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.

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];

A305795 Restricted growth sequence transform of A305794, a filter sequence constructed from the binary expansions of the divisors of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 11 2018

Keywords

Links

Crossrefs

Cf. A278222, A286622, A305794.
Cf. also A304105, A305793, A305815.

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; };
A305794(n) = { my(m=1); fordiv(n, d, if(d>1, m *= prime(A286622(d)-1))); (m); };
v305795 = rgs_transform(vector(up_to, n, A305794(n)));
A305795(n) = v305795[n];

Formula

For all i, j:
a(i) = a(j) => A000005(i) = A000005(j).
a(i) = a(j) => A007814(i) = A007814(j).
a(i) = a(j) => A093653(i) = A093653(j).
a(i) = a(j) => A154402(i) = A154402(j).
a(i) = a(j) => A305436(i) = A305436(j).

A304104 a(n) = Product_{d|n, d>1} prime(A304101(d)-1).

Original entry on oeis.org

1, 2, 2, 6, 2, 20, 3, 12, 10, 20, 3, 420, 2, 30, 20, 60, 11, 300, 11, 420, 12, 30, 5, 4200, 22, 20, 130, 990, 3, 11000, 11, 420, 102, 44, 30, 31500, 5, 242, 20, 10920, 11, 3000, 13, 1170, 1100, 190, 3, 231000, 33, 2420, 506, 420, 19, 66300, 12, 9900, 110, 30, 11, 8085000, 13, 242, 300, 5460, 52, 56100, 19, 660, 130, 19500, 13, 9135000, 11, 290, 4180, 2178, 99
Offset: 1

Views

Author

Antti Karttunen, May 13 2018

Keywords

Links

Crossrefs

Cf. A304101, A304102, A304105 (restricted growth sequence transform of this sequence).

Programs

  • PARI
    \\ Needs also code from A304101:
A304104(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(A304101(d)-1))); (m); };

Formula

a(n) = Product_{d|n, d>1} A000040(A304101(d)-1).
a(n) = (1/2) * A304102(n) * A000040(A304101(n)-1).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A001511(a(n)) = A005086(n).
A007949(a(n)) = A304096(n).

A304103 Restricted growth sequence transform of A304102, a filter sequence related to the proper divisors of n expressed in Fibonacci number system.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 5, 4, 7, 2, 8, 2, 6, 5, 5, 2, 9, 3, 4, 10, 11, 2, 12, 2, 6, 5, 13, 5, 14, 2, 13, 4, 9, 2, 15, 2, 11, 8, 10, 2, 16, 17, 18, 13, 6, 2, 19, 5, 20, 13, 5, 2, 21, 2, 13, 6, 22, 4, 23, 2, 24, 10, 25, 2, 26, 2, 10, 18, 27, 28, 12, 2, 29, 30, 13, 2, 31, 13, 32, 5, 33, 2, 34, 5, 35, 13, 5, 13, 21, 2, 36, 37, 38, 2, 39, 2, 9, 15
Offset: 1

Views

Author

Antti Karttunen, May 13 2018

Keywords

Comments

For all i, j: a(i) = a(j) => b(i) = b(j), where b can be any of {A000005, A293435, A304095 or A300836} for example.

Links

Crossrefs

Cf. A293435, A304095, A300836, A304102.
Cf. also A300835, A304105, A305800.
Cf. A305793 (analogous filter for base 2).

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])); }
A304102(n) = { my(m=1); fordiv(n,d,if(dA304101(d)-1))); (m); };
write_to_bfile(1,rgs_transform(vector(up_to,n,A304102(n))),"b304103.txt");

A374201 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A048679(A328845(i))) = A278222(A048679(A328845(j))), for all i, j >= 1, where A328845 is a Fibonacci-based variant of the arithmetic derivative.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 2, 4, 4, 5, 2, 4, 2, 6, 7, 4, 2, 6, 2, 8, 9, 10, 2, 4, 7, 8, 6, 11, 2, 8, 2, 11, 8, 12, 7, 4, 2, 12, 13, 7, 2, 11, 2, 8, 8, 13, 2, 7, 10, 7, 8, 14, 2, 15, 7, 13, 12, 10, 2, 8, 2, 12, 7, 10, 13, 5, 2, 16, 17, 5, 2, 7, 2, 13, 7, 15, 16, 18, 2, 7, 18, 12, 2, 8, 19, 20, 13, 7, 2, 8, 18, 16, 12, 10, 21, 13, 2, 8, 9, 16
Offset: 0

Views

Author

Antti Karttunen, Jul 02 2024

Keywords

Comments

Restricted growth sequence transform of A278222(A048679(A328845(n))), or equally, of A304101(A328845(n)).
Related to the Zeckendorf-representation (A014417) of A328845(n).
For all i, j >= 0: a(i) = a(j) => A328847(i) = A328847(j).

Links

Crossrefs

Cf. A000045, A014417, A048679, A278222, A304101, A328845, A328847.
Cf. also A304105, A373982.

Programs

  • PARI
    up_to = 75025;
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) = { my(s=0, i=0); while(n, if(2!=(n%4), s += (n%2)<>= 1); (s); };
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]); };
A278222(n) = A046523(A005940(1+n));
A328845(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]));
v374201 = rgs_transform(vector(1+up_to, n, A278222(A048679(A328845(n-1)))));
A374201(n) = v374201[1+n];

A305811 Filter sequence for a(Fibonacci prime) = constant sequences.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 5, 6, 7, 8, 9, 10, 2, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 2, 86, 87, 88, 89, 90
Offset: 1

Views

Author

Antti Karttunen, Jun 16 2018

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A305800(n), A305820(n)].
For all i, j: a(i) = a(j) => A304105(i) = A304105(j).

Links

Crossrefs

Cf. A005478, A010051, A010056, A304105, A305800, A305820.

Programs

  • PARI
    up_to = 100000;
A010056(n) = { my(k=n^2); k+=(k+1)<<2; (issquare(k) || (n>0 && issquare(k-8))) }; \\ From A010056
partialsums(f,up_to) = { my(v = vector(up_to), s=0); for(i=1,up_to,s += f(i); v[i] = s); (v); }
v_partsums = partialsums(x -> (isprime(x)&&A010056(x)), up_to);
A305811(n) = if(1==n,n,if(isprime(n)&&A010056(n),2,1+n-v_partsums[n]));

Formula

a(1) = 1; for n > 1, if A010051(n)==1 and A010056(n)==1 [when n is a Fibonacci prime, A005478], a(n) = 2, otherwise a(n) = running count from 3 onward.
Showing 1-6 of 6 results.

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