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.

Previous Showing 21-28 of 28 results.

A328477 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328469(A276086(i)) = A328469(A276086(j)) for all i, j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 8, 9, 10, 11, 12, 6, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 4, 8, 9, 10, 11, 28, 9, 29, 30, 31, 32, 33, 11, 31, 32, 34, 35, 36, 18, 37, 38, 39, 40, 41, 24, 42, 43, 44, 45, 46, 6, 10, 11, 13, 14, 33, 11, 31, 32, 34, 35, 47, 14, 34, 35, 48, 49, 50, 20, 39, 40, 51, 52, 53, 26, 44, 45, 54, 55, 56, 16, 17, 18, 19, 20, 36, 18
Offset: 0

Views

Author

Antti Karttunen, Oct 19 2019

Keywords

Comments

Restricted growth sequence transform of function f(n) = A328469(A276086(n)), or equally, of the function g(0) = 0; n > 0, g(n) = [A053669(n), A278226(n)], where in the ordered pair A053669(n) gives the smallest prime not dividing n, while A278226(n) gives the prime signature of A276086(n), i.e., a signature of the multiset of nonzero digits in the primorial base expansion of n. Note that A000720(A053669(n)) = A055396(A276086(n)) is one more than the number of trailing zeros in the primorial base expansion for n > 0.

Examples

			When written in primorial base (A049345), numbers 42 ("1200" as 42 = 1*A002110(3) + 2*A002110(2) + 0*A002110(1) + 0*A002110(0) = 1*30 + 2*6 + 0*2 + 0*1), 66 ("2100" as 66 = 2*30 + 1*6 + 0*2 + 0*1) and 222 ("10200" as 222 = 1*210 + 0*30 + 2*6 + 0*2 + 0*1) all have {1, 2} as their multiset of nonzero digits, and all have exactly two trailing zeros, thus they get an equal value in this sequence, namely a(42) = a(66) = a(222) = 33, where 33 is a running number allotted by the restricted growth sequence transform.

Links

Crossrefs

Cf. A020639, A046523, A049345, A053669, A276086, A278226, A286626, A328399, A328469.

Programs

  • PARI
    up_to = 32768;
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; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
Aux328469(n) = [A020639(n), A046523(n)];
Aux328477(n) = Aux328469(A276086(n));
v328477 = rgs_transform(vector(1+up_to, n, Aux328477(n-1)));
A328477(n) = v328477[1+n];

A328628 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A328624(i)) = A046523(A328624(j)) for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 2, 5, 3, 6, 7, 8, 4, 3, 9, 10, 3, 11, 12, 13, 5, 14, 15, 16, 17, 9, 13, 18, 19, 14, 2, 5, 5, 14, 3, 11, 3, 14, 6, 20, 21, 22, 7, 18, 23, 24, 18, 25, 26, 23, 16, 27, 28, 29, 30, 31, 32, 33, 34, 35, 4, 3, 3, 6, 19, 14, 9, 16, 10, 36, 37, 38, 39, 40, 41, 42, 40, 43, 13, 8, 11, 44, 10, 45, 26, 23, 46, 47, 21, 22, 12, 13, 13, 18, 7, 8, 48, 49, 40
Offset: 0

Views

Author

Antti Karttunen, Oct 25 2019

Keywords

Comments

Restricted growth sequence transform of function f(n) = A046523(A328624(n)) = A278226(A328625(n)).
For all i, j:
a(i) = a(j) => A328630(i) = A328630(j).
The scatter plot looks like a mound (or hive) of insects. - Antti Karttunen, Jan 09 2023

Links

Crossrefs

Cf. A046523, A278226, A328624, A328625, A328629, A328630.

Programs

  • PARI
    up_to = 32768;
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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A328624(n) = { my(m=1, p=2, e, g=1); while(n, e = (n%p); m *= (p^((g*e)%p)); g = e+1; n = n\p; p = nextprime(1+p)); (m); };
Aux328628(n) = A046523(A328624(n));
v328628 = rgs_transform(vector(1+up_to, n, Aux328628(n-1)));
A328628(n) = v328628[1+n];

A328395 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A276087(i)) = A046523(A276087(j)) for all i, j.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 5, 6, 1, 7, 4, 8, 9, 10, 11, 7, 5, 12, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 2, 5, 9, 5, 6, 8, 6, 23, 24, 1, 25, 4, 26, 27, 28, 11, 29, 8, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 9, 9, 42, 12, 24, 12, 7, 43, 44, 45, 46, 47, 48, 27, 49, 33, 50, 35, 51, 52, 53, 54, 55, 56, 57, 58, 59, 20, 60, 61, 30, 27, 62, 63, 64, 65, 66, 15, 67, 68, 69, 8
Offset: 0

Views

Author

Antti Karttunen, Oct 15 2019

Keywords

Comments

Restricted growth sequence transform of function f(n) = A278226(A276086(n)) = A046523(A276086(A276086(n))).
For all i, j:
a(i) = a(j) => A328397(i) = A328397(j) => A328389(i) = A328389(j).

Links

Crossrefs

Cf. A046523, A276086, A276087, A328389.
Cf. A143293 (positions of 1's after the initial one).
Cf. also A278226, A286626, A328388, A328396, A328397.

Programs

  • PARI
    up_to = 32589;
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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A276087(n) = A276086(A276086(n));
v328395 = rgs_transform(vector(1+up_to, n, A046523(A276087(n-1))));
A328395(n) = v328395[1+n];

A328396 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = A046523(A276086(A003415(A276086(n)))).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 5, 5, 6, 7, 3, 8, 7, 9, 5, 10, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2, 22, 3, 8, 23, 24, 25, 26, 11, 6, 27, 28, 29, 24, 11, 30, 31, 32, 27, 33, 10, 34, 35, 36, 37, 38, 39, 19, 40, 41, 3, 42, 43, 44, 25, 45, 46, 11, 9, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 4, 69, 70, 71, 72, 73, 74, 32
Offset: 0

Views

Author

Antti Karttunen, Oct 15 2019

Keywords

Comments

Restricted growth sequence transform of function f(n) = A278226(A327860(n)) = A046523(A276086(A003415(A276086(n)))).
For all i, j: a(i) = a(j) => A328392(i) = A328392(j).

Links

Crossrefs

Cf. A003415, A046523, A276086, A327860, A328392.
Cf. also A278226, A286626, A328388, A328395.

Programs

  • PARI
    up_to = 30030;
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; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
Aux328396(n) = A046523(A276086(A003415(A276086(n))));
v328396 = rgs_transform(vector(1+up_to, n, Aux328396(n-1)));
A328396(n) = v328396[1+n];

A328629 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A328627(i)) = A046523(A328627(j)) for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 2, 5, 6, 7, 8, 9, 4, 3, 5, 10, 11, 12, 13, 6, 14, 15, 3, 16, 17, 14, 3, 18, 19, 15, 2, 5, 5, 15, 3, 12, 14, 16, 18, 20, 21, 22, 23, 24, 16, 25, 26, 27, 19, 18, 28, 29, 10, 30, 31, 28, 24, 32, 33, 34, 4, 3, 3, 7, 8, 15, 5, 12, 28, 32, 33, 30, 19, 18, 12, 35, 36, 37, 31, 28, 38, 39, 15, 40, 41, 38, 18, 25, 26, 22, 13, 6, 6, 18, 19, 9, 42
Offset: 0

Views

Author

Antti Karttunen, Oct 25 2019

Keywords

Comments

Restricted growth sequence transform of function f(n) = A046523(A328627(n)), or equally, of function g(n) = A278226(A328626(n)).

Links

Crossrefs

Cf. A046523, A278226, A328626, A328627, A328628.

Programs

  • PARI
    up_to = 32768;
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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A328627(n) = { my(m=1, p=2, d=0); while(n, d = lift(Mod(n,p)/(d+1)); m *= (p^d); n = n\p; p = nextprime(1+p)); (m); };
Aux328629(n) = A046523(A328627(n));
v328629 = rgs_transform(vector(1+up_to, n, Aux328629(n-1)));
A328629(n) = v328629[1+n];

A329038 a(n) = A246277(A276086(n)).

Original entry on oeis.org

0, 1, 1, 3, 2, 9, 1, 5, 3, 15, 6, 45, 2, 25, 9, 75, 18, 225, 4, 125, 27, 375, 54, 1125, 8, 625, 81, 1875, 162, 5625, 1, 7, 5, 21, 10, 63, 3, 35, 15, 105, 30, 315, 6, 175, 45, 525, 90, 1575, 12, 875, 135, 2625, 270, 7875, 24, 4375, 405, 13125, 810, 39375, 2, 49, 25, 147, 50, 441, 9, 245, 75, 735, 150, 2205, 18, 1225, 225, 3675, 450, 11025, 36
Offset: 0

Views

Author

Antti Karttunen, Nov 08 2019

Keywords

Links

Crossrefs

Cf. A046523, A246277, A276086, A278226, A329048 (rgs-transform).
Cf. also A329345.

Programs

  • PARI
    A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329038(n) = A246277(A276086(n));

Formula

a(n) = A246277(A276086(n)).
For n >= 1, A046523(2*a(n)) = A278226(n).

A329048 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329038(i) = A329038(j) for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 2, 6, 3, 7, 8, 9, 4, 10, 5, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 2, 26, 6, 27, 28, 29, 3, 30, 7, 31, 32, 33, 8, 34, 9, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 4, 50, 10, 51, 52, 53, 5, 54, 11, 55, 56, 57, 12, 58, 13, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 14, 74, 15, 75, 76, 77, 16
Offset: 0

Views

Author

Antti Karttunen, Nov 11 2019

Keywords

Comments

Restricted growth sequence transform of A329038, i.e., of function f(n) = A246277(A276086(n)).
For all i, j:
a(i) = a(j) => A286626(i) = A286626(j),
a(i) = a(j) => A276088(i) = A276088(j),
a(i) = a(j) => A276153(i) = A276153(j),

Links

Crossrefs

Cf. A046523, A246277, A276086, A276088, A276153, A278226, A286626, A329038.
Cf. also A329345.

Programs

  • PARI
    up_to = 32768;
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; };
A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329038(n) = A246277(A276086(n));
v329048 = rgs_transform(vector(1+up_to, n, A329038(n-1)));
A329048(n) = v329048[1+n];

A328397 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328400(A276087(i)) = A328400(A276087(j)) for all i, j.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 3, 3, 1, 3, 2, 4, 5, 3, 6, 3, 3, 7, 3, 8, 9, 8, 10, 11, 12, 13, 14, 15, 16, 1, 3, 5, 3, 3, 4, 3, 3, 17, 1, 17, 2, 18, 19, 17, 6, 20, 4, 7, 21, 17, 22, 23, 24, 25, 26, 27, 28, 23, 21, 5, 5, 4, 7, 17, 7, 3, 3, 17, 29, 30, 31, 18, 19, 32, 22, 33, 24, 34, 35, 36, 37, 38, 39, 15, 40, 41, 14, 42, 43, 7, 19, 44, 45, 19, 46, 21, 8, 17, 47, 48, 4, 49
Offset: 0

Views

Author

Antti Karttunen, Oct 15 2019

Keywords

Comments

Restricted growth sequence transform of function f(n) = A328400(A276087(n)).
For all i, j:
A328395(i) = A328395(j) => a(i) = a(j) => A328389(i) = A328389(j).

Links

Crossrefs

Cf. A276086, A276087, A328389, A328400.
Cf. also A278226, A286626, A328388, A328395, A328396.

Programs

  • PARI
    up_to = 32589;
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; };
A007947(n) = factorback(factorint(n)[, 1]);
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); };
A328400(n) = A181821(A007947(A181819(n)));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A276087(n) = A276086(A276086(n));
v328397 = rgs_transform(vector(1+up_to, n, A328400(A276087(n-1))));
A328397(n) = v328397[1+n];
Previous Showing 21-28 of 28 results.

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