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 1-5 of 5 results.

A323901 a(n) = A002487(A163511(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 2, 3, 1, 8, 4, 7, 2, 4, 3, 3, 1, 14, 8, 11, 4, 18, 7, 9, 2, 12, 4, 9, 3, 8, 3, 5, 1, 22, 14, 43, 8, 34, 11, 47, 4, 16, 18, 23, 7, 26, 9, 13, 2, 16, 12, 23, 4, 18, 9, 17, 3, 6, 8, 11, 3, 6, 5, 5, 1, 64, 22, 127, 14, 112, 43, 97, 8, 84, 34, 121, 11, 26, 47, 111, 4, 66, 16, 89, 18, 40, 23, 57, 7, 36, 26, 57, 9, 50, 13, 29, 2, 50
Offset: 0

Views

Author

Antti Karttunen, Feb 09 2019

Keywords

Links

Crossrefs

Cf. A002487, A163511.
Cf. also A323902, A323903.

Programs

Formula

a(n) = A002487(A163511(n)).
a(2^n) = 1 for all n >= 0.

A323902 a(n) = A002487(A156552(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 3, 4, 1, 5, 1, 7, 4, 6, 1, 7, 2, 7, 3, 9, 1, 8, 1, 5, 5, 8, 3, 8, 1, 9, 6, 10, 1, 11, 1, 11, 5, 10, 1, 9, 2, 7, 7, 13, 1, 7, 4, 13, 8, 11, 1, 13, 1, 12, 7, 6, 5, 14, 1, 15, 9, 11, 1, 11, 1, 13, 5, 17, 3, 17, 1, 13, 4, 14, 1, 18, 6, 15, 10, 16, 1, 12, 4, 19, 11, 16, 7, 11, 1, 9, 9, 12, 1, 20, 1, 19, 8
Offset: 1

Views

Author

Antti Karttunen, Feb 09 2019

Keywords

Comments

Even though certain subset of terms of A156552 soon grow quite big, this sequence still has a quite moderate growth rate, thanks to the compensating effect of A002487.

Links

Crossrefs

Cf. A002487, A156552, A322993.
Cf. also A323240, A323244, A323901, A323903.

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
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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A323902(n) = A002487(A156552(n));

Formula

a(n) = A002487(A156552(n)) = A002487(A322993(n)).
a(p) = 1 for all primes p.

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)).

A324286 a(n) = A002487(A048675(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 4, 2, 1, 1, 3, 1, 2, 3, 5, 1, 3, 1, 6, 2, 3, 1, 3, 1, 3, 4, 7, 2, 2, 1, 8, 5, 3, 1, 5, 1, 4, 1, 9, 1, 2, 1, 4, 6, 5, 1, 3, 3, 5, 7, 10, 1, 1, 1, 11, 2, 2, 4, 7, 1, 6, 8, 5, 1, 3, 1, 12, 3, 7, 2, 9, 1, 1, 1, 13, 1, 2, 5, 14, 9, 7, 1, 4, 3, 8, 10, 15, 6, 3, 1, 5, 3, 3, 1, 11, 1, 9, 3
Offset: 1

Views

Author

Antti Karttunen, Feb 22 2019

Keywords

Comments

Like A323902 and A323903, this also has quite a moderate growth rate, even though some terms of A048675 soon grow quite big.

Links

Crossrefs

Cf. A002487, A048675, A283477, A322821, A323902, A323903, A324287.

Programs

  • PARI
    A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ From A048675
A324286(n) = A002487(A048675(n));

Formula

a(n) = A002487(A048675(n)) = A002487(A322821(n)).
a(A283477(n)) = A324287(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)).
Showing 1-5 of 5 results.

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