (A007949) and the prime signature (A046523) of n.

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A318500 Filter sequence combining A305897 and the parity of n.

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A318888 Filter sequence combining the 2-adic valuation of n (A007814) with the differences between odd primes in the prime factorization of n.

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A319347 Filter sequence combining A000035(n) (parity of n), A003557(n), and A046523(n) (prime signature of n).

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A365392 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = [A336158(n), A364255(n), A365425(n)].

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A358230 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j), A007949(i) = A007949(j) and A046523(i) = A046523(j), for all i, j, where A007814 and A007949 give the 2-adic and 3-adic valuation, and A046523 gives the prime signature of its argument.

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