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.

A331280 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278220(i) = A278220(j) for all i, j.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 17 2020

Keywords

Comments

Restricted growth sequence transform of A278220(n) (= A046523(A241909(n))).

Links

Crossrefs

Cf. A046523, A241909, A278220.
Cf. also A286621, A331299.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from A046523
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
A278220(n) = A046523(A241909(n));
v331280 = rgs_transform(vector(up_to, n, A278220(n)));
A331280(n) = v331280[n];

A336147 Lexicographically earliest infinite sequence such that a(i) = a(j) => A020639(i) = A020639(j) and A278221(i) = A278221(j), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 12 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A020639(n), A278221(n)].
For all i, j:
A324400(i) = A324400(j) => A336146(i) = A336146(j) => a(i) = a(j),
a(i) = a(j) => A243055(i) = A243055(j),
a(i) = a(j) => A336150(i) = A336150(j).

Links

Crossrefs

Cf. A020639, A046523, A122111, A278221.
Cf. also A243055, A286621, A318891, A324400, A336146, A336148, A336149, A336150.
First differs from A322590 at a(70) = 28 instead of 44.

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; };
A020639(n) = if(1==n, n, factor(n)[1, 1]);
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A278221(n) = A046523(A122111(n));
Aux336147(n) = [A020639(n),A278221(n)];
v336147 = rgs_transform(vector(up_to, n, Aux336147(n)));
A336147(n) = v336147[n];

A336151 Lexicographically earliest infinite sequence such that a(i) = a(j) => A001221(i) = A001221(j) and A006530(i) = A006530(j), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 11 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A001221(n), A006530(n)]. The first member of pair gives the number of distinct prime divisors of n, and the second member gives its largest prime factor.
For all i, j: A324400(i) = A324400(j) => A286621(i) = A286621(j) => a(i) = a(j).

Links

Crossrefs

Cf. A001221, A006530.
Cf. also A286621, A324400, A336150, A336152.

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; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
Aux336151(n) = [omega(n), A006530(n)];
v336151 = rgs_transform(vector(up_to, n, Aux336151(n)));
A336151(n) = v336151[n];

A336393 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336467(i) = A336467(j) and A278221(A000265(i)) = A278221(A000265(j)), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 10 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A336467(n), A278221(A000265(n))], or equally, of the ordered pair [A336467(n), A336395(n)].
For all i, j:
A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j).

Links

Crossrefs

Cf. A000265, A003602, A324400, A278221, A286621, A336390, A336395, A336467.

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; };
A000265(n) = (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
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A278221(n) = A046523(A122111(n));
A336467(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]+1))^f[k,2])); };
Aux336393(n) = [A336467(n), A278221(A000265(n))];
v336393 = rgs_transform(vector(up_to, n, Aux336393(n)));
A336393(n) = v336393[n];

A336316 The number of non-unitary divisors in the conjugated prime factorization of n: a(n) = A048105(A122111(n)).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 2, 4, 0, 5, 4, 2, 0, 6, 0, 7, 2, 5, 6, 8, 0, 2, 8, 1, 4, 9, 0, 10, 0, 8, 10, 4, 0, 11, 12, 11, 2, 12, 4, 13, 6, 2, 14, 14, 0, 3, 2, 14, 8, 15, 0, 8, 4, 17, 16, 16, 0, 17, 18, 5, 0, 12, 8, 18, 10, 20, 4, 19, 0, 20, 20, 2, 12, 6, 12, 21, 2, 1, 22, 22, 4, 16, 24, 23, 6, 23, 0, 11, 14, 26, 26, 20, 0, 24
Offset: 1

Views

Author

Antti Karttunen, Jul 18 2020

Keywords

Comments

Equally, the number of divisors in the conjugated prime factorization of n minus the number of its unitary divisors.
Note that A001221(A122111(n)) = A001221(n) for all n.

Links

Crossrefs

Cf. A000005, A001221, A034444, A048105, A122111, A286621, A336315.
Cf. A055932 (the positions of zeros).

Programs

  • PARI
    A336315(n) = if(1==n,n,my(p=apply(primepi,factor(n)[,1]~),m=1+p[1]); for(i=2, #p, m *= (1+p[i]-p[i-1])); (m));
A336316(n) = (A336315(n)-(2^omega(n)));

Formula

a(n) = A336315(n) - A034444(n) = A000005(A122111(n)) - 2^A001221(n).
a(n) = A048105(A122111(n)).

A336395 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278221(A000265(i)) = A278221(A000265(j)), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 10 2020

Keywords

Comments

Restricted growth sequence transform of the function f(n) = A278221(A000265(n)), the prime signature of the conjugated prime factorization of the odd part of n.
For all i, j:
A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j),
a(i) = a(j) => A005087(i) = A005087(j).

Links

Crossrefs

Cf. A000265, A003602, A005087, A278221, A286621, A324400.
Cf. also A336393.

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; };
A000265(n) = (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
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A278221(n) = A046523(A122111(n));
v336395 = rgs_transform(vector(up_to, n, A278221(A000265(n))));
A336395(n) = v336395[n];

A337201 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278221(A337194(i)) = A278221(A337194(j)), for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 3, 3, 4, 5, 1, 3, 1, 1, 1, 5, 4, 1, 1, 3, 1, 1, 1, 2, 1, 6, 3, 1, 1, 6, 5, 1, 4, 5, 3, 3, 1, 1, 7, 8, 3, 3, 2, 1, 3, 1, 4, 6, 1, 5, 1, 1, 2, 1, 5, 3, 4, 1, 1, 3, 3, 2, 9, 7, 1, 4, 1, 5, 4, 8, 10, 1, 5, 1, 2, 11, 1, 6, 6, 12, 1, 5, 1, 3, 1, 1, 3, 13, 3, 14, 15, 2, 2, 16, 1
Offset: 1

Views

Author

Antti Karttunen, Aug 20 2020

Keywords

Comments

Restricted growth sequence transform of f(n) = A278221(A337194(n)).
For all i, j: A324400(i) = A324400(j) => a(i) = a(j) => A337198(i) = A337198(j).

Links

Crossrefs

Cf. A000203, A000265, A278221, A161942, A336698, A337194, A337198.
Cf. also A286621, A324400, A337200.

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; };
A000265(n) = (n>>valuation(n, 2));
A337194(n) = (1+A000265(sigma(n)));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A278221(n) = A046523(A122111(n));
v337201 = rgs_transform(vector(up_to, n, A278221(A337194(n))));
A337201(n) = v337201[n];

A351955 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328571(A108951(i)) = A328571(A108951(j)) for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 03 2022

Keywords

Comments

Restricted growth sequence transform of A346091, or equally, of A346093.
For all i, j:
a(i) = a(j) => A006530(i) = A006530(j) [equally, A061395(i) = A061395(j)],
a(i) = a(j) => A329040(i) = A329040(j) => A351956(i) = A351956(j),
a(i) = a(j) => A329343(i) = A329343(j).
Interestingly, some of the rays in the scatter plot appear to be cut to discontinuous segments.

Links

Crossrefs

Cf. A108951, A328571, A346091, A346093.
Cf. also A006530, A061395, A329040, A329343, A351956, and A286621, A329045, A329345, A344594.

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; };
A002110(n) = prod(i=1,n,prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^f[i, 2]) };
A328571(n) = { my(m=1, p=2); while(n, m *= (p^!!(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A346091(n) = A328571(A108951(n));
v351955 = rgs_transform(vector(up_to, n, A346091(n)));
A351955(n) = v351955[n];
Previous Showing 21-28 of 28 results.

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