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.

A291751 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A048250(i) = A048250(j), for all i, j.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 06 2017

Keywords

Comments

Restricted growth sequence transform of A291750, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291750(i) = A291750(j) <=> A003557(i) = A003557(j) and A048250(i) = A048250(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.
Sigma (A000203) and psi (A001615) are functions of this sequence. See comments in A291750 for the reason. For example, to find the value of A001615(n) when we know just a(n), but without knowing n, let m be the least i for which a(i) = a(n); then A001615(n) = A003991(A291750(m)) = A003557(m) * A048250(m).

Links

Crossrefs

Cf. A001615, A003557, A048250, A291750, A291752, A294877, A295300, A295886, A295887, A295888, A319698, A322021.
Differs from A286603 for the first time at n = 25, where a(25) = 21, while A286603(25) = 14.

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; };
A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
v291751 = rgs_transform(vector(65537,n,A291750(n)));
A291751(n) = v291751[n];

Extensions

Name changed and comments added by Antti Karttunen, Nov 24 2018

A323238 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A291750(n) for all n, except for odd numbers n > 1, f(n) = 0.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 08 2019

Keywords

Comments

For all i, j:
A319701(i) = A319701(j) => a(i) = a(j),
a(i) = a(j) => A146076(i) = A146076(j),
a(i) = a(j) => A319697(i) = A319697(j).

Links

Crossrefs

Cf. A003557, A048250, A146076, A291750, A291751, A319698, A319697, A319701, A322588, A323237, A323241.

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; };
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
Aux323238(n) = if((n>1)&&(n%2),0,(1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n)));
v323238 = rgs_transform(vector(up_to, n, Aux323238(n)));
A323238(n) = v323238[n];

A319697 Sum of even squarefree divisors of n.

Original entry on oeis.org

0, 2, 0, 2, 0, 8, 0, 2, 0, 12, 0, 8, 0, 16, 0, 2, 0, 8, 0, 12, 0, 24, 0, 8, 0, 28, 0, 16, 0, 48, 0, 2, 0, 36, 0, 8, 0, 40, 0, 12, 0, 64, 0, 24, 0, 48, 0, 8, 0, 12, 0, 28, 0, 8, 0, 16, 0, 60, 0, 48, 0, 64, 0, 2, 0, 96, 0, 36, 0, 96, 0, 8, 0, 76, 0, 40, 0, 112, 0, 12, 0, 84, 0, 64, 0, 88, 0, 24, 0, 48, 0, 48, 0, 96
Offset: 1

Views

Author

Antti Karttunen, Oct 31 2018

Keywords

Links

Crossrefs

Cf. A048250, A146076, A183063, A206787, A319698.

Programs

  • Mathematica
    Table[Total[Select[Divisors[n],EvenQ[#]&&SquareFreeQ[#]&]],{n,100}] (* Harvey P. Dale, May 18 2019 *)
f[2, e_] := 2; f[p_, e_] := p + 1; a[n_] := If[OddQ[n], 0, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 30 2022 *)
  • PARI
    A319697(n) = sumdiv(n, d, (!(d%2))*issquarefree(d)*d);

Formula

a(n) = Sum_{d|n} A059841(d)*A008966(d)*d.
a(n) = A048250(n) - A206787(n).

A322022 Lexicographically earliest such sequence a that a(i) = a(j) => A305891(i) = A305891(j) and A319697(i) = A319697(j), for all i, j.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 29 2018

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A305891(n), A319697(n)], or equally, of the triple [A007814(n), A046523(n), A319697(n)].

Links

Crossrefs

Cf. A007814, A046523, A097272, A305891, A319697, A319698, A322021.

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; };
A007814(n) = valuation(n,2);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A319697(n) = sumdiv(n, d, (!(d%2))*issquarefree(d)*d);
v322022 = rgs_transform(vector(up_to, n, [A007814(n), A046523(n), A319697(n)]));
A322022(n) = v322022[n];
Showing 1-4 of 4 results.

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