A336149 -id:A336149 - cp's OEIS frontend

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.

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];

A336146 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000035(i) = A000035(j) and A000265(i) = A000265(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 5, 10, 11, 12, 2, 13, 14, 15, 8, 16, 17, 18, 5, 19, 20, 21, 11, 22, 23, 24, 2, 25, 26, 27, 14, 28, 29, 30, 8, 31, 32, 33, 17, 34, 35, 36, 5, 37, 38, 39, 20, 40, 41, 42, 11, 43, 44, 45, 23, 46, 47, 48, 2, 49, 50, 51, 26, 52, 53, 54, 14, 55, 56, 57, 29, 58, 59, 60, 8, 61, 62, 63, 32, 64, 65, 66, 17, 67, 68, 69, 35, 70, 71, 72, 5, 73, 74, 75, 38, 76, 77, 78, 20, 79

Offset: 1

Antti Karttunen, Jul 12 2020

nonn

Restricted growth sequence transform of the ordered pair [A000035(n), A000265(n)] (parity and the odd part of n), or equally, of the ordered pair [A000265(n), A278221(n)].
For all i, j:
  A324400(i) = A324400(j) => a(i) = a(j),
  a(i) = a(j) => A336126(i) = A336126(j),
  a(i) = a(j) => A336147(i) = A336147(j),
  a(i) = a(j) => A336148(i) = A336148(j),
  a(i) = a(j) => A336149(i) = A336149(j).

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

Cf. A000035, A000265, A046523, A064989, A122111, A278221.

Cf. also A286621, A324400, A336126, A336147, A336148, A336149.

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; };
A000035(n) = (n%2);
A000265(n) = (n>>valuation(n,2));
Aux336146(n) = [A000035(n), A000265(n)];
v336146 = rgs_transform(vector(up_to, n, Aux336146(n)));
A336146(n) = v336146[n];

A336148 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278221(i) = A278221(j) and A336158(i) = A336158(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 5, 10, 11, 12, 2, 13, 14, 15, 8, 16, 17, 18, 5, 19, 20, 21, 11, 22, 23, 24, 2, 25, 26, 27, 14, 28, 29, 30, 8, 31, 32, 33, 17, 34, 35, 36, 5, 37, 38, 39, 20, 40, 41, 25, 11, 42, 43, 44, 23, 45, 46, 47, 2, 48, 49, 50, 26, 51, 32, 52, 14, 53, 54, 34, 29, 55, 56, 57, 8, 58, 59, 60, 32, 61, 62, 63, 17, 64, 65, 30, 35, 66, 67, 68, 5, 69, 70, 71, 38, 72, 73, 74, 20, 75

Offset: 1

Antti Karttunen, Jul 12 2020

nonn

Restricted growth sequence transform of the ordered pair [A278221(n), A336158(n)], i.e., of the ordered pair [A046523(A122111(n)), A046523(A000265(n))].

For all i, j: A324400(i) = A324400(j) => A336146(i) = A336146(j) => a(i) = a(j).

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

Cf. A000265, A046523, A064989, A122111, A278221, A336158.

Cf. also A286621, A318890, A324400, A336146, A336149, A336159.

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; };
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));
A000265(n) = (n>>valuation(n,2));
A336158(n) = A046523(A000265(n));
Aux336148(n) = [A278221(n),A336158(n)];
v336148 = rgs_transform(vector(up_to, n, Aux336148(n)));
A336148(n) = v336148[n];

A035531 a(n) = A000120(n) + A001221(n) - 1.

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 3, 1, 2, 3, 3, 3, 3, 4, 5, 1, 2, 3, 3, 3, 4, 4, 4, 3, 3, 4, 4, 4, 4, 6, 5, 1, 3, 3, 4, 3, 3, 4, 5, 3, 3, 5, 4, 4, 5, 5, 5, 3, 3, 4, 5, 4, 4, 5, 6, 4, 5, 5, 5, 6, 5, 6, 7, 1, 3, 4, 3, 3, 4, 5, 4, 3, 3, 4, 5, 4, 5, 6, 5, 3, 3, 4, 4, 5, 5, 5, 6, 4, 4, 6, 6, 5, 6, 6, 7, 3, 3, 4, 5, 4, 4, 6, 5, 4, 6, 5, 5, 5, 5, 7, 7

Offset: 1

Daniele Parisse

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from G. C. Greubel)

Cf. A000120, A001221.

Cf. also A336149.

A035531 := proc(n)
    A000120(n)+A001221(n)-1 ;
end proc:
seq(A035531(n),n=1..100) ; # R. J. Mathar, Mar 12 2018

Table[DigitCount[n, 2, 1] + PrimeNu[n] - 1, {n, 1, 100}] (* G. C. Greubel, Apr 24 2017 *)

a(n) = hammingweight(n) + omega(n) - 1; \\ Michel Marcus, Apr 25 2017

from sympy import primefactors
def a(n): return 0 if n<2 else bin(n)[2:].count("1") + len(primefactors(n)) - 1 # Indranil Ghosh, Apr 25 2017

G.f.: Sum a(n) x^n = Sum A000120(p)*x^p/(1-x^p), p = prime.

More terms from David W. Wilson.

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

Antti Karttunen, Jul 12 2020

nonn

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).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

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.

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];

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A336146 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000035(i) = A000035(j) and A000265(i) = A000265(j), for all i, j >= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A336148 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278221(i) = A278221(j) and A336158(i) = A336158(j), for all i, j >= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A035531 a(n) = A000120(n) + A001221(n) - 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI