A324343 -id:A324343 - cp's OEIS frontend

A324400 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = -1 if n = 2^k and k > 0, and f(n) = n for all other numbers.

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

Offset: 1

Antti Karttunen, Mar 01 2019

nonn

In the following, A stands for this sequence, A324400, and S -> T (where S and T are sequence A-numbers) indicates that for all i, j >= 1: S(i) = S(i) => T(i) = T(j).
For example, the following chains of implications hold:
  A -> A286619 -> A005811,
and
  A -> A003602 -> A286622 -> A000120,
               -> A286378 -> A002487,
               -> A323889,
               -> A000593,
               -> A001227,
among many others.

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

Cf. A000523.
Cf. A000120, A000265, A000593, A001227, A002487, A003602, A005811, A209229, A286378, A286619, A286622, A294898, A297159, A318311, A323234, A323889, A323904, A324343, A324345 (some of the matching sequences).
Cf. also A103391, A295300, A305800, A305801, A324401.

A000523(n) = if(n<1, 0, #binary(n)-1);
A324400(n) = if(n<4,n,if(!bitand(n,n-1),2,1+n-A000523(n)));

If n <= 3, a(n) = n; and for n >= 4, if A209229(n) = 1, then a(n) = 2, otherwise a(n) = 1 + n - A000523(n).

A283477 If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 180, 360, 210, 420, 1260, 2520, 6300, 12600, 37800, 75600, 2310, 4620, 13860, 27720, 69300, 138600, 415800, 831600, 485100, 970200, 2910600, 5821200, 14553000, 29106000, 87318000, 174636000, 30030, 60060, 180180, 360360, 900900, 1801800, 5405400, 10810800, 6306300, 12612600, 37837800, 75675600

Offset: 0

Antti Karttunen, Mar 16 2017

nonn

a(n) = Product of distinct primorials larger than one, obtained as Product_{i} A002110(1+i), where i ranges over the zero-based positions of the 1-bits present in the binary representation of n.
This sequence can be represented as a binary tree. Each child to the left is obtained as A283980(k), and each child to the right is obtained as 2*A283980(k), when their parent contains k:
                                      1
                                      |
                   ...................2....................
                  6                                       12
       30......../ \........60                 180......../ \......360
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
  210       420      1260       2520     6300       12600   37800       75600
etc.

Cf. A129912 (same terms, but sorted into ascending order).
Cf. A000120, A001221, A001222, A002110, A005187, A005361, A006068, A007814, A007913, A019565, A029931, A030308, A046660, A048675, A051903, A056169, A065120, A070939, A072411, A090880, A097248, A108951, A124757, A248663, A276075, A276085, A283475, A283483, A283980, A283984, A283985, A284001, A284002, A284003, A284005, A324287, A324289, A324341, A324342, A324343.
Cf. A005940, A052330, A322827, A323505 for other similar trees.
Cf. also A260443.

Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]], {n, 0, 43}] (* Michael De Vlieger, Mar 18 2017 *)

A283477(n) = prod(i=0,exponent(n),if(bittest(n,i),vecprod(primes(1+i)),1)) \\ Edited by M. F. Hasler, Nov 11 2019

from sympy import prime, primerange, factorint
from operator import mul
from functools import reduce
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a108951(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # after Chai Wah Wu
def a(n): return a108951(a019565(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 22 2017

from sympy import primorial
from math import prod
def A283477(n): return prod(primorial(i) for i, b in enumerate(bin(n)[:1:-1],1) if b =='1') # Chai Wah Wu, Dec 08 2022

(define (A283477 n) (A108951 (A019565 n)))
;; Recursive "binary tree" implementation, using memoization-macro definec:
(definec (A283477 n) (cond ((zero? n) 1) ((even? n) (A283980 (A283477 (/ n 2)))) (else (* 2 (A283980 (A283477 (/ (- n 1) 2)))))))

a(0) = 1; a(2n) = A283980(a(n)), a(2n+1) = 2*A283980(a(n)).
Other identities. For all n >= 0 (or for n >= 1):
a(2n+1) = 2*a(2n).
a(n) = A108951(A019565(n)).
A097248(a(n)) = A283475(n).
A007814(a(n)) = A051903(a(n)) = A000120(n).
A001221(a(n)) = A070939(n).
A001222(a(n)) = A029931(n).
A048675(a(n)) = A005187(n).
A248663(a(n)) = A006068(n).
A090880(a(n)) = A283483(n).
A276075(a(n)) = A283984(n).
A276085(a(n)) = A283985(n).
A046660(a(n)) = A124757(n).
A056169(a(n)) = A065120(n). [seems to be]
A005361(a(n)) = A284001(n).
A072411(a(n)) = A284002(n).
A007913(a(n)) = A284003(n).
A000005(a(n)) = A284005(n).
A324286(a(n)) = A324287(n).
A276086(a(n)) = A324289(n).
A267263(a(n)) = A324341(n).
A276150(a(n)) = A324342(n). [subsequences in the latter are converging towards this sequence]
G.f.: Product_{k>=0} (1 + prime(k + 1)# * x^(2^k)), where prime()# = A002110. - Ilya Gutkovskiy, Aug 19 2019

More formulas and the binary tree illustration added by Antti Karttunen, Mar 19 2017

Four more linking formulas added by Antti Karttunen, Feb 25 2019

A324344 Lexicographically earliest positive sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A324342(i) = A324342(j), for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 24 2019

nonn

Restricted growth sequence transform of the ordered pair [A000120(n), A324342(n)].

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000120, A324342, A324343.

Cf. also A318310.

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; };
A002110(n) = prod(i=1,n,prime(i));
A030308(n,k) = bittest(n,k);
A283477(n) = prod(i=0,#binary(n),if(0==A030308(n,i),1,A030308(n,i)*A002110(1+i)));
A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A324342(n) = A276150(A283477(n));
A324344aux(n) = [hammingweight(n), A324342(n)];
v324344 = rgs_transform(vector(1+up_to,n,A324344aux(n-1)));
A324344(n) = v324344[1+n];

A324390 Lexicographically earliest positive sequence such that a(i) = a(j) => A278219(i) = A278219(j) and A324386(i) = A324386(j), for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 27 2019

nonn

Restricted growth sequence transform of the ordered pair [A278219(n), A324386(n)].

Cf. A278219, A324386.

Cf. also A286619, A324343, A324344, A324380 (compare the scatter-plots).

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; };
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));
A003188(n) = bitxor(n, n>>1);
A278219(n) = A278222(A003188(n));
Aux324390(n) = [A278219(n), A324386(n)]; \\ See code for A324386 in that entry.
v324390 = rgs_transform(vector(1+up_to,n,Aux324390(n-1)));
A324390(n) = v324390[1+n];

a(A000225(n)) = 2 for all n >= 1.

A324345 Lexicographically earliest positive sequence such that a(i) = a(j) => A005811(i) = A005811(j) and A278222(i) = A278222(j), for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 24 2019

nonn

Restricted growth sequence transform of the ordered pair [A005811(n), A278222(n)], or equally, of [A005811(n), A286622(n)].
For all i, j >= 1:
  a(i) = a(j) => A033264(i) = A033264(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A005811, A278222, A286622.

Cf. also A033264, A323889, A324343, A324346.

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; };
A005811(n) = hammingweight(bitxor(n, n>>1)); \\ From A005811
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));
Aux324345(n) = [A005811(n), A278222(n)];
v324345 = rgs_transform(vector(1+up_to,n,Aux324345(n-1)));
A324345(n) = v324345[1+n];

a(2^n) = 3 for all n >= 1.

A324380 Lexicographically earliest positive sequence such that a(i) = a(j) => A069010(i) = A069010(j) and A324386(i) = A324386(j), for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 27 2019

nonn

Restricted growth sequence transform of the ordered pair [A069010(n), A324386(n)].

Cf. A069010, A324386.

Cf. also A324343, A324344, A324390.

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; };
A069010(n) = ((1 + (hammingweight(bitxor(n, n>>1)))) >> 1); \\ From A069010
Aux324380(n) = [A069010(n), A324386(n)]; \\ Code for A324386 available in that entry.
v324380 = rgs_transform(vector(1+up_to,n,Aux324380(n-1)));
A324380(n) = v324380[1+n];

a(A000225(n)) = 2 for all n >= 1.

A324725 a(n) = sign(A324543(n)) * A001511(A324543(n)), with a(n) = 0 if A324543(n) = 0.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 4, 5, 1, -1, 1, 3, 1, 5, 1, 4, 1, 3, 3, 1, 1, 2, 1, 4, 3, 5, 3, 3, 1, 2, 2, 6, 1, -2, 1, 1, 1, 6, 1, 4, 1, -1, 2, 1, 1, 3, 1, 3, 2, 2, 1, 2, 1, 5, 3, 4, 4, 1, 1, 1, 3, -4, 1, 3, 1, 3, 1, 1, 3, -1, 1, 4, 5, 5, 1, 1, 2, 4, 2, 4, 1, 1, 1, 1, 7, 5, 2, 7, 1, -2, 1, 2, 1, 1, 1, 8, 2

Offset: 1

Antti Karttunen, Mar 17 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A001511, A324343, A324543, A324724, A324817, A324828.

A324543(n) = sumdiv(n,d,moebius(n/d)*A323243(d));
A001511ext(n) = if(!n,n,sign(n)*(1+valuation(n,2))); \\ Like A001511 but gives 0 for 0 and -A001511(-n) for negative numbers.
A324725(n) = A001511ext(A324543(n));

If A324543(n) = 0, then a(n) = 0, otherwise a(n) = sign(A324543(n)) * A001511(A324543(n)).
a(p) = 1 for all primes p.
A324828(n) = [1 == abs(a(n))], where [ ] is the Iverson bracket.

cp's OEIS Frontend

A324400 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = -1 if n = 2^k and k > 0, and f(n) = n for all other numbers.

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A283477 If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).

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A324344 Lexicographically earliest positive sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A324342(i) = A324342(j), for all i, j >= 0.

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A324390 Lexicographically earliest positive sequence such that a(i) = a(j) => A278219(i) = A278219(j) and A324386(i) = A324386(j), for all i, j >= 0.

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A324345 Lexicographically earliest positive sequence such that a(i) = a(j) => A005811(i) = A005811(j) and A278222(i) = A278222(j), for all i, j >= 0.

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A324380 Lexicographically earliest positive sequence such that a(i) = a(j) => A069010(i) = A069010(j) and A324386(i) = A324386(j), for all i, j >= 0.

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A324725 a(n) = sign(A324543(n)) * A001511(A324543(n)), with a(n) = 0 if A324543(n) = 0.

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