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 11-19 of 19 results.

A328835 Prime shadow of primorial base exp-function: a(n) = A181819(A276086(n)).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 2, 4, 4, 8, 6, 12, 3, 6, 6, 12, 9, 18, 5, 10, 10, 20, 15, 30, 7, 14, 14, 28, 21, 42, 2, 4, 4, 8, 6, 12, 4, 8, 8, 16, 12, 24, 6, 12, 12, 24, 18, 36, 10, 20, 20, 40, 30, 60, 14, 28, 28, 56, 42, 84, 3, 6, 6, 12, 9, 18, 6, 12, 12, 24, 18, 36, 9, 18, 18, 36, 27, 54, 15, 30, 30, 60, 45, 90, 21, 42, 42, 84, 63, 126, 5, 10, 10, 20, 15, 30, 10, 20, 20
Offset: 0

Views

Author

Antti Karttunen, Oct 29 2019

Keywords

Comments

From Antti Karttunen, Apr 30 2022: (Start)
These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the number of times a nonzero digit k occurs in the primorial base representation of n.
Note that this sequence, and all the sequences derived from it as b(n) = f(a(n)), [where f is any integer-valued function] can be represented as b(n) = g(A278226(n)), where g(n) = f(A181819(n)). E.g., if f is the identity function (so that b(n) is this sequence), then g(n) is A181819(n). See the comment and formulas in the latter sequence.
(End)

Links

Crossrefs

Cf. A001222, A007814, A061395, A181819, A181821, A267263, A276086, A278226 (A286626), A328114, A328614.
Cf. A275735 for analogous sequence.

Programs

  • PARI
    A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328835(n) = A181819(A276086(n));

Formula

a(n) = A181819(A276086(n)).
A001222(a(n)) = A267263(n).
A007814(a(n)) = A328614(n).
A061395(a(n)) = A328114(n).
For all n >= 0, a(n) = A181819(A278226(n)) and A181821(a(n)) = A278226(n). - Antti Karttunen, Apr 30 2022

A328388 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A327860(i)) = A046523(A327860(j)) for all i, j >= 0.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 2, 3, 5, 3, 4, 4, 4, 6, 4, 4, 7, 8, 6, 6, 9, 6, 9, 9, 10, 11, 11, 10, 12, 13, 2, 14, 4, 3, 4, 8, 6, 3, 3, 4, 8, 4, 4, 4, 9, 6, 8, 8, 9, 9, 6, 6, 15, 9, 13, 11, 13, 11, 16, 13, 4, 4, 4, 4, 17, 8, 4, 8, 8, 4, 9, 8, 12, 8, 8, 18, 19, 18, 9, 9, 20, 21, 17, 17, 12, 12, 13, 12, 22, 23, 6, 6, 24, 6, 9, 9, 9, 6, 6, 6, 25, 17, 9, 17, 17, 9
Offset: 0

Views

Author

Antti Karttunen, Oct 15 2019

Keywords

Comments

Restricted growth sequence transform of A046523(A327860(n)).

Links

Crossrefs

Cf. A003415, A046523, A276086, A327860.
Cf. also A286626 (compare the scatter plots).

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); };
A327860(n) = A003415(A276086(n));
Aux328388(n) = if(!n,0,A046523(A327860(n)));
v328388 = rgs_transform(vector(1+up_to, n, Aux328388(n-1)));
A328388(n) = v328388[1+n];

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];

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];

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];

A331172 a(n) = min(n, A289234(n)), where A289234 is primorial base "reciprocal" flip.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 12 2020

Keywords

Comments

For all i, j:
a(i) = a(j) => A267263(i) = A267263(j).
For all i, j > 0:
a(i) = a(j) => A053669(i) = A053669(j).

Links

Crossrefs

Cf. A053669, A267263, A289234.
Cf. also A286626, A328477, A330740, A331171.

Programs

  • PARI
    A289234(n) = { my(pr=1, p=2, v=0); while(n>0, my (d=n%p); if(d>0, v += pr * lift(1/Mod(d, p))); pr *= p; n \= p; p = nextprime(p+1)); return(v); }; \\ From A289234.
A331172(n) = min(n, A289234(n));

Formula

a(n) = min(n, A289234(n)).

A342022 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342002(i) = A342002(j) for all i, j >= 0.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Mar 12 2021

Keywords

Comments

Restricted growth sequence transform of A342002.
For all i, j >= 1:
a(i) = a(j) => A342017(i) = A342017(j) => A342019(i) = A342019(j).

Links

Crossrefs

Cf. A003415, A276086, A327860, A328572, A342002, A342017, A342019.
Cf. also A286626, A328388.

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; };
A342002(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= p^(e>0); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
v342022 = rgs_transform(vector(1+up_to,n,A342002(n-1)));
A342022(n) = v342022[1+n];
Previous Showing 11-19 of 19 results.

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