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.

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A331747 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A278222(i) = A278222(j) for all i, j.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Feb 04 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A009194(n), A278222(n)].
For all i, j:
A331746(i) = A331746(j) => a(i) = a(j).

Links

Crossrefs

Cf. A009194, A278222.
Cf. also A324400, A286622, A318310, A318311, A324389, A331744, A331745, A331746.

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; };
A009194(n) = gcd(n, sigma(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));
Aux331747(n) = [A009194(n),A278222(n)];
v331747 = rgs_transform(vector(up_to, n, Aux331747(n)));
A331747(n) = v331747[n];

Formula

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

A323240 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = [A001222(n), A323244(n)] for n > 1, and f(1) = 0.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 10 2019

Keywords

Comments

Restricted growth sequence transform of function f, where f(1) = 0 and f(n) = [A001222(n), A323244(n)] for n > 1.
Equally, restricted growth sequence transform of function f, where f(1) = 0 and f(n) = A318310(A156552(n)) for n > 1.
For all i, j:
a(i) = a(j) => A323245(i) = A323245(j), [Equally: A323244(i) = A323244(j)]
a(i) = a(j) => A323246(i) = A323246(j). [Equally: A323248(i) = A323248(j)]

Links

Crossrefs

Cf. A001222, A156552, A318310, A323244, A323245, A323246, A323248.
Cf. also A323168.

Programs

  • PARI
    up_to = 1024;
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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k));
A323240aux(n) = if(1==n,-1,[bigomega(n),A323244(n)]);
v323240 = rgs_transform(vector(up_to, n, A323240aux(n)));
A323240(n) = v323240[n];

A323904 Lexicographically earliest sequence such that a(i) = a(j) => A033879(i) = A033879(j) and A083254(i) = A083254(j), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Feb 10 2019

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A033879(n), A083254(n)], the deficiency of n, and its Möbius transform.
For all i, j: a(i) = a(j) => A297159(i) = A297159(j).

Examples

			For n=39, we have A033879(39) = 2*39 - A000203(39) = 22, and A083254(39) = 2*A000010(39)-39 = 9. For n=63 the results are same, with A033879(63) = 22 and A083254(63) = 9, thus a(39) and a(63) are allotted the same number by the restricted growth sequence transform, which in this case is 35, thus a(39) = a(63) = 35.
For n=42 and 54, we have A033879(42) = -12, A083254(42) = -18 and A033879(54) = -12, A083254(54) = -18, thus a(42) = a(54) (= 38).

Links

Crossrefs

Cf. A000010, A000203, A033879, A083254.
Cf. also A318310, A323892, A323897, A323898.

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; };
A033879(n) = (2*n-sigma(n));
A083254(n) = (2*eulerphi(n)-n);
A323904aux(n) = [A033879(n), A083254(n)];
v323904 = rgs_transform(vector(up_to,n,A323904aux(n)));
A323904(n) = v323904[n];

Formula

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

A318308 Restricted growth sequence transform of A294898.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 26 2018

Keywords

Links

Crossrefs

Cf. A294898, A286449, A318309, A318310.

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; };
A294898(n) = (2*n - sigma(n) - hammingweight(n));
v318308 = rgs_transform(vector(up_to,n,A294898(n)));
A318308(n) = v318308[n];
Previous Showing 11-14 of 14 results.

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