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.

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A356307 The nearest common ancestor of A161942(n) and gcd(A000265(n), sigma(n)) in the A253563-tree.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 1, 1, 3, 1, 1, 9, 7, 1, 1, 1, 3, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3
Offset: 1

Views

Author

Antti Karttunen, Aug 04 2022

Keywords

Links

Crossrefs

Cf. A000203, A000265, A161942, A253553, A355931, A356300, A356306, A356307.
Cf. also A356157.

Programs

  • PARI
    A000265(n) = (n>>valuation(n,2));
A161942(n) = A000265(sigma(n));
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A356300sq(x,y) = if(1==x||1==y,1, my(lista=List([]), i, k=x, stemvec, stemlen, h=y); while(k>1, listput(lista,k); k = A253553(k)); stemvec = Vecrev(Vec(lista)); stemlen = #stemvec; while(1, if((i=vecsearch(stemvec,h))>0, return(stemvec[i])); h = A253553(h)));
A356307(n) = A356300sq(A161942(n), gcd(n, A161942(n)));

Formula

a(n) = A356300(A161942(n), A355931(n)) = A356300(A161942(n), gcd(n, A161942(n))).

A336313 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A336125(i)) = A278222(A336125(j)) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 4, 3, 2, 2, 2, 3, 3, 2, 2, 5, 2, 2, 3, 3, 2, 2, 2, 3, 4, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 3, 3, 2, 2, 5, 2, 2, 4, 5, 2, 2, 2, 3, 3, 3, 2, 5, 2, 2, 3, 3, 3, 2, 2, 3, 6, 2, 2, 5, 2, 2, 3, 3, 2, 2, 3, 3, 3, 2, 2, 5, 2, 2, 4, 3, 2, 2, 2, 3, 3
Offset: 1

Views

Author

Antti Karttunen, Jul 17 2020

Keywords

Comments

Restricted growth sequence transform of A278222(A336125(n)).
For all i, j: A336311(i) = A336311(j) => a(i) = a(j) => A336123(i) = A336123(j).

Links

Crossrefs

Cf. A253553, A278222, A336124, A336125.
Cf. also A292585, A336311, A336312, A336123.

Programs

  • PARI
    \\ Needs also code from A336124:
up_to = 1024; \\ 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; };
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A336125(n) = if(n<=2,n-1,(1==A336124(n))+(2*A336125(A253553(n))));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
v336313 = rgs_transform(vector(up_to,n,A278222(A336125(n))));
A336313(n) = v336313[n];

A356301 The nearest common ancestor of A000265(sigma(n)) and A000265(n) in the tree depicted in A253563.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 1, 3, 3, 3, 3, 7, 3, 3, 1, 3, 9, 5, 3, 1, 3, 3, 3, 5, 3, 3, 7, 3, 9, 1, 1, 3, 3, 3, 3, 19, 3, 3, 3, 3, 3, 11, 3, 9, 3, 3, 3, 3, 3, 9, 7, 3, 9, 3, 3, 3, 3, 3, 15, 31, 3, 3, 1, 3, 9, 17, 3, 3, 3, 3, 9, 37, 3, 3, 5, 3, 21, 5, 3, 3, 3, 3, 3, 3, 3, 15, 3, 3, 45, 7, 3, 1, 3, 3, 3, 7, 3, 9, 5, 3, 9, 13, 3, 3
Offset: 1

Views

Author

Antti Karttunen, Aug 03 2022

Keywords

Links

Crossrefs

Cf. A000203, A000265, A161942, A356300, A356301, A356306, A356307, A356308.
Cf. also A347879.
Positions of 1's in this sequence is given by the union of A000079 and A046528.

Programs

  • PARI
    A000265(n) = (n>>valuation(n,2));
A161942(n) = A000265(sigma(n));
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A356300sq(x,y) = if(1==x||1==y,1, my(lista=List([]), i, k=x, stemvec, stemlen, h=y); while(k>1, listput(lista,k); k = A253553(k)); stemvec = Vecrev(Vec(lista)); stemlen = #stemvec; while(1, if((i=vecsearch(stemvec,h))>0, return(stemvec[i])); h = A253553(h)));
A356301(n) = A356300sq(A161942(n),A000265(n));

Formula

a(n) = A356300(A161942(n), A000265(n)).

A356308 a(n) = gcd(n, A356301(n)), where A356301(n) is the nearest common ancestor of A000265(sigma(n)) and A000265(n) in the A253563-tree.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 1, 1, 1, 3, 5, 1, 3, 7, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 15, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 45, 7, 1, 1, 1, 1, 3, 1, 1, 9, 5, 1, 3, 1, 1, 3
Offset: 1

Views

Author

Antti Karttunen, Aug 04 2022

Keywords

Links

Crossrefs

Cf. A000203, A000265, A161942, A253553, A356300, A356301, A356306, A356307.
Cf. also A356158.

Programs

  • PARI
    A000265(n) = (n>>valuation(n,2));
A161942(n) = A000265(sigma(n));
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A356300sq(x,y) = if(1==x||1==y,1, my(lista=List([]), i, k=x, stemvec, stemlen, h=y); while(k>1, listput(lista,k); k = A253553(k)); stemvec = Vecrev(Vec(lista)); stemlen = #stemvec; while(1, if((i=vecsearch(stemvec,h))>0, return(stemvec[i])); h = A253553(h)));
A356301(n) = A356300sq(A161942(n),A000265(n));
A356308(n) = gcd(n, A356301(n));

Formula

a(n) = gcd(n, A356301(n)).
Previous Showing 11-14 of 14 results.

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