A305897 -id:A305897 - cp's OEIS frontend

A101296 n has the a(n)-th distinct prime signature.

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

Offset: 1

David Wasserman, Dec 21 2004

From Antti Karttunen, May 12 2017: (Start)
Restricted growth sequence transform of A046523, the least representative of each prime signature. Thus this partitions the natural numbers to the same equivalence classes as A046523, i.e., for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j), and for that reason satisfies in that respect all the same conditions as A046523. For example, we have, for all i, j: if a(i) = a(j), then:
    A000005(i) = A000005(j), A008683(i) = A008683(j), A286605(i) = A286605(j).
So, this sequence (instead of A046523) can be used for finding sequences where a(n)'s value is dependent only on the prime signature of n, that is, only on the multiset of prime exponents in the factorization of n. (End)
This is also the restricted growth sequence transform of many other sequences, for example, that of A181819. See further comments there. - Antti Karttunen, Apr 30 2022

			From _David A. Corneth_, May 12 2017: (Start)
1 has prime signature (), the first distinct prime signature. Therefore, a(1) = 1.
2 has prime signature (1), the second distinct prime signature after (1). Therefore, a(2) = 2.
3 has prime signature (1), as does 2. Therefore, a(3) = a(2) = 2.
4 has prime signature (2), the third distinct prime signature after () and (1). Therefore, a(4) = 3. (End)
From _Antti Karttunen_, May 12 2017: (Start)
Construction of restricted growth sequences: In this case we start with a(1) = 1 for A046523(1) = 1, and thereafter, for all n > 1, we use the least so far unused natural number k for a(n) if A046523(n) has not been encountered before, otherwise [whenever A046523(n) = A046523(m), for some m < n], we set a(n) = a(m).
For n = 2, A046523(2) = 2, which has not been encountered before (first prime), thus we allot for a(2) the least so far unused number, which is 2, thus a(2) = 2.
For n = 3, A046523(2) = 2, which was already encountered as A046523(1), thus we set a(3) = a(2) = 2.
For n = 4, A046523(4) = 4, not encountered before (first square of prime), thus we allot for a(4) the least so far unused number, which is 3, thus a(4) = 3.
For n = 5, A046523(5) = 2, as for the first time encountered at n = 2, thus we set a(5) = a(2) = 2.
For n = 6, A046523(6) = 6, not encountered before (first semiprime pq with distinct p and q), thus we allot for a(6) the least so far unused number, which is 4, thus a(6) = 4.
For n = 8, A046523(8) = 8, not encountered before (first cube of a prime), thus we allot for a(8) the least so far unused number, which is 5, thus a(8) = 5.
For n = 9, A046523(9) = 4, as for the first time encountered at n = 4, thus a(9) = 3.
(End)
From _David A. Corneth_, May 12 2017: (Start)
(Rough) description of an algorithm of computing the sequence:
Suppose we want to compute a(n) for n in [1..20].
We set up a vector of 20 elements, values 0, and a number m = 1, the minimum number we haven't checked and c = 0, the number of distinct prime signatures we've found so far.
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
We check the prime signature of m and see that it's (). We increase c with 1 and set all elements up to 20 with prime signature () to 1. In the process, we adjust m. This gives:
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]. The least number we haven't checked is m = 2. 2 has prime signature (1). We increase c with 1 and set all elements up to 20 with prime signature (1) to 2. In the process, we adjust m. This gives:
[1, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0]
We check the prime signature of m = 4 and see that its prime signature is (2). We increase c with 1 and set all numbers up to 20 with prime signature (2) to 3. This gives:
[1, 2, 2, 3, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0]
Similarily, after m = 6, we get
[1, 2, 2, 3, 2, 4, 2, 0, 3, 4, 2, 0, 2, 4, 4, 0, 2, 0, 2, 0], after m = 8 we get:
[1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 0, 2, 4, 4, 0, 2, 0, 2, 0], after m = 12 we get:
[1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 0, 2, 6, 2, 0], after m = 16 we get:
[1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 0], after m = 20 we get:
[1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 8]. Now, m > 20 so we stop. (End)
The above method is inefficient, because the step "set all elements a(n) up to n = Nmax with prime signature s(n) = S[c] to c" requires factoring all integers up to Nmax (or at least comparing their signature, once computed, with S[c]) again and again. It is much more efficient to run only once over each m = 1..Nmax, compute its prime signature s(m), add it to an ordered list in case it did not occur earlier, together with its "rank" (= new size of the list), and assign that rank to a(m). The list of prime signatures is much shorter than [1..Nmax]. One can also use m'(m) := the smallest n with the prime signature of m (which is faster to compute than to search for the signature) as representative for s(m), and set a(m) := a(m'(m)). Then it is sufficient to have just one counter (number of prime signatures seen so far) as auxiliary variable, in addition to the sequence to be computed. - _M. F. Hasler_, Jul 18 2019

Michel Marcus (terms 1..10000) & Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A025487, A046523, A064839 (ordinal transform of this sequence), A181819, and arrays A095904, A179216.
Cf. A000005, A008683.
Sequences that are unions of finite number (>= 2) of equivalence classes determined by the values that this sequence obtains (i.e., sequences mentioned in David A. Corneth's May 12 2017 formula): A001358 (A001248 U A006881, values 3 & 4), A007422 (values 1, 4, 5), A007964 (2, 3, 4, 5), A014612 (5, 6, 9), A030513 (4, 5), A037143 (1, 2, 3, 4), A037144 (1, 2, 3, 4, 5, 6, 9), A080258 (6, 7), A084116 (2, 4, 5), A167171 (2, 4), A217856 (6, 9).
Cf. also A077462, A305897 (stricter variants, with finer partitioning) and A254524, A286603, A286605, A286610, A286619, A286621, A286622, A286626, A286378 for other similarly constructed sequences.

A101296 := proc(n)
    local a046523, a;
    a046523 := A046523(n) ;
    for a from 1 do
        if A025487(a) = a046523 then
            return a;
        elif A025487(a) > a046523 then
            return -1 ;
        end if;
    end do:
end proc: # R. J. Mathar, May 26 2017

With[{nn = 120}, Function[s, Table[Position[Keys@s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1], {n, nn}] ] (* Michael De Vlieger, May 12 2017, Version 10 *)

find(ps, vps) = {for (k=1, #vps, if (vps[k] == ps, return(k)););}
lisps(nn) = {vps = []; for (n=1, nn, ps = vecsort(factor(n)[,2]); ips = find(ps, vps); if (! ips, vps = concat(vps, ps); ips = #vps); print1(ips, ", "););} \\ Michel Marcus, Nov 15 2015; edited by M. F. Hasler, Jul 16 2019

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,rgs_transform(vector(100000,n,A046523(n))),"b101296.txt");
\\ Antti Karttunen, May 12 2017

A025487(a(n)) = A046523(n).
Indices of records give A025487. - Michel Marcus, Nov 16 2015
From David A. Corneth, May 12 2017: (Start) [Corresponding characteristic function in brackets]
a(A000012(n)) = 1 (sig.: ()).      [A063524]
a(A000040(n)) = 2 (sig.: (1)).     [A010051]
a(A001248(n)) = 3 (sig.: (2)).     [A302048]
a(A006881(n)) = 4 (sig.: (1,1)).   [A280710]
a(A030078(n)) = 5 (sig.: (3)).
a(A054753(n)) = 6 (sig.: (1,2)).   [A353472]
a(A030514(n)) = 7 (sig.: (4)).
a(A065036(n)) = 8 (sig.: (1,3)).
a(A007304(n)) = 9 (sig.: (1,1,1)). [A354926]
a(A050997(n)) = 10 (sig.: (5)).
a(A085986(n)) = 11 (sig.: (2,2)).
a(A178739(n)) = 12 (sig.: (1,4)).
a(A085987(n)) = 13 (sig.: (1,1,2)).
a(A030516(n)) = 14 (sig.: (6)).
a(A143610(n)) = 15 (sig.: (2,3)).
a(A178740(n)) = 16 (sig.: (1,5)).
a(A189975(n)) = 17 (sig.: (1,1,3)).
a(A092759(n)) = 18 (sig.: (7)).
a(A189988(n)) = 19 (sig.: (2,4)).
a(A179643(n)) = 20 (sig.: (1,2,2)).
a(A189987(n)) = 21 (sig.: (1,6)).
a(A046386(n)) = 22 (sig.: (1,1,1,1)).
a(A162142(n)) = 23 (sig.: (2,2,2)).
a(A179644(n)) = 24 (sig.: (1,1,4)).
a(A179645(n)) = 25 (sig.: (8)).
a(A179646(n)) = 26 (sig.: (2,5)).
a(A163569(n)) = 27 (sig.: (1,2,3)).
a(A179664(n)) = 28 (sig.: (1,7)).
a(A189982(n)) = 29 (sig.: (1,1,1,2)).
a(A179666(n)) = 30 (sig.: (3,4)).
a(A179667(n)) = 31 (sig.: (1,1,5)).
a(A179665(n)) = 32 (sig.: (9)).
a(A189990(n)) = 33 (sig.: (2,6)).
a(A179669(n)) = 34 (sig.: (1,2,4)).
a(A179668(n)) = 35 (sig.: (1,8)).
a(A179670(n)) = 36 (sig.: (1,1,1,3)).
a(A179671(n)) = 37 (sig.: (3,5)).
a(A162143(n)) = 38 (sig.: (2,2,2)).
a(A179672(n)) = 39 (sig.: (1,1,6)).
a(A030629(n)) = 40 (sig.: (10)).
a(A179688(n)) = 41 (sig.: (1,3,3)).
a(A179689(n)) = 42 (sig.: (2,7)).
a(A179690(n)) = 43 (sig.: (1,1,2,2)).
a(A189991(n)) = 44 (sig.: (4,4)).
a(A179691(n)) = 45 (sig.: (1,2,5)).
a(A179692(n)) = 46 (sig.: (1,9)).
a(A179693(n)) = 47 (sig.: (1,1,1,4)).
a(A179694(n)) = 48 (sig.: (3,6)).
a(A179695(n)) = 49 (sig.: (2,2,3)).
a(A179696(n)) = 50 (sig.: (1,1,7)).
(End)

Data section extended to 120 terms by Antti Karttunen, May 12 2017

Minor edits/corrections by M. F. Hasler, Jul 18 2019

A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 5, 11, 1, 12, 2, 13, 4, 14, 1, 15, 1, 16, 7, 17, 3, 18, 1, 19, 11, 20, 1, 21, 1, 22, 6, 23, 1, 24, 2, 25, 13, 26, 1, 27, 5, 28, 17, 29, 1, 30, 1, 31, 10, 32, 7, 33, 1, 34, 19, 35, 1, 36, 1, 37, 9, 38, 3, 39, 1, 40, 8, 41, 1, 42

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

If n >= 2, n occurs in column a(n) of A246278.

By convention, a(1) = 0 because 1 does not occur in A246278.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Terms of A348717 halved. A305897 is the restricted growth sequence transform.
Positions of terms 1 .. 8 in this sequence are given by the following sequences: A000040, A001248, A006094, A030078, A090076, A251720, A090090, A030514.
Cf. A078898 (has the same role with array A083221 as this sequence has with A246278).
Cf. A000040, A001222, A001358, A003961, A055396, A064989, A064216, A243055, A246272, A249810, A249820, A249735, A252463.
This sequence is also used in the definition of the following permutations: A246274, A246276, A246675, A246677, A246683, A249815, A249817 (A249818), A249823, A249825, A250244, A250245, A250247, A250249.
Also in the definition of arrays A249821, A251721, A251722.
Sum of prime indices of a(n) is A359358(n) + A001222(n) - 1, cf. A326844.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A005940, A241916, A253565, A356958, A358195.

a246277[n_Integer] := Module[{f, p, a064989, a},
  f[x_] := Transpose@FactorInteger[x];
  p[x_] := Which[
    x == 1, 1,
    x == 2, 1,
    True, NextPrime[x, -1]];
  a064989[x_] := Times @@ Power[p /@ First[f[x]], Last[f[x]]];
  a[1] = 0;
  a[x_] := If[EvenQ[x], x/2, NestWhile[a064989, x, OddQ]/2];
a/@Range[n]]; a246277[84] (* Michael De Vlieger, Dec 19 2014 *)

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)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };

A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2); \\ Antti Karttunen, Apr 30 2022

from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a064989(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
def a(n): return 0 if n==1 else n//2 if n%2==0 else a(a064989(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

;; two different variants, the second one employing memoizing definec-macro)
(define (A246277 n) (if (= 1 n) 0 (let loop ((n n)) (if (even? n) (/ n 2) (loop (A064989 n))))))
(definec (A246277 n) (cond ((= 1 n) 0) ((even? n) (/ n 2)) (else (A246277 (A064989 n)))))

a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)) = a(A064216(n+1)). [Cf. the formula for A252463.]
Instead of the equation for a(2n+1) above, we may write a(A003961(n)) = a(n). - Peter Munn, May 21 2022
Other identities. For all n >= 1, the following holds:
For all w >= 0, a(p_{i} * p_{j} * ... * p_{k}) = a(p_{i+w} * p_{j+w} * ... * p_{k+w}).
For all n >= 2, A001222(a(n)) = A001222(n)-1. [a(n) has one less prime factor than n. Thus each semiprime (A001358) is mapped to some prime (A000040), etc.]
For all n >= 2, a(n) = A078898(A249817(n)).
For semiprimes n = p_i * p_j, j >= i, a(n) = A000040(1+A243055(n)) = p_{1+j-i}.
a(n) = floor(A348717(n)/2). - Antti Karttunen, Apr 30 2022
If n has prime factorization Product_{i=1..k} prime(x_i), then a(n) = Product_{i=2..k} prime(x_i-x_1+1). The opposite version is A358195, prime indices A358172, even bisection A241916. - Gus Wiseman, Dec 29 2022

A348717 a(n) is the least k such that A003961^i(k) = n for some i >= 0 (where A003961^i denotes the i-th iterate of A003961).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 10, 2, 12, 2, 14, 6, 16, 2, 18, 2, 20, 10, 22, 2, 24, 4, 26, 8, 28, 2, 30, 2, 32, 14, 34, 6, 36, 2, 38, 22, 40, 2, 42, 2, 44, 12, 46, 2, 48, 4, 50, 26, 52, 2, 54, 10, 56, 34, 58, 2, 60, 2, 62, 20, 64, 14, 66, 2, 68, 38, 70, 2, 72, 2

Offset: 1

Rémy Sigrist, Oct 31 2021

nonn

All terms except a(1) = 1 are even.
To compute a(n) for n > 1:
- if n = Product_{j = 1..o} prime(p_j)^e_j (where prime(i) denotes the i-th prime number, p_1 < ... < p_o and e_1 > 0)
- then a(n) = Product_{j = 1..o} prime(p_j + 1 - p_1)^e_j.
This sequence has similarities with A304776: here we shift down prime indexes, there prime exponents.
Smallest number generated by uniformly decrementing the indices of the prime factors of n. Thus, for n > 1, the smallest m > 1 such that the first differences of the indices of the ordered prime factors (including repetitions) are the same for m and n. As a function, a(.) preserves properties such as prime signature. - Peter Munn, May 12 2022

Positions of particular values (see formula section): A000040, A001248, A006094, A030078, A030514, A046301, A050997, A090076, A090090, A166329, A251720.
Also see formula section for the relationship to: A000265, A003961, A004277, A005940, A020639, A046523, A055396, A071364, A122111, A156552, A243055, A243074, A297845, A322993.
Sequences with comparable definitions: A304776, A316437.
Cf. A246277 (terms halved), A305897 (restricted growth sequence transform), A354185 (Möbius transform), A354186 (Dirichlet inverse), A354187 (sum with it).

a[1] = 1; a[n_] := Module[{f = FactorInteger[n], d}, d = PrimePi[f[[1, 1]]] - 1; Times @@ ((Prime[PrimePi[#[[1]]] - d]^#[[2]]) & /@ f)]; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)

a(n) = { my (f=factor(n)); if (#f~>0, my (pi1=primepi(f[1,1])); for (k=1, #f~, f[k,1] = prime(primepi(f[k,1])-pi1+1))); factorback(f) }

a(n) = n iff n belongs to A004277.
A003961^(A055396(n)-1)(a(n)) = n for any n > 1.
a(n) = 2 iff n belongs to A000040 (prime numbers).
a(n) = 4 iff n belongs to A001248 (squares of prime numbers).
a(n) = 6 iff n belongs to A006094 (products of two successive prime numbers).
a(n) = 8 iff n belongs to A030078 (cubes of prime numbers).
a(n) = 10 iff n belongs to A090076.
a(n) = 12 iff n belongs to A251720.
a(n) = 14 iff n belongs to A090090.
a(n) = 16 iff n belongs to A030514.
a(n) = 30 iff n belongs to A046301.
a(n) = 32 iff n belongs to A050997.
a(n) = 36 iff n belongs to A166329.
a(1) = 1, for n > 1, a(n) = 2*A246277(n). - Antti Karttunen, Feb 23 2022
a(n) = A122111(A243074(A122111(n))). - Peter Munn, Feb 23 2022
From Peter Munn and Antti Karttunen, May 12 2022: (Start)
a(1) = 1; a(2n) = 2n; a(A003961(n)) = a(n). [complete definition]
a(n) = A005940(1+A322993(n)) = A005940(1+A000265(A156552(n))).
Equivalently, A156552(a(n)) = A000265(A156552(n)).
A297845(a(n), A020639(n)) = n.
A046523(a(n)) = A046523(n).
A071364(a(n)) = A071364(n).
a(n) >= A071364(n).
A243055(a(n)) = A243055(n).
(End)

A305800 Filter sequence for a(prime) = constant sequences.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 57, 58, 2, 59, 60, 61, 2, 62, 63, 64, 65, 66, 2, 67, 68, 69, 70, 71, 72, 73, 2, 74, 75, 76, 2, 77, 2, 78, 79, 80, 2, 81, 2, 82, 83, 84, 2, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96

Offset: 1

Antti Karttunen, Jun 14 2018

nonn

Restricted growth sequence transform of A239968.
In the following, A stands for this sequence, A305800, and S -> T (where S and T are sequence A-numbers) indicates that for all i, j: S(i) = S(i) => T(i) = T(j).
For example, the following implications hold:
  A -> A300247 -> A305897 -> A077462 -> A101296,
  A -> A290110 -> A300250 -> A101296.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000720, A239968.
Differs from A296073 for the first time at n=125, as a(125) = 96, while A296073(125) = 33.
Cf. also A305900, A305801, A295300, A289626 for other "upper level" filters.

Join[{1},Table[If[PrimeQ[n],2,1+n-PrimePi[n]],{n,2,150}]] (* Harvey P. Dale, Jul 12 2019 *)

A305800(n) = if(1==n,n,if(isprime(n),2,1+n-primepi(n)));

a(1) = 1; for n > 1, a(n) = 2 for prime n, and a(n) = 1+n-A000720(n) for composite n.

A322993 a(1) = 0; for n > 1, a(n) = A000265(A156552(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 3, 9, 1, 11, 1, 17, 5, 15, 1, 13, 1, 19, 9, 33, 1, 23, 3, 65, 7, 35, 1, 21, 1, 31, 17, 129, 5, 27, 1, 257, 33, 39, 1, 37, 1, 67, 11, 513, 1, 47, 3, 25, 65, 131, 1, 29, 9, 71, 129, 1025, 1, 43, 1, 2049, 19, 63, 17, 69, 1, 259, 257, 41, 1, 55, 1, 4097, 13, 515, 5, 133, 1, 79, 15, 8193, 1, 75, 33, 16385, 513, 135, 1, 45, 9

Offset: 1

Antti Karttunen, Jan 02 2019

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000265, A156552, A246277, A305897 (restricted growth sequence transform), A322994 (Möbius transform).

Cf. also A322995.

Array[#/2^IntegerExponent[#, 2] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 91] (* Michael De Vlieger, Jan 03 2019 *)

A000265(n) = (n/2^valuation(n, 2));
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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A322993(n) = if(1==n,0,A000265(A156552(n)));

a(1) = 0; for n > 1, a(n) = A000265(A156552(n)).
For n > 1, a(n) = A156552(2*A246277(n)).
A000120(a(n)) = A001222(n) for all n >= 1.

A350065 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350063(i) = A350063(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 5, 2, 3, 2, 3, 4, 5, 2, 3, 3, 5, 3, 5, 2, 5, 2, 3, 3, 5, 3, 6, 2, 3, 5, 5, 2, 3, 2, 3, 3, 7, 2, 3, 3, 4, 5, 3, 2, 3, 4, 3, 5, 8, 2, 3, 2, 5, 3, 8, 3, 5, 2, 5, 3, 3, 2, 5, 2, 5, 3, 5, 3, 5, 2, 3, 5, 5, 2, 8, 5, 9, 7, 7, 2, 8, 4, 5, 8, 10, 5, 5, 2, 4, 5, 5, 2, 8, 2, 3, 5

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Restricted growth sequence transform of A350063.
For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j) => A324117(i) = A324117(j).
For all i, j >= 2: a(i) = a(j) => A342656(i) = A342656(j).

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

Cf. A000265, A046523, A156552, A322993, A350063, A350064, A350067, A350068.

Cf. also A305897, A324117, A342656.

up_to = 3000;
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));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
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 };
A350063(n) = if(1==n,0,A046523(A000265(A156552(n))));
v350065 = rgs_transform(vector(up_to, n, A350063(n)));
A350065(n) = v350065[n];

\\ Version using the factorization file available at https://oeis.org/A156552/a156552.txt
v156552sigs = readvec("a156552.txt");
up_to = #v156552sigs;
A350063(n) = if(n<=2,n-1,my(es=v156552sigs[n][2]); if(n%2, es = vector(#es-1,i,es[1+i])); my(f=vecsort(es, , 4), p=0); prod(i=1, #f, (p=nextprime(p+1))^f[i]));
v350065 = rgs_transform(vector(up_to, n, A350063(n)));
A350065(n) = v350065[n]; \\ Antti Karttunen, Jan 29 2022

A350067 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A342666(n), A350063(n)] for n > 1, with f(1) = 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Restricted growth sequence transform of the ordered pair [A342666(n), A350063(n)], when assuming that A342666(1) = 0.
Restricted growth sequence transform of the function f(1) = 0, f(n) = A336470(A156552(n)) for n > 1.
For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j) => A350065(i) = A350065(j).
For all i, j >= 2: a(i) = a(j) => A342651(i) = A342651(j).

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

Cf. A156552, A322993, A336466, A342666, A350063, A336470.

Cf. also A305897, A350065, A342651.

up_to = 3003;
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));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
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 };
A336466(n) = { my(f=factor(n)); prod(k=1, #f~, A000265(f[k, 1]-1)^f[k, 2]); };
Aux350067(n) = if(1==n,1,my(u=A000265(A156552(n))); [A046523(u),A336466(u)]);
v350067 = rgs_transform(vector(up_to, n, Aux350067(n)));
A350067(n) = v350067[n];

A352898 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A046523(n), A352892(n)], except f(n) = -n when <= 2.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 08 2022

nonn

For all i, j: A305801(i) = A305801(j) => a(i) = a(j) => A352897(i) = A352897(j).

Cf. A046523, A101296, A305801, A305897, A352892, A352897, A352899.

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); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
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)};
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 };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A352892(n) = A348717(A341515(n));
Aux352898(n) = if(n<=2,-n,[A046523(n),A352892(n)]);
v352898 = rgs_transform(vector(up_to, n, Aux352898(n)));
A352898(n) = v352898[n];

A355833 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A348717(i) = A348717(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A342671(n), A348717(n)].

Terms that occur in positions given by A349166 may occur only a finite number of times in this sequence. See also the array A355924.

			a(100) = a(3025) [= 66 as allotted by the rgs-transform] because 3025 = A003961(A003961(100)), therefore it is in the same column of the prime shift array A246278 as 100 is], and as A342671(100) = A342671(3025) = 7.
a(300) = a(21175) [= 200 as allotted by the rgs-transform], as 21175 = A003961(A003961(300)) and as A342671(300) = A342671(21175) = 7.
a(1215) = a(21875) [= 831 as allotted by the rgs-transform] because 21875 = A003961(1215), therefore it is in the same column of the prime shift array A246278 as 1215 is, and as A342671(1215) = A342671(21875) = 7.
a(2835) = a(48125) [= 1953 as allotted by the rgs-transform] because 48125 = A003961(2835) and as A342671(2835) = A342671(48125) = 11.

Cf. A000203, A003961, A246278, A342671, A348717, A349165, A349166, A355924.

Cf. also A305801, A305897, A355834, A355835.

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; };
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
Aux355833(n) = [A342671(n), A348717(n)];
v355833 = rgs_transform(vector(up_to,n,Aux355833(n)));
A355833(n) = v355833[n];

cp's OEIS Frontend

A318500 Filter sequence combining A305897 and the parity of n.

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A101296 n has the a(n)-th distinct prime signature.

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A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)).

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A348717 a(n) is the least k such that A003961^i(k) = n for some i >= 0 (where A003961^i denotes the i-th iterate of A003961).

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A305800 Filter sequence for a(prime) = constant sequences.

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A322993 a(1) = 0; for n > 1, a(n) = A000265(A156552(n)).

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A350065 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350063(i) = A350063(j), for all i, j >= 1.

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