A305801 -id:A305801 - cp's OEIS frontend

A319350 Filter sequence which records the number of cyclotomic cosets of 2 mod p for odd primes p, and for any other number assigns a unique number.

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

Offset: 1

Antti Karttunen, Sep 26 2018

nonn

Restricted growth sequence transform of function f defined as f(n) = A006694((n-1)/2) when n is an odd prime, otherwise -n.
For all i, j:
  a(i) = a(j) => A305801(i) = A305801(j),
  a(i) = a(j) => A319351(i) = A319351(j).

			a(3) = a(5) = a(11) = a(13) = a(19) = a(29) = a(37) because 3, 5, 11, 13, 19, 29, 37 are primes p for which A006694((p-1)/2) = 1 (are in A001122).
a(7) = a(17) = a(23) = a(41) = a(47) because 7, 17, 23, 41, 47 are primes p for which A006694((p-1)/2) = 2 (are in A115591).

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

Cf. A001917, A006694, A286573, A305801, A319351.
Cf. A001122 (positions of 3's), A115591 (positions of 6's).
Cf. also A319704, A319705, 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; };
A319350aux(n) = if((n<=2)||!isprime(n),n,-((n-1)/znorder(Mod(2, n))));
v319350 = rgs_transform(vector(up_to,n,A319350aux(n)));
A319350(n) = v319350[n];

A322810 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = 0 if n is an odd squarefree number > 1, and f(n) = n for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A305980(i) = A305980(j),
  a(i) = a(j) => A322808(i) = A322808(j).

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

Cf. A305801, A305980, A322808.

Cf. A056911 (after the initial 1, gives the positions of 3's in this sequence).

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; };
A322810aux(n) = if((n>1)&&(n%2)&&issquarefree(n),0,n);
v322810 = rgs_transform(vector(up_to,n,A322810aux(n)));
A322810(n) = v322810[n];

a(1) = 1, a(2) = 2, for n > 2, if n is an odd squarefree number (in A056911), a(n) = 3, otherwise a(n) = running count from 4 onward.

A323078 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = 0 if n is an odd prime, and f(n) = A097246(n) for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 03 2019

nonn

For all i, j: a(i) = a(j) => A322869(i) = A322869(j).

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

Cf. A097246, A305801, A322808, A322869.

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; };
A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A323078aux(n) = if((n>2)&&isprime(n),0,A097246(n));
v323078 = rgs_transform(vector(up_to,n,A323078aux(n)));
A323078(n) = v323078[n];

A323082 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -(n mod 2) if n is a prime, and f(n) = A300840(n) for any other number.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

For all i, j: A323074(i) = A323074(j) => a(i) = a(j).
Like the related A322822 also this filter sequence satisfies the following two implications, for all i, j >= 1:
  a(i) = a(j) => A322356(i) = A322356(j),
  a(i) = a(j) => A290105(i) = A290105(j).

Antti Karttunen, Table of n, a(n) for n = 1..65539

Cf. A052330, A300840, A305801, A322822, A323074.

Cf. also A290105, A322356.

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; };
ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A050376list(up_to) = { my(v=vector(up_to), i=0); for(n=1,oo,if(A302777(n), i++; v[i] = n); if(i == up_to,return(v))); };
v050376 = A050376list(up_to);
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&A302777(n/d), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A300840(n) = A052330(A052331(n)>>1);
A323082aux(n) = if(isprime(n),-(n%2),A300840(n));
v323082 = rgs_transform(vector(up_to,n,A323082aux(n)));
A323082(n) = v323082[n];

A326199 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(n) = 0 for odd primes.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 13 2019

nonn

For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A294877(i) = A294877(j).

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

Cf. A003557, A046523, A048250, A294877, A295300, A305801.

Differs from A323401 for the first time at n = 382 where a(382) = 253, while A323401(382) = 140.

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) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
Aux326199(n) = if((n>2)&&isprime(n),0,(1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n)));
v326199 = rgs_transform(vector(up_to,n,Aux326199(n)));
A326199(n) = v326199[n];

A326201 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n > 1 and gcd(n,A000005(n)) = 1, with f(n) = n for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 11 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A009191(i) = A009191(j) => A322980(i) = A322980(j).

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

Cf. A000005, A009191, A046642, A305801, A322980.

Cf. also A326202, A326203.

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; };
Aux326201(n) = if((n>1) && (1==gcd(n,numdiv(n))),0,n);
v326201 = rgs_transform(vector(up_to, n, Aux326201(n)));
A326201(n) = v326201[n];

A326202 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n > 2 and gcd(n,sigma(n)) = 1, with f(n) = n for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 11 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => A324401(i) = A324401(j) => a(i) = a(j),
  a(i) = a(j) => A009194(i) = A009194(j) => A325964(i) = A325964(j).

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

Cf. A000203, A009194, A014567, A305801, A324401, A325964.

Cf. also A326200, A326201, A326203.

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; };
Aux326202(n) = if((n>2) && (1==gcd(n,sigma(n))),0,n);
v326202 = rgs_transform(vector(up_to, n, Aux326202(n)));
A326202(n) = v326202[n];

A326203 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n > 2 and gcd(n,phi(n)) = 1, with f(n) = n for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 11 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A009195(i) = A009195(j) => A297086(i) = A297086(j),
  a(i) = a(j) => A000001(i) = A000001(j) => A297086(i) = A297086(j).

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

Cf. A000001, A000010, A003277, A009195, A297086, A305801.

Cf. also A326201, A326202.

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; };
Aux326203(n) = if((n>2) && (1==gcd(n,eulerphi(n))),0,n);
v326203 = rgs_transform(vector(up_to, n, Aux326203(n)));
A326203(n) = v326203[n];

A328470 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A053669(i) = A053669(j) for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 19 2019

nonn

Restricted growth sequence transform of A286142, or equally, of the ordered pair [A046523(n), A053669(n)], where A053669(n) gives the smallest prime not dividing n, while A046523(n) gives the prime signature of n.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j) => A291761(i) = A291761(j).

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

Cf. A046523, A053669, A286142, A291761, A305801, A328469.

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]); }; \\ From A046523
A053669(n) = forprime(p=2, , if(n%p, return(p))); \\ From A053669
Aux328470(n) = [A046523(n), A053669(n)];
v328470 = rgs_transform(vector(up_to, n, Aux328470(n)));
A328470(n) = v328470[n];

A331300 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = min(n, A057889(n)), and A057889 is a bijective base-2 reverse.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 18 2020

Restricted growth sequence transform of A331166. See comments in that sequence.

Antti Karttunen, Table of n, a(n) for n = 0..100000
Index entries for sequences related to binary expansion of n

Cf. A057889, A331166.

Cf. also A324400, A331303, A305801, A305801, A305900, A295300 for other "top level" filtering sequences.

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; };
A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
A331166(n) = min(n, A057889(n));
v331300 = rgs_transform(vector(1+up_to,n,A331166(n-1)));
A331300(n) = v331300[1+n];
for(n=0,up_to,write("b331300.txt", n, " ", A331300(n)));

cp's OEIS Frontend

A319350 Filter sequence which records the number of cyclotomic cosets of 2 mod p for odd primes p, and for any other number assigns a unique number.

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A322810 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = 0 if n is an odd squarefree number > 1, and f(n) = n for all other numbers.

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A323078 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = 0 if n is an odd prime, and f(n) = A097246(n) for all other numbers.

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A323082 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -(n mod 2) if n is a prime, and f(n) = A300840(n) for any other number.

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A326199 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(n) = 0 for odd primes.

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A326201 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n > 1 and gcd(n,A000005(n)) = 1, with f(n) = n for all other numbers.

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A326202 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n > 2 and gcd(n,sigma(n)) = 1, with f(n) = n for all other numbers.

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A326203 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n > 2 and gcd(n,phi(n)) = 1, with f(n) = n for all other numbers.

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A328470 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A053669(i) = A053669(j) for all i, j.

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