cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A331600 a(n) = A002487(A331595(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 4, 3, 1, 4, 1, 3, 4, 2, 1, 3, 2, 2, 3, 3, 1, 4, 1, 5, 4, 2, 4, 3, 1, 2, 4, 3, 1, 4, 1, 3, 7, 2, 1, 5, 2, 12, 4, 3, 1, 3, 8, 3, 4, 2, 1, 3, 1, 2, 7, 5, 8, 4, 1, 3, 4, 12, 1, 5, 1, 2, 4, 3, 4, 4, 1, 5, 3, 2, 1, 3, 8, 2, 4, 3, 1, 3, 8, 3, 4, 2, 8, 5, 1, 16, 7, 3, 1, 4, 1, 3, 18
Offset: 1

Views

Author

Antti Karttunen, Jan 22 2020

Keywords

Links

Crossrefs

Cf. A002487, A122111, A241909, A331595, A331731.
Cf. also A323901, A323902, A323903, A331601.

Programs

  • Mathematica
    Array[If[# == 1, 1, NestWhile[If[OddQ[#3], {#1, #1 + #2, #4}, {#1 + #2, #2, #4}] & @@ Append[#, Floor[#[[-1]]/2]] &, {1, 0, #}, #[[-1]] > 0 &][[2]] &@ Apply[GCD, {Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]}] &@ FactorInteger[#]] &, 105] (* Michael De Vlieger, Jan 25 2020, after JungHwan Min at A122111 *)
  • PARI
    A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
A331595(n) = gcd(A122111(n), A241909(n));
A331600(n) = A002487(A331595(n));

Formula

a(n) = A002487(A331595(n)) = A002487(gcd(A122111(n), A241909(n))).
a(n) = A002487(A331731(n)).

A331597 a(n) = A007947(A331595(n)).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 5, 3, 3, 2, 5, 2, 3, 6, 7, 2, 15, 2, 5, 6, 3, 2, 7, 3, 3, 5, 5, 2, 15, 2, 11, 6, 3, 6, 7, 2, 3, 6, 7, 2, 15, 2, 5, 10, 3, 2, 11, 3, 15, 6, 5, 2, 7, 6, 7, 6, 3, 2, 7, 2, 3, 10, 13, 6, 15, 2, 5, 6, 15, 2, 11, 2, 3, 15, 5, 6, 15, 2, 11, 7, 3, 2, 7, 6, 3, 6, 7, 2, 7, 6, 5, 6, 3, 6, 13, 2, 15, 10, 7, 2, 15, 2, 7, 30
Offset: 1

Views

Author

Antti Karttunen, Jan 22 2020

Keywords

Links

Crossrefs

Cf. A007947, A064989, A122111, A241909, A331595, A331596.

Programs

  • Mathematica
    Array[Times @@ FactorInteger[#][[All, 1]] &@ If[# == 1, 1, GCD @@ {Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]} &@ FactorInteger[#]] &, 105] (* Michael De Vlieger, Jan 24 2020, after JungHwan Min at A122111 *)
  • PARI
    A331597(n) = factorback(factorint(gcd(A122111(n), A241909(n)))[, 1]);

Formula

a(n) = A007947(A331595(n)) = A007947(gcd(A122111(n), A241909(n))).

A331731 Odd part of A331595(n), where A331595(n) = gcd(A122111(n), A241909(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 5, 1, 3, 9, 7, 1, 15, 1, 5, 9, 3, 1, 7, 3, 3, 5, 5, 1, 15, 1, 11, 9, 3, 9, 7, 1, 3, 9, 7, 1, 15, 1, 5, 25, 3, 1, 11, 3, 45, 9, 5, 1, 7, 27, 7, 9, 3, 1, 7, 1, 3, 25, 13, 27, 15, 1, 5, 9, 45, 1, 11, 1, 3, 15, 5, 9, 15, 1, 11, 7, 3, 1, 7, 27, 3, 9, 7, 1, 7, 27, 5, 9, 3, 27, 13, 1, 135, 25, 7, 1, 15, 1, 7, 75
Offset: 1

Views

Author

Antti Karttunen, Jan 25 2020

Keywords

Links

Crossrefs

Cf. A122111, A241909, A331595, A331600.
Cf. also A322865, A331732.

Programs

Formula

a(n) = A000265(A331595(n)).

A331730 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A331595(n) for all other n, except for odd primes p, f(p) = 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 3, 5, 4, 4, 3, 5, 3, 4, 6, 7, 3, 8, 3, 5, 6, 4, 3, 7, 4, 4, 5, 5, 3, 8, 3, 9, 6, 4, 6, 7, 3, 4, 6, 7, 3, 8, 3, 5, 10, 4, 3, 9, 4, 11, 6, 5, 3, 7, 12, 7, 6, 4, 3, 7, 3, 4, 10, 13, 12, 8, 3, 5, 6, 11, 3, 9, 3, 4, 8, 5, 6, 8, 3, 9, 7, 4, 3, 7, 12, 4, 6, 7, 3, 7, 12, 5, 6, 4, 12, 13, 3, 14, 10, 7, 3, 8, 3, 7, 15
Offset: 1

Views

Author

Antti Karttunen, Jan 25 2020

Keywords

Comments

For all i, j:
A305801(i) = A305801(j) => a(i) = a(j),
a(i) = a(j) => A331597(i) = A331597(j) => A331596(i) = A331596(j),
a(i) = a(j) => A331731(i) = A331731(j) => A331600(i) = A331600(j).

Links

Crossrefs

Cf. A122111, A241909, A305801, A331594, A331595, A331596, A331597, A331600, A331731.

Programs

  • PARI
    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; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
A331595(n) = gcd(A122111(n), A241909(n));
Aux331730(n) = if((n%2)&&isprime(n),0,A331595(n));
v331730 = rgs_transform(vector(up_to, n, Aux331730(n)));
A331730(n) = v331730[n];

A331598 a(n) = A122111(n) / gcd(A122111(n),A241909(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 2, 1, 8, 1, 1, 1, 1, 1, 4, 2, 16, 1, 2, 9, 32, 5, 8, 1, 2, 1, 1, 4, 64, 3, 3, 1, 128, 8, 4, 1, 4, 1, 16, 1, 256, 1, 2, 27, 1, 16, 32, 1, 5, 1, 8, 32, 512, 1, 6, 1, 1024, 2, 1, 2, 8, 1, 64, 64, 2, 1, 3, 1, 2048, 5, 128, 9, 16, 1, 4, 7, 4096, 1, 12, 4, 8192, 128, 16, 1, 10, 3, 256, 256, 16384, 8, 2, 1, 1, 4, 9, 1, 32, 1, 32, 1
Offset: 1

Views

Author

Antti Karttunen, Jan 22 2020

Keywords

Comments

It appears that these and the terms of A331599 have the same prime signatures, that is, A046523(a(n)) = A046523(A331599(n)) seems to hold for all n. However, the sequences are not equivalence-class-wise same: a(6) = a(12) = 2, whereas A331599(6) = 3 and A331599(12) = 5.

Links

Crossrefs

Cf. A122111, A241909, A241916, A331594, A331595, A331599.

Programs

  • Mathematica
    Array[If[# == 1, 1, #1/GCD[#1, #2] & @@ {Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]} &@ FactorInteger[#]] &, 90] (* Michael De Vlieger, Jan 25 2020, after JungHwan Min at A122111 *)
  • PARI
    A331598(n) = { my(u=A122111(n)); u/gcd(u, A241909(n)); };

Formula

a(n) = A122111(n)/A331598(n) = A122111(n) / gcd(A122111(n),A241909(n)).
a(n) = A331599(A241916(n)).

A331599 a(n) = A241909(n) / gcd(A122111(n),A241909(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 2, 9, 1, 5, 1, 27, 1, 1, 1, 1, 1, 25, 3, 81, 1, 7, 4, 243, 2, 125, 1, 5, 1, 1, 9, 729, 2, 5, 1, 2187, 27, 49, 1, 25, 1, 625, 1, 6561, 1, 11, 8, 1, 81, 3125, 1, 3, 1, 343, 243, 19683, 1, 35, 1, 59049, 5, 1, 3, 125, 1, 15625, 729, 5, 1, 7, 1, 177147, 2, 78125, 4, 625, 1, 121, 2, 531441, 1, 245, 9, 1594323, 2187, 2401, 1, 21
Offset: 1

Views

Author

Antti Karttunen, Jan 22 2020

Keywords

Comments

It appears that these and the terms of A331598 have the same prime signatures, that is, A046523(a(n)) = A046523(A331598(n)) seems to hold for all n.

Links

Crossrefs

Cf. A122111, A241909, A241916, A331595, A331598.

Programs

  • Mathematica
    Array[If[# == 1, 1, #2/GCD[#1, #2] & @@ {Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]} &@ FactorInteger[#]] &, 90] (* Michael De Vlieger, Jan 24 2020, after JungHwan Min at A122111 *)
  • PARI
    A331599(n) = { my(u=A241909(n)); u/gcd(A122111(n), u); };

Formula

a(n) = A241909(n) / A331595(n) = A241909(n) / gcd(A122111(n),A241909(n)).
a(n) = A331598(A241916(n)).
Showing 1-6 of 6 results.

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