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 1-10 of 10 results.

A331744 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A323901(i) = A323901(j) for all i, j.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Feb 04 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A009194(n), A323901(n)].

Links

Crossrefs

Cf. A002487, A009194, A163511, A323901.
Cf. also A324400, A318310, A324389, A331745, A331746, A331747.

Programs

  • PARI
    \\ Needs also code from A323901.
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; };
A009194(n) = gcd(n, sigma(n));
Aux331744(n) = [A009194(n),A323901(n)];
v331744 = rgs_transform(vector(up_to, n, Aux331744(n)));
A331744(n) = v331744[n];

Formula

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

A331743 Lexicographically earliest infinite sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A323901(i) = A323901(j) for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 14, 8, 15, 5, 16, 9, 17, 2, 18, 10, 19, 6, 20, 11, 21, 4, 22, 12, 23, 7, 24, 13, 25, 3, 26, 14, 27, 8, 28, 15, 29, 5, 30, 16, 31, 9, 32, 17, 33, 2, 34, 18, 35, 10, 36, 19, 37, 6, 38, 20, 39, 11, 40, 21, 41, 4, 42, 22, 43, 12, 44, 23, 45, 7, 46, 24, 47, 13, 48, 25, 49, 3, 50, 26, 51, 14, 52, 27, 53, 8, 54
Offset: 0

Views

Author

Antti Karttunen, Feb 05 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A002487(n), A002487(A163511(n))].
For all i, j:
a(i) = a(j) => A331748(i) = A331748(j),
a(i) = a(j) => A331749(i) = A331749(j).

Links

Crossrefs

Cf. A002487, A163511, A323901, A324400, A331742, A331744, A331748, A331749.
Differs from A331745 for the first time at n=77, where a(77) = 40, while A331745(77) = 24.
Differs from A103391(1+n) for the first time at n=191, where a(191) = 23, while A103391(192) = 97.

Programs

  • PARI
    \\ Needs also code from A323901.
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; };
Aux331743(n) = [A002487(n), A323901(n)];
v331743 = rgs_transform(vector(1+up_to, n, Aux331743(n-1)));
A331743(n) = v331743[1+n];

Formula

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

A331745 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A323901(i) = A323901(j) for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 14, 8, 15, 5, 16, 9, 17, 2, 18, 10, 19, 6, 20, 11, 21, 4, 22, 12, 23, 7, 24, 13, 25, 3, 26, 14, 27, 8, 28, 15, 29, 5, 30, 16, 31, 9, 32, 17, 33, 2, 34, 18, 35, 10, 36, 19, 37, 6, 38, 20, 39, 11, 24, 21, 40, 4, 41, 22, 42, 12, 43, 23, 44, 7, 45, 24, 46, 13, 47, 25, 48, 3, 49, 26, 50, 14, 51, 27, 52, 8, 45
Offset: 0

Views

Author

Antti Karttunen, Feb 04 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A278222(n), A323901(n)].

Links

Crossrefs

Cf. A278222, A323901.
Cf. also A324400, A286622, A318310, A324389, A331744, A331746.

Programs

  • PARI
    \\ Needs also code from A323901.
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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1)));
t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
Aux331745(n) = [A278222(n),A323901(n)];
v331745 = rgs_transform(vector(1+up_to, n, Aux331745(n-1)));
A331745(n) = v331745[1+n];

Formula

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

A331742 Lexicographically earliest infinite sequence such that a(i) = a(j) => A323901(i) = A323901(j) for all i, j.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 3, 4, 4, 1, 7, 5, 8, 3, 9, 6, 10, 2, 11, 3, 10, 4, 5, 4, 12, 1, 13, 7, 14, 5, 15, 8, 16, 3, 17, 9, 18, 6, 19, 10, 20, 2, 17, 11, 18, 3, 9, 10, 21, 4, 22, 5, 8, 4, 22, 12, 12, 1, 23, 13, 24, 7, 25, 14, 26, 5, 27, 15, 28, 8, 19, 16, 29, 3, 30, 17, 31, 9, 32, 18, 33, 6, 34, 19, 33, 10, 35, 20, 36, 2, 35, 17, 37, 11, 38, 18, 14, 3, 34
Offset: 0

Views

Author

Antti Karttunen, Feb 05 2020

Keywords

Comments

Restricted growth sequence transform of function A323901(n) = A002487(A163511(n)).

Links

Crossrefs

Cf. A002487, A163511, A323901.
Cf. also A286378, A305899, A331743, A331744, A331745.

Programs

  • PARI
    \\ Needs also code from A323901.
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; };
v331742 = rgs_transform(vector(1+up_to, n, A323901(n-1)));
A331742(n) = v331742[1+n];

Formula

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)).

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)).

A331748 a(n) = A002487(n) XOR A002487(A163511(n)).

Original entry on oeis.org

1, 0, 0, 0, 0, 7, 0, 0, 0, 12, 7, 2, 0, 1, 0, 7, 0, 11, 12, 12, 7, 26, 2, 14, 0, 11, 1, 1, 0, 15, 7, 0, 0, 16, 11, 34, 12, 41, 12, 37, 7, 27, 26, 26, 2, 22, 14, 4, 0, 25, 11, 27, 1, 31, 1, 26, 0, 12, 15, 0, 7, 15, 0, 3, 0, 71, 16, 116, 11, 126, 34, 108, 12, 91, 41, 107, 12, 11, 37, 98, 7, 76, 27, 74, 26, 61, 26, 43, 2, 53, 22, 42, 14, 34, 4, 22, 0, 57
Offset: 0

Views

Author

Antti Karttunen, Feb 05 2020

Keywords

Links

Crossrefs

Cf. A002487, A003987, A163511, A323901, A331743, A331749.

Programs

Formula

a(n) = A002487(n) XOR A323901(n) = A002487(n) XOR A002487(A163511(n)).
a(2^n) = 0 for all n >= 0.

A331749 a(n) = A002487(A163511(n)) - A002487(n).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 4, 1, 2, 0, -1, 0, -1, 0, 9, 4, 4, 1, 10, 2, 2, 0, 5, -1, 1, 0, 1, -1, 0, 0, 16, 9, 34, 4, 23, 4, 37, 1, 5, 10, 10, 2, 14, 2, 4, 0, 7, 5, 11, -1, 5, 1, 6, 0, -4, 1, 0, -1, -3, 0, -1, 0, 57, 16, 116, 9, 98, 34, 84, 4, 69, 23, 103, 4, 9, 37, 98, 1, 52, 5, 70, 10, 19, 10, 39, 2, 19, 14, 38, 2, 34, 4, 18, 0, 39
Offset: 0

Views

Author

Antti Karttunen, Feb 05 2020

Keywords

Links

Crossrefs

Cf. A002487, A163511, A323901, A331743, A331748.

Programs

Formula

a(n) = A323901(n) - A002487(n) = A002487(A163511(n)) - A002487(n).
a(2^n) = 0 for all n >= 0.

A335420 a(n) = A000120(A163511(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 2, 3, 2, 4, 2, 3, 1, 3, 4, 6, 2, 4, 3, 3, 2, 4, 4, 3, 2, 3, 3, 3, 1, 6, 3, 5, 4, 7, 6, 6, 2, 4, 4, 6, 3, 4, 3, 5, 2, 4, 4, 6, 4, 4, 3, 4, 2, 6, 3, 5, 3, 2, 3, 3, 1, 6, 6, 6, 3, 7, 5, 5, 4, 5, 7, 7, 6, 3, 6, 6, 2, 5, 4, 5, 4, 8, 6, 7, 3, 6, 4, 6, 3, 6, 5, 4, 2, 5, 4, 7, 4, 4, 6, 5, 4, 6
Offset: 0

Views

Author

Antti Karttunen, Jun 09 2020

Keywords

Links

Crossrefs

Cf. A000079 (positions of ones), A000120, A001222, A163511, A335421, A335422.
Cf. also A323901, A334204.

Programs

  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A054429(n) = ((3<<#binary(n\2))-n-1);
A163511(n) = if(!n,1,A005940(1+A054429(n)));
A335420(n) = hammingweight(A163511(n));
  • Python
    from sympy import nextprime
def A335420(n):
    c, p, k = 1, 1, n
    while k:
        c *= (p:=nextprime(p))**(s:=(~k&k-1).bit_length())
        k >>= s+1
    return (c*p).bit_count() # Chai Wah Wu, Jul 25 2023

Formula

a(n) = A000120(A163511(n)).
a(n) = A001222(A335422(n)).
a(n) = a(2n) = a(A000265(n)).
For all n >= 0, a(2^n) = 1.
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.