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

A295300 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)].

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 19 2017

Keywords

Comments

Restricted growth sequence transform of A291752.
For all i, j:
a(i) = a(j) => A291751(i) = A291751(j),
a(i) = a(j) => A326199(i) = A326199(j) => A294877(i) = A294877(j),
a(i) = a(j) => A322021(i) = A322021(j),
a(i) = a(j) => A295888(i) = A295888(j),
a(i) = a(j) => A296090(i) = A296090(j).

Links

Crossrefs

Cf. A003557, A046523, A048250, A101296, A286360, A291750, A291751, A291752, A291757, A291758, A295888, A296090, A322021, A326199.
Cf. also A305801, A305900, A323368, A324400.

Programs

  • PARI
    up_to = 100000;
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
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
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));
Aux295300(n) = (1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n));
v295300 = rgs_transform(vector(up_to,n,Aux295300(n)));
A295300(n) = v295300[n];

Extensions

Name changed and the comments section added by Antti Karttunen, Jul 13 2019

A294877 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 4, 8, 2, 9, 2, 7, 4, 4, 2, 10, 11, 4, 12, 7, 2, 13, 2, 14, 4, 4, 4, 15, 2, 4, 4, 10, 2, 13, 2, 7, 9, 4, 2, 16, 17, 18, 4, 7, 2, 19, 4, 10, 4, 4, 2, 20, 2, 4, 9, 21, 4, 13, 2, 7, 4, 13, 2, 22, 2, 4, 18, 7, 4, 13, 2, 16, 23, 4, 2, 20, 4, 4, 4, 10, 2, 24, 4, 7, 4, 4, 4, 25, 2, 26, 9, 27, 2, 13, 2, 10, 13, 4, 2, 28, 2, 13
Offset: 1

Views

Author

Antti Karttunen, Nov 11 2017

Keywords

Comments

Restricted growth sequence transform of A291757, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291757(i) = A291757(j) <=> A003557(i) = A003557(j) and A046523(i) = A046523(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.
Also the restricted growth sequence transform of A294876, Product_{d|n, d>1} prime(gcd(d,n/d)). (This was the original definition).
For all i, j:
A295300(i) = A295300(j) => a(i) = a(j),
A319347(i) = A319347(j) => a(i) = a(j),
a(i) = a(j) => A055155(i) = A055155(j).

Links

Crossrefs

Cf. also A291751, A291757, A294876, A295300, A295888, A319347, A322021.
Cf. A000188, A055155, A294897, A295666, A322020 (a few of the matched sequences).

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; };
A294876(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(gcd(d,n/d)))); m; };
v294877 = rgs_transform(vector(up_to,n,A294876(n)));
A294877(n) = v294877[n];
  • PARI
    A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
v294877 = rgs_transform(vector(up_to,n,[A003557(n),A046523(n)]));
A294877(n) = v294877[n]; \\ Antti Karttunen, Nov 28 2018

Extensions

Name changed and comments added by Antti Karttunen, Nov 28 2018

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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