A336156 -id:A336156 - cp's OEIS frontend

A336158 The least number with the prime signature of the odd part of n: a(n) = A046523(A000265(n)).

1, 1, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 6, 1, 2, 4, 2, 2, 6, 2, 2, 2, 4, 2, 8, 2, 2, 6, 2, 1, 6, 2, 6, 4, 2, 2, 6, 2, 2, 6, 2, 2, 12, 2, 2, 2, 4, 4, 6, 2, 2, 8, 6, 2, 6, 2, 2, 6, 2, 2, 12, 1, 6, 6, 2, 2, 6, 6, 2, 4, 2, 2, 12, 2, 6, 6, 2, 2, 16, 2, 2, 6, 6, 2, 6, 2, 2, 12, 6, 2, 6, 2, 6, 2, 2, 4, 12, 4, 2, 6, 2, 2, 30

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

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

Cf. A000265, A001227, A005087, A046523, A064989, A087436, A336156, A336157, A336159, A336160.

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
A336158(n) = A046523(A000265(n));

from math import prod
from sympy import factorint, prime
def A336158(n): return prod(prime(i+1)**e for i,e in enumerate(sorted(factorint(n>>(~n&n-1).bit_length()).values(),reverse=True))) # Chai Wah Wu, Sep 16 2022

a(n) = A046523(A000265(n)) = A046523(A064989(n)).
A000005(a(n)) = A001227(n).
A001221(a(n)) = A005087(n).
A001222(a(n)) = A087436(n).

A336159 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A336158(i) = A336158(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 11 2020

Restricted growth sequence transform of the ordered pair [A278222(n), A336158(n)], i.e., of the ordered pair [A046523(A005940(1+n)), A046523(A000265(n))].

For all i, j: A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j).

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

Cf. A000265, A005940, A046523, A278222, A336158.

Cf. also A003602, A286163, A286622, A291762, A324400, A336149, A336156, A336157, A336160, A336162.

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));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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));
A336158(n) = A046523(A000265(n));
Aux336159(n) = [A278222(n), A336158(n)];
v336159 = rgs_transform(vector(up_to, n, Aux336159(n)));
A336159(n) = v336159[n];

A336155 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A335915(i) = A335915(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 2, 1, 6, 7, 2, 8, 6, 9, 2, 10, 2, 11, 6, 4, 12, 13, 2, 10, 14, 3, 6, 15, 6, 16, 2, 17, 18, 19, 2, 20, 21, 22, 6, 15, 6, 23, 12, 4, 24, 25, 2, 10, 18, 19, 14, 26, 2, 27, 6, 28, 29, 30, 6, 31, 12, 32, 2, 33, 12, 34, 18, 35, 18, 36, 2, 37, 38, 19, 21, 28, 14, 39, 6, 1, 29, 40, 6, 41, 42, 43, 12, 44, 6, 45, 24, 17, 46, 47, 2, 48, 18, 7, 18, 49, 18, 50, 14, 10

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A007814(1+n), A335915(n)]. Note that A007814(1+n) gives the number of trailing 1-bits in the binary expansion of n.

For all i, j: A324400(i) = A324400(j) => a(i) = a(j).

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

Cf. A000265, A007814, A335915.

Cf. also A324400, A336154, A336156, A336162.

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; };
A007814(n) = valuation(n,2);
A000265(n) = (n>>valuation(n,2));
A335915(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1)*A000265(f[k,1]+1))^f[k,2])); };
Aux336155(n) = [A007814(1+n), A335915(n)];
v336155 = rgs_transform(vector(up_to, n, Aux336155(n)));
A336155(n) = v336155[n];

A336152 Lexicographically earliest infinite sequence such that a(i) = a(j) => A001221(i) = A001221(j) and A007814(1+i) = A007814(1+j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A001221(n), A007814(1+n)]. The first member of pair gives the number of distinct prime divisors of n, and the second member gives the number of trailing 1-bits in its binary expansion.

For all i, j: A324400(i) = A324400(j) => a(i) = a(j).

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

Cf. A001221, A007814.

Cf. also A324400, A336150, A336151, A336156.

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; };
A007814(n) = valuation(n,2);
Aux336152(n) = [omega(n), A007814(1+n)];
v336152 = rgs_transform(vector(up_to, n, Aux336152(n)));
A336152(n) = v336152[n];

A336154 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A278222(i) = A278222(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A007814(1+n), A278222(n)]. Note that A007814(1+n) gives the number of trailing 1-bits in the binary expansion of n.

For all i, j: A324400(i) = A324400(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A007814, A278222.

Cf. also A286622, A324400, A336153, A336155, A336156.

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; };
A007814(n) = valuation(n,2);
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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));
Aux336154(n) = [A007814(1+n), A278222(n)];
v336154 = rgs_transform(vector(up_to, n, Aux336154(n)));
A336154(n) = v336154[n];

A336153 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A009194(i) = A009194(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 5, 2, 1, 6, 3, 7, 1, 6, 8, 2, 1, 9, 3, 6, 1, 6, 5, 10, 1, 6, 3, 11, 1, 4, 12, 2, 13, 6, 3, 2, 1, 6, 5, 14, 1, 4, 3, 7, 13, 6, 15, 7, 1, 2, 16, 6, 1, 4, 5, 17, 1, 6, 3, 10, 1, 6, 18, 2, 1, 4, 3, 6, 13, 6, 5, 9, 1, 6, 3, 7, 1, 4, 15, 6, 1, 6, 3, 11, 1, 6, 19, 7, 1, 20, 21, 7, 1, 6, 22, 10, 1, 2, 16, 2, 1, 4, 5, 6, 13

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A007814(1+n), A009194(n)]. Note that A007814(1+n) gives the number of trailing 1-bits in the binary expansion of n.

For all i, j: A324400(i) = A324400(j) => a(i) = a(j).

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

Cf. A007814, A009194.

Cf. also A324400, A331747, A336154, A336155, A336156.

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; };
A007814(n) = valuation(n,2);
A009194(n) = gcd(n, sigma(n));
Aux336153(n) = [A007814(1+n), A009194(n)];
v336153 = rgs_transform(vector(up_to, n, Aux336153(n)));
A336153(n) = v336153[n];

cp's OEIS Frontend

A336158 The least number with the prime signature of the odd part of n: a(n) = A046523(A000265(n)).

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A336159 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A336158(i) = A336158(j), for all i, j >= 1.

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A336155 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A335915(i) = A335915(j), for all i, j >= 1.

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A336152 Lexicographically earliest infinite sequence such that a(i) = a(j) => A001221(i) = A001221(j) and A007814(1+i) = A007814(1+j), for all i, j >= 1.

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A336154 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A278222(i) = A278222(j), for all i, j >= 1.

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A336153 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A009194(i) = A009194(j), for all i, j >= 1.

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