A322313 -id:A322313 - cp's OEIS frontend

A322312 a(n) = Product_{d|n, d+1 is prime} prime(1+A286561(n,d+1)), where A286561(n,k) gives the k-valuation of n (for k > 1).

2, 6, 2, 20, 2, 18, 2, 28, 2, 12, 2, 120, 2, 6, 2, 88, 2, 60, 2, 60, 2, 12, 2, 168, 2, 6, 2, 40, 2, 72, 2, 104, 2, 6, 2, 800, 2, 6, 2, 168, 2, 54, 2, 40, 2, 12, 2, 528, 2, 12, 2, 40, 2, 84, 2, 56, 2, 12, 2, 1440, 2, 6, 2, 136, 2, 72, 2, 20, 2, 24, 2, 2240, 2, 6, 2, 20, 2, 36, 2, 528, 2, 12, 2, 720, 2, 6, 2, 112, 2

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

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

Cf. A067513, A185633, A286561, A322313 (rgs-transform), A322314.

Cf. also A293514, A322310.

A322312(n) = { my(m=1,p); fordiv(n,d,p=1+d; if(isprime(p), for(i=0,oo,if(n%(p^i),m *= prime(i);break)))); (m); };

a(n) = Product_{d|n} A000040(1+A286561(n,1+d))^A010051(1+d).
a(n) = A181819(A185633(n)).
For all n, A001222(a(n)) = A067513(n).

A322315 Lexicographically earliest such sequence a that a(i) = a(j) => A185633(i) = A185633(j) for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Restricted growth sequence transform of A185633.

For all i, j: a(i) = a(j) => A322313(i) = A322313(j).

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

Cf. A185633, A322313.

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; };
A185633(n) = { my(m=1); fordiv(n, d, if(isprime(1+d), m *= (1+d)^(1+valuation(n,1+d)))); (m); };
v322315 = rgs_transform(vector(up_to, n, A185633(n)));
A322315(n) = v322315[n];

A322311 Lexicographically earliest such sequence a that a(i) = a(j) => A322310(i) = A322310(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 3, 3, 9, 3, 5, 3, 10, 3, 7, 3, 11, 3, 3, 3, 7, 3, 7, 3, 12, 3, 3, 3, 13, 3, 3, 3, 14, 3, 5, 3, 2, 3, 7, 3, 15, 3, 3, 3, 7, 3, 7, 3, 2, 3, 7, 3, 16, 3, 3, 3, 17, 3, 7, 3, 3, 3, 7, 3, 18, 3, 3, 3, 3, 3, 7, 3, 19, 3, 7, 3, 20, 3, 3, 3, 8, 3, 3, 3, 2, 3, 3, 3, 21, 3, 3, 3, 5, 3, 7, 3, 2, 3

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Restricted growth sequence transform of A322310.

For all i, j: a(i) = a(j) => A014197(i) = A014197(j).

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

Cf. A014197, A320000, A322310, A322313.

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; };
A320000sq(n, k) = if(1==n, if(1==k,2,1), sumdiv(n, d, if(d>=k && isprime(d+1), my(p=d+1, q=n/d); sum(i=0, valuation(n, p), A320000sq(q/(p^i), p))))); \\ Cf. A320000
A322310(n) = if(1==n,3,my(m=1); fordiv(n,d, my(s, p=d+1, q=n/d); if(isprime(p) && (s = sum(i=0,valuation(n, p), A320000sq(q/(p^i),p))), m *= prime(s))); (m));
v322311 = rgs_transform(vector(up_to, n, A322310(n)));
A322311(n) = v322311[n];

A322314 Lexicographically earliest such sequence a that a(i) = a(j) => A046523(i) = A046523(j) and A322312(i) = A322312(j), for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A322312(n)].

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

Cf. A046523, A322312, A322313.

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]); };  \\ From A046523
A322312(n) = { my(m=1,p); fordiv(n,d,p=1+d; if(isprime(p), for(i=0,oo,if(n%(p^i),m *= prime(i);break)))); (m); };
v322314 = rgs_transform(vector(up_to, n, [A046523(n), A322312(n)]));
A322314(n) = v322314[n];

cp's OEIS Frontend

A322312 a(n) = Product_{d|n, d+1 is prime} prime(1+A286561(n,d+1)), where A286561(n,k) gives the k-valuation of n (for k > 1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A322315 Lexicographically earliest such sequence a that a(i) = a(j) => A185633(i) = A185633(j) for all i, j.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A322311 Lexicographically earliest such sequence a that a(i) = a(j) => A322310(i) = A322310(j) for all i, j.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A322314 Lexicographically earliest such sequence a that a(i) = a(j) => A046523(i) = A046523(j) and A322312(i) = A322312(j), for all i, j.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI