A332824 -id:A332824 - cp's OEIS frontend

A332826 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A332824(i)) = A046523(A332824(j)) for all i, j.

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

Offset: 1

Antti Karttunen, Feb 25 2020

nonn

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

Cf. A000010, A019565, A046523, A332824, A332827.
Cf. A019434 (positions of 3's).
Cf. also A318835.

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; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A332824(n) = { my(m=1); fordiv(n,d,m *= A019565(eulerphi(d))); (m); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
v332826 = rgs_transform(vector(up_to,n,A046523(A332824(n))));
A332826(n) = v332826[n];

A332827 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A003557(A332824(n)) for all other numbers, except f(1) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 25 2020

nonn

For all i, j:

A305801(i) = A305801(j) => a(i) = a(j).

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

Cf. A000010, A003557, A019565, A305801, A332824, A332826.

Cf. also A324538.

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; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A332824(n) = { my(m=1); fordiv(n,d,m *= A019565(eulerphi(d))); (m); };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
Aux332827(n) = if(1==n,0,A003557(A332824(n)));
v332827 = rgs_transform(vector(up_to,n,Aux332827(n)));
A332827(n) = v332827[n];

A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 8, 3, 4, 5, 16, 4, 32, 9, 6, 4, 64, 5, 128, 6, 10, 17, 256, 5, 8, 33, 6, 10, 512, 7, 1024, 5, 18, 65, 12, 6, 2048, 129, 34, 7, 4096, 11, 8192, 18, 8, 257, 16384, 6, 16, 9, 66, 34, 32768, 7, 20, 11, 130, 513, 65536, 8, 131072, 1025, 12, 6, 36, 19

Offset: 1

Antti Karttunen, Jul 14 1999

nonn

The original motivation for this sequence was to encode the prime factorization of n in the binary representation of a(n), each such representation being unique as long as this map is restricted to A005117 (squarefree numbers, resulting a permutation of nonnegative integers A048672) or any of its subsequence, resulting an injective function like A048623 and A048639.
However, also the restriction to A260443 (not all terms of which are squarefree) results a permutation of nonnegative integers, namely A001477, the identity permutation.
When a polynomial with nonnegative integer coefficients is encoded with the prime factorization of n (e.g., as in A206296, A260443), then a(n) gives the evaluation of that polynomial at x=2.
The primitive completely additive integer sequence that satisfies a(n) = a(A225546(n)), n >= 1. By primitive, we mean that if b is another such sequence, then there is an integer k such that b(n) = k * a(n) for all n >= 1. - Peter Munn, Feb 03 2020
If the binary rank of an integer partition y is given by Sum_i 2^(y_i-1), and the Heinz number is Product_i prime(y_i), then a(n) is the binary rank of the integer partition with Heinz number n. Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices), and the function taking a multiset m to Product_i prime(m_i) is the inverse of A112798 (prime indices). - Gus Wiseman, May 22 2024

			From _Gus Wiseman_, May 22 2024: (Start)
The A018819(7) = 6 cases of binary rank 7 are the following, together with their prime indices:
   30: {1,2,3}
   40: {1,1,1,3}
   54: {1,2,2,2}
   72: {1,1,1,2,2}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
(End)

Antti Karttunen, Table of n, a(n) for n = 1..1024
Hans Havermann, Factorization of the first 10000 terms, in format [[primes], [exponents]]

Row 2 of A104244.
Similar logarithmic functions: A001414, A056239, A090880, A289506, A293447.
Left inverse of the following sequences: A000079, A019565, A038754, A068911, A134683, A260443, A332824.
A003961, A028234, A032742, A055396, A064989, A067029, A225546, A297845 are used to express relationship between terms of this sequence.
Cf. also A048623, A048676, A099884, A277896 and tables A277905, A285325.
Cf. A297108 (Möbius transform), A332813 and A332823 [= a(n) mod 3].
Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000203,A331750), (A005940,A087808), (A007913,A248663), (A007947,A087207), (A097248,A048675), (A206296,A000129), (A248692,A056239), (A283477,A005187), (A284003,A006068), (A285101,A028362), (A285102,A068052), (A293214,A001065), (A318834,A051953), (A319991,A293897), (A319992,A293898), (A320017,A318674), (A329352,A069359), (A332461,A156552), (A332462,A156552), (A332825,A000010) and apparently (A163511,A135529).
See comments/formulas in A277333, A331591, A331740 giving their relationship to this sequence.
The formula section details how the sequence maps the terms of A329050, A329332.
A277892, A322812, A322869, A324573, A324575 give properties of the n-th term of this sequence.
The term k appears A018819(k) times.
The inverse transformation is A019565 (Heinz number of binary indices).
The version for distinct prime indices is A087207.
Numbers k such that a(k) is prime are A277319, counts A372688.
Grouping by image gives A277905.
A014499 lists binary indices of prime numbers.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A000720, A035100, A071814, A096111, A225620, A243055, A304818, A355536, A358136, A372890.
Cf. A087207, A097248.

nthprime := proc(n) local i; if(isprime(n)) then for i from 1 to 1000000 do if(ithprime(i) = n) then RETURN(i); fi; od; else RETURN(0); fi; end; # nthprime(2) = 1, nthprime(3) = 2, nthprime(5) = 3, etc. - this is also A049084.
A048675 := proc(n) local s,d; s := 0; for d in ifactors(n)[ 2 ] do s := s + d[ 2 ]*(2^(nthprime(d[ 1 ])-1)); od; RETURN(s); end;
# simpler alternative
f:= n -> add(2^(numtheory:-pi(t[1])-1)*t[2], t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Oct 10 2016

a[1] = 0; a[n_] := Total[ #[[2]]*2^(PrimePi[#[[1]]]-1)& /@ FactorInteger[n] ]; Array[a, 100] (* Jean-François Alcover, Mar 15 2016 *)

a(n) = my(f = factor(n)); sum(k=1, #f~, f[k,2]*2^primepi(f[k,1]))/2; \\ Michel Marcus, Oct 10 2016

\\ The following program reconstructs terms (e.g. for checking purposes) from the factorization file prepared by Hans Havermann:
v048675sigs = readvec("a048675.txt");
A048675(n) = if(n<=2,n-1,my(prsig=v048675sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,ps[i]^es[i])); \\ Antti Karttunen, Feb 02 2020

from sympy import factorint, primepi
def a(n):
    if n==1: return 0
    f=factorint(n)
    return sum([f[i]*2**(primepi(i) - 1) for i in f])
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jun 19 2017

a(1) = 0, a(n) = 1/2 * (e1*2^i1 + e2*2^i2 + ... + ez*2^iz) if n = p_{i1}^e1*p_{i2}^e2*...*p_{iz}^ez, where p_i is the i-th prime. (e.g. p_1 = 2, p_2 = 3).
Totally additive with a(p^e) = e * 2^(PrimePi(p)-1), where PrimePi(n) = A000720(n). [Missing factor e added to the comment by Antti Karttunen, Jul 29 2015]
From Antti Karttunen, Jul 29 2015: (Start)
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A032742(n)). [Where A055396(n) gives the index of the smallest prime dividing n and A032742(n) gives the largest proper divisor of n.]
a(1) = 0; for n > 1, a(n) = (A067029(n) * (2^(A055396(n)-1))) + a(A028234(n)).
Other identities. For all n >= 0:
a(A019565(n)) = n.
a(A260443(n)) = n.
a(A206296(n)) = A000129(n).
a(A005940(n+1)) = A087808(n).
a(A007913(n)) = A248663(n).
a(A007947(n)) = A087207(n).
a(A283477(n)) = A005187(n).
a(A284003(n)) = A006068(n).
a(A285101(n)) = A028362(1+n).
a(A285102(n)) = A068052(n).
Also, it seems that a(A163511(n)) = A135529(n) for n >= 1. (End)
a(1) = 0, a(2n) = 1+a(n), a(2n+1) = 2*a(A064989(2n+1)). - Antti Karttunen, Oct 11 2016
From Peter Munn, Jan 31 2020: (Start)
a(n^2) = a(A003961(n)) = 2 * a(n).
a(A297845(n,k)) = a(n) * a(k).
a(n) = a(A225546(n)).
a(A329332(n,k)) = n * k.
a(A329050(n,k)) = 2^(n+k).
(End)
From Antti Karttunen, Feb 02-25 2020, Feb 01 2021: (Start)
a(n) = Sum_{d|n} A297108(d) = Sum_{d|A225546(n)} A297108(d).
a(n) = a(A097248(n)).
For n >= 2:
A001221(a(n)) = A322812(n), A001222(a(n)) = A277892(n).
A000203(a(n)) = A324573(n), A033879(a(n)) = A324575(n).
For n >= 1, A331750(n) = a(A000203(n)).
For n >= 1, the following chains hold:
A293447(n) >= a(n) >= A331740(n) >= A331591(n).
a(n) >= A087207(n) >= A248663(n).
(End)
a(n) = A087207(A097248(n)). - Flávio V. Fernandes, Jul 16 2025

Entry revised by Antti Karttunen, Jul 29 2015

More linking formulas added by Antti Karttunen, Apr 18 2017

A097248 a(n) is the eventual stable point reached when iterating k -> A097246(k), starting from k = n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 6, 5, 10, 11, 5, 13, 14, 15, 5, 17, 10, 19, 15, 21, 22, 23, 10, 7, 26, 15, 21, 29, 30, 31, 10, 33, 34, 35, 15, 37, 38, 39, 30, 41, 42, 43, 33, 7, 46, 47, 15, 11, 14, 51, 39, 53, 30, 55, 42, 57, 58, 59, 7, 61, 62, 35, 15, 65, 66, 67, 51, 69, 70, 71, 30, 73, 74, 21

Offset: 1

Reinhard Zumkeller, Aug 03 2004

nonn

a(n) = r(n,m) with m such that r(n,m)=r(n,m+1), where r(n,k) = A097246(r(n,k-1)), r(n,0)=n. (The original definition.)
A097248(n) = r(n,a(n)).
From Antti Karttunen, Nov 15 2016: (Start)
The above remark could be interpreted to mean that A097249(n) <= a(n).
All terms are squarefree, and the squarefree numbers are the fixed points.
These are also fixed points eventually reached when iterating A277886.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Range of values is A005117.
Cf. A008683, A019565, A048675, A097246, A097247, A097249, A277905, A332824.
A003961, A225546, A277885, A277886, A331590 are used to express relationship between terms of this sequence.
The formula section also details how the sequence maps the terms of A007913, A260443, A329050, A329332.
See comments/formulas in A283475, A283478, A331751 giving their relationship to this sequence.

Table[FixedPoint[Times @@ Map[#1^#2 & @@ # &, Partition[#, 2, 2] &@ Flatten[FactorInteger[#] /. {p_, e_} /; e >= 2 :> {If[OddQ@ e, {p, 1}, {1, 1}], {NextPrime@ p, Floor[e/2]}}]] &, n], {n, 75}] (* Michael De Vlieger, Mar 18 2017 *)

A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
\\ Antti Karttunen, Mar 18 2017

from sympy import factorint, nextprime
from operator import mul
def a097246(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(nextprime(i)**int(f[i]/2))*(i**(f[i]%2)) for i in f])
def a(n):
    k=a097246(n)
    while k!=n:
        n=k
        k=a097246(k)
    return k # Indranil Ghosh, May 15 2017

;; with memoization-macro definec
;; Two implementations:
(definec (A097248 n) (if (not (zero? (A008683 n))) n (A097248 (A097246 n))))
(definec (A097248 n) (if (zero? (A277885 n)) n (A097248 (A277886 n))))
;; Antti Karttunen, Nov 15 2016

a(A005117(n)) = A005117(n).
From Antti Karttunen, Nov 15 2016: (Start)
If A008683(n) <> 0 [when n is squarefree], a(n) = n, otherwise a(n) = a(A097246(n)).
If A277885(n) = 0, a(n) = n, otherwise a(n) = a(A277886(n)).
A007913(a(n)) = a(n).
a(A007913(n)) = A007913(n).
A048675(a(n)) = A048675(n).
a(A260443(n)) = A019565(n).
(End)
From Peter Munn, Feb 06 2020: (Start)
a(1) = 1; a(p) = p, for prime p; a(m*k) = A331590(a(m), a(k)).
a(A331590(m,k)) = A331590(a(m), a(k)).
a(n^2) = a(A003961(n)) = A003961(a(n)).
a(A225546(n)) = a(n).
a(n) = A225546(2^A048675(n)) = A019565(A048675(n)).
a(A329050(n,k)) = prime(n+k-1) = A000040(n+k-1).
a(A329332(n,k)) = A019565(n * k).
Equivalently, a(A019565(n)^k) = A019565(n * k).
(End)
From Antti Karttunen, Feb 22-25 & Mar 01 2020: (Start)
a(A019565(x)*A019565(y)) = A019565(x+y).
a(A332461(n)) = A332462(n).
a(A332824(n)) = A019565(n).
a(A277905(n,k)) = A277905(n,1) = A019565(n), for all n >= 1, and 1 <= k <= A018819(n).
(End)

Name changed and the original definition moved to the Comments section by Antti Karttunen, Nov 15 2016

cp's OEIS Frontend

A332826 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A332824(i)) = A046523(A332824(j)) for all i, j.

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A332827 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A003557(A332824(n)) for all other numbers, except f(1) = 0.

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A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

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A097248 a(n) is the eventual stable point reached when iterating k -> A097246(k), starting from k = n.

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A332825 a(n) = A019565(phi(n)).

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