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.

A304101 Restricted growth sequence transform of A278222(A048679(n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 13 2018

Keywords

Comments

Positions of 2's is given by the positive Fibonacci numbers: 1, 2, 3, 5, 8, 13, 21, ..., that is, A000045(n) from n >= 2 onward.
Positions of 3's is given by Lucas numbers larger than 3: 4, 7, 11, 18, ..., that is, A000032(n) from n >= 3 onward.
Sequence allots a distinct value for each distinct multiset formed from the lengths of 1-runs in the binary representation of A048679(n). Compare to the scatter plot of A286622.

Links

Crossrefs

Cf. A000032, A000045, A003603, A007895, A035513, A048679, A278222, A304100, A304102, A304103, A304104, A304105.
Cf. also A286622 (compare the scatter-plots).

Programs

  • PARI
    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; };
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A106151(n) = if(n<=1, n, if(n%2, 1+(2*A106151((n-1)/2)), A106151(n>>valuation(n, 2))<<(valuation(n, 2)-1)));
A048679(n) = if(!n,n,A106151(2*A003714(n)));
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));
v304101 = rgs_transform(vector(1+up_to, n, A278222(A048679(n-1))));
A304101(n) = v304101[1+n];

A305793 Restricted growth sequence transform of A305792, a filter sequence constructed from binary expansions of the proper divisors of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 11 2018

Keywords

Links

Crossrefs

Cf. A278222, A286622, A305792, A305795.
Cf. also A293215, A300833, A304103, A305813.

Programs

  • PARI
    \\ Needs also code from A286622:
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; };
A305792(n) = { my(m=1); fordiv(n,d,if(dA286622(d)-1))); (m); };
v305793 = rgs_transform(vector(up_to, n, A305792(n)));
A305793(n) = v305793[n];

Formula

For all i, j:
a(i) = a(j) => A000005(i) = A000005(j).
a(i) = a(j) => A292257(i) = A292257(j).
a(i) = a(j) => A305426(i) = A305426(j).
a(i) = a(j) => A305435(i) = A305435(j).

A304102 a(n) = Product_{d|n, dA304101(d)-1).

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 12, 4, 8, 2, 120, 2, 12, 8, 24, 2, 200, 2, 120, 12, 12, 2, 1680, 4, 8, 20, 180, 2, 2000, 2, 120, 12, 44, 12, 12600, 2, 44, 8, 1680, 2, 1200, 2, 180, 200, 20, 2, 42000, 6, 440, 44, 120, 2, 7800, 12, 3960, 44, 12, 2, 3234000, 2, 44, 120, 840, 8, 10200, 2, 264, 20, 3000, 2, 630000, 2, 20, 440, 1452, 18, 2000, 2, 109200, 260, 44, 2, 1386000
Offset: 1

Views

Author

Antti Karttunen, May 13 2018

Keywords

Links

Crossrefs

Cf. A304101, A304103 (restricted growth sequence transform of this sequence), A304104.

Programs

  • PARI
    \\ Needs also code from A304101:
A304102(n) = { my(m=1); fordiv(n,d,if(dA304101(d)-1))); (m); };

Formula

a(n) = Product_{d|n, dA000040(A304101(d)-1).
a(n) = 2*A304104(n) / A000040(A304101(n)-1).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A293435(n).
A007949(a(n)) = A304095(n).

A304105 Restricted growth sequence transform of A304104, a filter sequence related to how the divisors of n are expressed in Fibonacci number system.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 13 2018

Keywords

Links

Crossrefs

Cf. A005086, A304096, A300837, A304101, A304103, A304104.

Programs

  • PARI
    \\ Needs also code from A304101:
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A304104(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(A304101(d)-1))); (m); };
write_to_bfile(1,rgs_transform(vector(up_to,n,A304104(n))),"b304105.txt");

Formula

For all i, j: a(i) = a(j) => b(i) = b(j), where b can be any of {A000005, A005086, A304096 or A300837} for example.
Showing 1-4 of 4 results.

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