A356306 -id:A356306

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A356307

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

FORMULA

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

PROG

(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)));

CROSSREFS

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

Cf. also A356157.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 04 2022

STATUS

approved

A356156

The nearest common ancestor of n and gcd(n, sigma(n)) in the Doudna tree (A005940).

+10
4

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 10, 1, 3, 1, 2, 3, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 1, 1, 1, 3, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 5, 1, 2, 3, 2, 1, 2, 5, 2, 1, 2, 5, 12, 1, 1, 3, 1, 1, 3, 1, 2, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

FORMULA

a(n) = A348041(n, A009194(n)) = A348041(n, gcd(n, A000203(n))).

PROG

(PARI)

Abincompreflen(n, m) = { my(x=binary(n), y=binary(m), u=min(#x, #y)); for(i=1, u, if(x[i]!=y[i], return(i-1))); (u); };

Abinprefix(n, k) = { my(digs=binary(n)); fromdigits(vector(k, i, digs[i]), 2); };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };

A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552

A348040sq(x, y) = Abincompreflen(A156552(x), A156552(y));

A348041sq(x, y) = A005940(1+Abinprefix(A156552(x), A348040sq(x, y)));

A356156(n) = A348041sq(n, gcd(n, sigma(n)));

CROSSREFS

Cf. A000203, A007691 (fixed points), A009194, A348040, A348041.

Cf. also A347879, A356157, A356158, A356306.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jul 30 2022

STATUS

approved

A356301

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

FORMULA

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

PROG

(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));

A356301(n) = A356300sq(A161942(n), A000265(n));

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.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 03 2022

STATUS

approved

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.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

FORMULA

a(n) = gcd(n, A356301(n)).

PROG

(PARI)

A000265(n) = (n>>valuation(n, 2));