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 1-4 of 4 results.

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

Original entry on oeis.org

1, 2, 4, 3, 8, 3, 16, 5, 3, 3, 32, 5, 64, 3, 18, 7, 128, 15, 256, 5, 18, 3, 512, 7, 3, 3, 5, 5, 1024, 15, 2048, 11, 18, 3, 18, 7, 4096, 3, 18, 7, 8192, 15, 16384, 5, 50, 3, 32768, 11, 3, 45, 18, 5, 65536, 7, 108, 7, 18, 3, 131072, 7, 262144, 3, 50, 13, 108, 15, 524288, 5, 18, 45, 1048576, 11, 2097152, 3, 15, 5, 18, 15, 4194304, 11, 7, 3
Offset: 1

Views

Author

Antti Karttunen, Jan 22 2020

Keywords

Links

Crossrefs

Cf. A122111, A241909, A241916, A331596 (number of distinct prime factors), A331597, A331598, A331599, A331600.
Cf. also A280489, A280491.

Programs

  • Mathematica
    Array[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[#]] &, 82] (* Michael De Vlieger, Jan 24 2020, after JungHwan Min at A122111 *)
  • PARI
    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));

Formula

a(n) = gcd(A122111(n), A241909(n)).
a(A241916(n)) = a(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)).

A331601 a(n) = A002487(A241909(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 3, 2, 8, 1, 7, 1, 14, 4, 3, 1, 4, 1, 11, 8, 22, 1, 9, 2, 64, 3, 43, 1, 18, 1, 5, 14, 110, 4, 9, 1, 162, 22, 47, 1, 34, 1, 127, 7, 440, 1, 13, 2, 12, 64, 191, 1, 8, 8, 97, 110, 1002, 1, 23, 1, 752, 11, 5, 14, 112, 1, 1249, 162, 16, 1, 17, 1, 610, 4, 897, 4, 220, 1, 111, 3, 4882, 1, 121, 22, 5494, 440, 281, 1, 26, 8, 7623, 1002
Offset: 1

Views

Author

Antti Karttunen, Jan 22 2020

Keywords

Links

Crossrefs

Cf. A002487, A241909, A331732.
Cf. also A323901, A323902, A323903, A331600.

Programs

  • PARI
    A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
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));
A331601(n) = A002487(A241909(n));

Formula

a(n) = A002487(A241909(n)).
a(n) = A002487(A331732(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];
Showing 1-4 of 4 results.

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