A097248 -id:A097248 - cp's OEIS frontend

A283478 a(n) = A097248(A108951(n)).

1, 2, 6, 3, 30, 5, 210, 6, 15, 7, 2310, 10, 30030, 11, 21, 5, 510510, 30, 9699690, 14, 33, 13, 223092870, 15, 105, 17, 14, 22, 6469693230, 42, 200560490130, 10, 39, 19, 165, 7, 7420738134810, 23, 51, 21, 304250263527210, 66, 13082761331670030, 26, 70, 29, 614889782588491410, 30, 1155, 210, 57, 34, 32589158477190044730, 21, 195, 33, 69, 31

Offset: 1

Antti Karttunen, Mar 16 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024

Cf. A019565, A097248, A108951, A283475.

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]}}]] &, #] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]], {n, 58}] (* Michael De Vlieger, Mar 18 2017 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From Charles R Greathouse IV, Jun 28 2015
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; };
A283478(n) = A097248(A108951(n));

from sympy import primerange, factorint, nextprime
from operator import mul
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 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 a097248(n):
    k=a097246(n)
    while k!=n:
        n=k
        k=a097246(k)
    return k
def a(n): return a097248(a108951(n)) # Indranil Ghosh, May 15 2017

(define (A283478 n) (A097248 (A108951 n)))

a(n) = A097248(A108951(n)).
Other identities:
For all n >= 0, a(A019565(n)) = A283475(n).

A019565 The squarefree numbers ordered lexicographically by their prime factorization (with factors written in decreasing order). a(n) = Product_{k in I} prime(k+1), where I is the set of indices of nonzero binary digits in n = Sum_{k in I} 2^k.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 42, 35, 70, 105, 210, 11, 22, 33, 66, 55, 110, 165, 330, 77, 154, 231, 462, 385, 770, 1155, 2310, 13, 26, 39, 78, 65, 130, 195, 390, 91, 182, 273, 546, 455, 910, 1365, 2730, 143, 286, 429, 858, 715, 1430, 2145, 4290

Offset: 0

Marc LeBrun

A permutation of the squarefree numbers A005117. The missing positive numbers are in A013929. - Alois P. Heinz, Sep 06 2014
From Antti Karttunen, Apr 18 & 19 2017: (Start)
Because a(n) toggles the parity of n there are neither fixed points nor any cycles of odd length.
Conjecture: there are no finite cycles of any length. My grounds for this conjecture: any finite cycle in this sequence, if such cycles exist at all, must have at least one member that occurs somewhere in A285319, the terms that seem already to be quite rare. Moreover, any such a number n should satisfy in addition to A019565(n) < n also that A048675^{k}(n) is squarefree, not just for k=0, 1 but for all k >= 0. As there is on average a probability of only 6/(Pi^2) = 0.6079... that any further term encountered on the trajectory of A048675 is squarefree, the total chance that all of them would be squarefree (which is required from the elements of A019565-cycles) is soon minuscule, especially as A048675 is not very tightly bounded (many trajectories seem to skyrocket, at least initially). I am also assuming that usually there is no significant correlation between the binary expansions of n and A048675(n) (apart from their least significant bits), or, for that matter, between their prime factorizations.
See also the slightly stronger conjecture in A285320, which implies that there would neither be any two-way infinite cycles.
If either of the conjectures is false (there are cycles), then certainly neither sequence A285332 nor its inverse A285331 can be a permutation of natural numbers. (End)
The conjecture made in A087207 (see also A288569) implies the two conjectures mentioned above. A further constraint for cycles is that in any A019565-trajectory which starts from a squarefree number (A005117), every other term is of the form 4k+2, while every other term is of the form 6k+3. - Antti Karttunen, Jun 18 2017
The sequence satisfies the exponential function identity, a(x + y) = a(x) * a(y), whenever x and y do not have a 1-bit in the same position, i.e., when A004198(x,y) = 0. See also A283475. - Antti Karttunen, Oct 31 2019
The above identity becomes unconditional if binary exclusive OR, A003987(.,.), is substituted for addition, and A059897(.,.), a multiplicative equivalent of A003987, is substituted for multiplication. This gives us a(A003987(x,y)) = A059897(a(x), a(y)). - Peter Munn, Nov 18 2019
Also the Heinz number of the binary indices of n, where the Heinz number of a sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k), and a number's binary indices (A048793) are the positions of 1's in its reversed binary expansion. - Gus Wiseman, Dec 28 2022

			5 = 2^2+2^0, e_1 = 2, e_2 = 0, prime(2+1) = prime(3) = 5, prime(0+1) = prime(1) = 2, so a(5) = 5*2 = 10.
From _Philippe Deléham_, Jun 03 2015: (Start)
This sequence regarded as a triangle withs rows of lengths 1, 1, 2, 4, 8, 16, ...:
   1;
   2;
   3,  6;
   5, 10, 15, 30;
   7, 14, 21, 42, 35,  70, 105, 210;
  11, 22, 33, 66, 55, 110, 165, 330, 77, 154, 231, 462, 385, 770, 1155, 2310;
  ...
(End)
From _Peter Munn_, Jun 14 2020: (Start)
The initial terms are shown below, equated with the product of their prime factors to exhibit the lexicographic order. We start with 1, since 1 is factored as the empty product and the empty list is first in lexicographic order.
   n     a(n)
   0     1 = .
   1     2 = 2.
   2     3 = 3.
   3     6 = 3*2.
   4     5 = 5.
   5    10 = 5*2.
   6    15 = 5*3.
   7    30 = 5*3*2.
   8     7 = 7.
   9    14 = 7*2.
  10    21 = 7*3.
  11    42 = 7*3*2.
  12    35 = 7*5.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191

Row 1 of A285321.
Equivalent sequences for k-th-power-free numbers: A101278 (k=3), A101942 (k=4), A101943 (k=5), A054842 (k=10).
Cf. A007088, A030308, A000040, A013929, A005117, A103785, A103786, A110765, A064273, A246353, A283475, A283477, A285319, A285331, A285332, A288569, A293442.
Cf. A109162 (iterates).
Cf. also A048675 (a left inverse), A087207, A097248, A260443, A054841.
Cf. A285315 (numbers for which a(n) < n), A285316 (for which a(n) > n).
Cf. A276076, A276086 (analogous sequences for factorial and primorial bases), A334110 (terms squared).
For partial sums see A288570.
A003961, A003987, A004198, A059897, A089913, A331590, A334747 are used to express relationships between sequence terms.
Column 1 of A329332.
Even bisection (which contains the odd terms): A332382.
A160102 composed with A052330, and subsequence of the latter.
Related to A000079 via A225546, to A057335 via A122111, to A008578 via A336322.
Least prime index of a(n) is A001511.
Greatest prime index of a(n) is A029837 or A070939.
Taking prime indices gives A048793, reverse A272020, row sums A029931.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A000120, A000720, A005940, A066099, A358137, A358170.

a019565 n = product $ zipWith (^) a000040_list (a030308_row n)
-- Reinhard Zumkeller, Apr 27 2013

a:= proc(n) local i, m, r; m:=n; r:=1;
      for i while m>0 do if irem(m,2,'m')=1
        then r:=r*ithprime(i) fi od; r
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 06 2014

Do[m=1;o=1;k1=k;While[ k1>0, k2=Mod[k1, 2];If[k2\[Equal]1, m=m*Prime[o]];k1=(k1-k2)/ 2;o=o+1];Print[m], {k, 0, 55}] (* Lei Zhou, Feb 15 2005 *)
Table[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2], {n, 0, 55}]  (* Michael De Vlieger, Aug 27 2016 *)
b[0] := {1}; b[n_] := Flatten[{ b[n - 1], b[n - 1] * Prime[n] }];
  a = b[6] (* Fred Daniel Kline, Jun 26 2017 *)

a(n)=factorback(vecextract(primes(logint(n+!n,2)+1),n))  \\ M. F. Hasler, Mar 26 2011, updated Aug 22 2014, updated Mar 01 2018

from operator import mul
from functools import reduce
from sympy import prime
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
# Chai Wah Wu, Dec 25 2014

(define (A019565 n) (let loop ((n n) (i 1) (p 1)) (cond ((zero? n) p) ((odd? n) (loop (/ (- n 1) 2) (+ 1 i) (* p (A000040 i)))) (else (loop (/ n 2) (+ 1 i) p))))) ;; (Requires only the implementation of A000040 for prime numbers.) - Antti Karttunen, Apr 20 2017

G.f.: Product_{k>=0} (1 + prime(k+1)*x^2^k), where prime(k)=A000040(k). - Ralf Stephan, Jun 20 2003
a(n) = f(n, 1, 1) with f(x, y, z) = if x > 0 then f(floor(x/2), y*prime(z)^(x mod 2), z+1) else y. - Reinhard Zumkeller, Mar 13 2010
For all n >= 0: A048675(a(n)) = n; A013928(a(n)) = A064273(n). - Antti Karttunen, Jul 29 2015
a(n) = a(2^x)*a(2^y)*a(2^z)*... = prime(x+1)*prime(y+1)*prime(z+1)*..., where n = 2^x + 2^y + 2^z + ... - Benedict W. J. Irwin, Jul 24 2016
From Antti Karttunen, Apr 18 2017 and Jun 18 2017: (Start)
a(n) = A097248(A260443(n)), a(A005187(n)) = A283475(n), A108951(a(n)) = A283477(n).
A055396(a(n)) = A001511(n), a(A087207(n)) = A007947(n). (End)
a(2^n - 1) = A002110(n). - Michael De Vlieger, Jul 05 2017
a(n) = A225546(A000079(n)). - Peter Munn, Oct 31 2019
From Peter Munn, Mar 04 2022: (Start)
a(2n) = A003961(a(n)); a(2n+1) = 2*a(2n).
a(x XOR y) = A059897(a(x), a(y)) = A089913(a(x), a(y)), where XOR denotes bitwise exclusive OR (A003987).
a(n+1) = A334747(a(n)).
a(x+y) = A331590(a(x), a(y)).
a(n) = A336322(A008578(n+1)).
(End)

Definition corrected by Klaus-R. Löffler, Aug 20 2014

New name from Peter Munn, Jun 14 2020

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

A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 210, 8, 36, 60, 2310, 24, 30030, 420, 180, 16, 510510, 72, 9699690, 120, 1260, 4620, 223092870, 48, 900, 60060, 216, 840, 6469693230, 360, 200560490130, 32, 13860, 1021020, 6300, 144, 7420738134810, 19399380, 180180, 240, 304250263527210, 2520

Offset: 1

Paul Boddington, Jul 21 2005

This sequence is a permutation of A025487.
And thus also a permutation of A181812, see the formula section. - Antti Karttunen, Jul 21 2014
A previous description of this sequence was: "Multiplicative with a(p^e) equal to the product of the e-th powers of all primes at most p" (see extensions), Giuseppe Coppoletta, Feb 28 2015

			a(12) = a(2^2) * a(3) = (2#)^2 * (3#) = 2^2 * 6 = 24
a(45) = (3#)^2 * (5#) = (2*3)^2 * (2*3*5) = 1080 (as 45 = 3^2 * 5).

Amiram Eldar, Table of n, a(n) for n = 1..2370 (terms 1..256 from Antti Karttunen)
Index to divisibility sequences.
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to primorial numbers.

Cf. A319626, A329900 (left inverses).

Cf. A034386, A002110, A025487, A048673, A064216, A064989, A085082, A122111, A124859, A161360, A181811, A181812, A181814, A181815, A181817, A181819, A181822, A238690, A283477, A283478, A307035, A324886, A324887, A324888, A324896, A325226, A329040, A329046, A329047, A329344, A329348, A329349, A329378, A329382, A329600, A329602, A329605, A329607, A329615, A329616, A329617, A329619, A329622, A319627, A329647, A331292, A337474, A346108, A346109, A344698, A344699.

a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f]>1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Feb 24 2015 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}], {n, 42}] (* Michael De Vlieger, Mar 18 2017 *)

primorial(n)=prod(i=1,primepi(n),prime(i))
a(n)=my(f=factor(n)); prod(i=1,#f~, primorial(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015

from sympy import primerange, factorint
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
def p(f):
    return sharp_primorial(f[0])^f[1]
[prod(p(f) for f in factor(n)) for n in range (1,51)]
# Giuseppe Coppoletta, Feb 07 2015

Dirichlet g.f.: 1/(1-2*2^(-s))/(1-6*3^(-s))/(1-30*5^(-s))...
Completely multiplicative with a(p_i) = A002110(i) = prime(i)#. [Franklin T. Adams-Watters, Jun 24 2009; typos corrected by Antti Karttunen, Jul 21 2014]
From Antti Karttunen, Jul 21 2014: (Start)
a(1) = 1, and for n > 1, a(n) = n * a(A064989(n)).
a(n) = n * A181811(n).
a(n) = A002110(A061395(n)) * A331188(n). - [added Jan 14 2020]
a(n) = A181812(A048673(n)).
Other identities:
A006530(a(n)) = A006530(n). [Preserves the largest prime factor of n.]
A071178(a(n)) = A071178(n). [And also its exponent.]
a(2^n) = 2^n. [Fixes the powers of two.]
A067029(a(n)) = A007814(a(n)) = A001222(n). [The exponent of the least prime of a(n), that prime always being 2 for n>1, is equal to the total number of prime factors in n.]
(End)
From Antti Karttunen, Nov 19 2019: (Start)
Further identities:
a(A307035(n)) = A000142(n).
a(A003418(n)) = A181814(n).
a(A025487(n)) = A181817(n).
a(A181820(n)) = A181822(n).
a(A019565(n)) = A283477(n).
A001221(a(n)) = A061395(n).
A001222(a(n)) = A056239(n).
A181819(a(n)) = A122111(n).
A124859(a(n)) = A181821(n).
A085082(a(n)) = A238690(n).
A328400(a(n)) = A329600(n). (smallest number with the same set of distinct prime exponents)
A000188(a(n)) = A329602(n). (square root of the greatest square divisor)
A072411(a(n)) = A329378(n). (LCM of exponents of prime factors)
A005361(a(n)) = A329382(n). (product of exponents of prime factors)
A290107(a(n)) = A329617(n). (product of distinct exponents of prime factors)
A000005(a(n)) = A329605(n). (number of divisors)
A071187(a(n)) = A329614(n). (smallest prime factor of number of divisors)
A267115(a(n)) = A329615(n). (bitwise-AND of exponents of prime factors)
A267116(a(n)) = A329616(n). (bitwise-OR of exponents of prime factors)
A268387(a(n)) = A329647(n). (bitwise-XOR of exponents of prime factors)
A276086(a(n)) = A324886(n). (prime product form of primorial base expansion)
A324580(a(n)) = A324887(n).
A276150(a(n)) = A324888(n). (digit sum in primorial base)
A267263(a(n)) = A329040(n). (number of distinct nonzero digits in primorial base)
A243055(a(n)) = A329343(n).
A276088(a(n)) = A329348(n). (least significant nonzero digit in primorial base)
A276153(a(n)) = A329349(n). (most significant nonzero digit in primorial base)
A328114(a(n)) = A329344(n). (maximal digit in primorial base)
A062977(a(n)) = A325226(n).
A097248(a(n)) = A283478(n).
A324895(a(n)) = A324896(n).
A324655(a(n)) = A329046(n).
A327860(a(n)) = A329047(n).
A329601(a(n)) = A329607(n).
(End)
a(A181815(n)) = A025487(n), and A319626(a(n)) = A329900(a(n)) = n. - Antti Karttunen, Dec 29 2019
From Antti Karttunen, Jul 09 2021: (Start)
a(n) = A346092(n) + A346093(n).
a(n) = A346108(n) - A346109(n).
a(A342012(n)) = A004490(n).
a(A337478(n)) = A336389(n).
A336835(a(n)) = A337474(n).
A342002(a(n)) = A342920(n).
A328571(a(n)) = A346091(n).
A328572(a(n)) = A344592(n).
(End)
Sum_{n>=1} 1/a(n) = A161360. - Amiram Eldar, Aug 04 2022

More terms computed by Antti Karttunen, Jul 21 2014
The name of the sequence was changed for more clarity, in accordance with the above remark of Franklin T. Adams-Watters (dated Jun 24 2009). It is implicitly understood that a(n) is then uniquely defined by completely multiplicative extension. - Giuseppe Coppoletta, Feb 28 2015
Name "Primorial inflation" (coined by Matthew Vandermast in A181815) prefixed to the name by Antti Karttunen, Jan 14 2020

A225546 Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).

Original entry on oeis.org

1, 2, 4, 3, 16, 8, 256, 6, 9, 32, 65536, 12, 4294967296, 512, 64, 5, 18446744073709551616, 18, 340282366920938463463374607431768211456, 48, 1024, 131072, 115792089237316195423570985008687907853269984665640564039457584007913129639936, 24, 81, 8589934592, 36, 768

Offset: 1

Paul Tek, May 10 2013

This is a multiplicative self-inverse permutation of the integers.
A225547 gives the fixed points.
From Antti Karttunen and Peter Munn, Feb 02 2020: (Start)
This sequence operates on the Fermi-Dirac factors of a number. As arranged in array form, in A329050, this sequence reflects these factors about the main diagonal of the array, substituting A329050[j,i] for A329050[i,j], and this results in many relationships including significant homomorphisms.
This sequence provides a relationship between the operations of squaring and prime shift (A003961) because each successive column of the A329050 array is the square of the previous column, and each successive row is the prime shift of the previous row.
A329050 gives examples of how significant sets of numbers can be formed by choosing their factors in relation to rows and/or columns. This sequence therefore maps equivalent derived sets by exchanging rows and columns. Thus odd numbers are exchanged for squares, squarefree numbers for powers of 2 etc.
Alternative construction: For n > 1, form a vector v of length A299090(n), where each element v[i] for i=1..A299090(n) is a product of those distinct prime factors p(i) of n whose exponent e(i) has the bit (i-1) "on", or 1 (as an empty product) if no such exponents are present. a(n) is then Product_{i=1..A299090(n)} A000040(i)^A048675(v[i]). Note that because each element of vector v is squarefree, it means that each exponent A048675(v[i]) present in the product is a "submask" (not all necessarily proper) of the binary string A087207(n).
This permutation effects the following mappings:
A000035(a(n)) = A010052(n), A010052(a(n)) = A000035(n). [Odd numbers <-> Squares]
A008966(a(n)) = A209229(n), A209229(a(n)) = A008966(n). [Squarefree numbers <-> Powers of 2]
(End)
From Antti Karttunen, Jul 08 2020: (Start)
Moreover, we see also that this sequence maps between A016825 (Numbers of the form 4k+2) and A001105 (2*squares) as well as between A008586 (Multiples of 4) and A028983 (Numbers with even sum of the divisors).
(End)

			  7744  = prime(1)^2^(2-1)*prime(1)^2^(3-1)*prime(5)^2^(2-1).
a(7744) = prime(2)^2^(1-1)*prime(3)^2^(1-1)*prime(2)^2^(5-1) = 645700815.

Cf. A225547 (fixed points) and the subsequences listed there.
Transposes A329050, A329332.
An automorphism of positive integers under the binary operations A059895, A059896, A059897, A306697, A329329.
An automorphism of A059897 subgroups: A000379, A003159, A016754, A122132.
Permutes lists where membership is determined by number of Fermi-Dirac factors: A000028, A050376, A176525, A268388.
Also permutes: A036554, A059404, A066427, A066428, A072774, A331593.
Sequences f that satisfy f(a(n)) = f(n): A048675, A064179, A064547, A097248, A302777, A331592.
Pairs of sequences (f,g) that satisfy a(f(n)) = g(a(n)): (A000265,A008833), (A000290,A003961), (A005843,A334747), (A006519,A007913), (A008586,A334748).
Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000040,A001146), (A000079,A019565).
Pairs of sequences (f,g) that satisfy f(a(n)) = g(n), possibly with offset change: (A000035, A010052), (A008966, A209229), (A007814, A248663), (A061395, A299090), (A087207, A267116), (A225569, A227291).
Cf. A331287 [= gcd(a(n),n)].
Cf. A331288 [= min(a(n),n)], see also A331301.
Cf. A331309 [= A000005(a(n)), number of divisors].
Cf. A331590 [= a(a(n)*a(n))].
Cf. A331591 [= A001221(a(n)), number of distinct prime factors], see also A331593.
Cf. A331740 [= A001222(a(n)), number of prime factors with multiplicity].
Cf. A331733 [= A000203(a(n)), sum of divisors].
Cf. A331734 [= A033879(a(n)), deficiency].
Cf. A331735 [= A009194(a(n))].
Cf. A331736 [= A000265(a(n)) = a(A008833(n)), largest odd divisor].
Cf. A335914 [= A038040(a(n))].
A self-inverse isomorphism between pairs of A059897 subgroups: (A000079,A005117), (A000244,A062503), (A000290\{0},A005408), (A000302,A056911), (A000351,A113849 U {1}), (A000400,A062838), (A001651,A252895), (A003586,A046100), (A007310,A000583), (A011557,A113850 U {1}), (A028982,A042968), (A053165,A065331), (A262675,A268390).
A bijection between pairs of sets: (A001248,A011764), (A007283,A133466), (A016825, A001105), (A008586, A028983).
Cf. also A336321, A336322 (compositions with another involution, A122111).

Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 28] (* Michael De Vlieger, Jan 21 2020 *)

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
a(n) = {my(f=factor(n)); for (i=1, #f~, my(p=f[i,1]); f[i,1] = A019565(f[i,2]); f[i,2] = 2^(primepi(p)-1);); factorback(f);} \\ Michel Marcus, Nov 29 2019

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A225546(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=1,u,prime(i)^A048675(prods[i]))); \\ Antti Karttunen, Feb 02 2020

from math import prod
from sympy import prime, primepi, factorint
def A225546(n): return prod(prod(prime(i) for i, v in enumerate(bin(e)[:1:-1],1) if v == '1')**(1<Chai Wah Wu, Mar 17 2023

Multiplicative, with a(prime(i)^j) = A019565(j)^A000079(i-1).
a(prime(i)) = 2^(2^(i-1)).
From Antti Karttunen and Peter Munn, Feb 06 2020: (Start)
a(A329050(n,k)) = A329050(k,n).
a(A329332(n,k)) = A329332(k,n).
Equivalently, a(A019565(n)^k) = A019565(k)^n. If n = 1, this gives a(2^k) = A019565(k).
a(A059897(n,k)) = A059897(a(n), a(k)).
The previous formula implies a(n*k) = a(n) * a(k) if A059895(n,k) = 1.
a(A000040(n)) = A001146(n-1); a(A001146(n)) = A000040(n+1).
a(A000290(a(n))) = A003961(n); a(A003961(a(n))) = A000290(n) = n^2.
a(A000265(a(n))) = A008833(n); a(A008833(a(n))) = A000265(n).
a(A006519(a(n))) = A007913(n); a(A007913(a(n))) = A006519(n).
A007814(a(n)) = A248663(n); A248663(a(n)) = A007814(n).
A048675(a(n)) = A048675(n) and A048675(a(2^k * n)) = A048675(2^k * a(n)) = k + A048675(a(n)).
(End)
From Antti Karttunen and Peter Munn, Jul 08 2020: (Start)
For all n >= 1, a(2n) = A334747(a(n)).
In particular, for n = A003159(m), m >= 1, a(2n) = 2*a(n). [Note that A003159 includes all odd numbers]
(End)

Name edited by Peter Munn, Feb 14 2020

"Tek's flip" prepended to the name by Antti Karttunen, Jul 08 2020

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

A332823 A 3-way classification indicator generated by the products of two consecutive primes and the cubes of primes. a(n) is -1, 0, or 1 such that a(n) == A048675(n) (mod 3).

Original entry on oeis.org

0, 1, -1, -1, 1, 0, -1, 0, 1, -1, 1, 1, -1, 0, 0, 1, 1, -1, -1, 0, 1, -1, 1, -1, -1, 0, 0, 1, -1, 1, 1, -1, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, -1, 0, -1, -1, 1, 0, 1, 0, 0, 1, -1, 1, -1, -1, 1, 0, 1, -1, -1, -1, 0, 0, 0, 1, 1, 0, 0, 1, -1, 1, 1, 0, 1, 1, 0, -1, -1, -1, -1, -1, 1, 0, -1, 0, 1, 1, -1, 0

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

sign

Completely additive modulo 3.
The equivalent sequence modulo 2 is A096268 (with offset 1), which produces the {A003159, A036554} classification.
Let H be the multiplicative subgroup of the positive rational numbers generated by the products of two consecutive primes and the cubes of primes. a(n) indicates the coset of H containing n. a(n) = 0 if n is in H. a(n) = 1 if n is in 2H. a(n) = -1 if n is in (1/2)H.
The properties of this classification can usefully be compared to two well-studied classifications. With the {A026424, A028260} classes, multiplying a member of one class by a prime gives a member of the other class. With the {A000028, A000379} classes, adding a factor to the Fermi-Dirac factorization of a member of one class gives a member of the other class. So, if 4 is not a Fermi-Dirac factor of k, k and 4k will be in different classes of the {A000028, A000379} set; but k and 4k will be in the same class of the {A026424, A028260} set. For two numbers to necessarily be in different classes when they differ in either of the 2 ways described above, 3 classes are needed.
With the classes defined by this sequence, no two of k, 2k and 4k are in the same class. This is a consequence of the following stronger property: if k is any positive integer and m is a member of A050376 (often called Fermi-Dirac primes), then no two of k, k * m, k * m^2 are in the same class. Also, if p and q are consecutive primes, then k * p and k * q are in different classes.
Further properties are given in the sequences that list the classes: A332820, A332821, A332822.
The scaled imaginary part of the Eisenstein integer-valued function, f, defined in A353445. - Peter Munn, Apr 27 2022

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

Cf. A332813 (0,1,2 version of this sequence), A353350.
Cf. A353354 (inverse Möbius transform, gives another 3-way classification indicator function).
Cf. A102283, A048675.
Cf. A332820, A332821, A332822 for positions of 0's, 1's and -1's in this sequence; also A003159, A036554 for the modulo 2 equivalents.
Comparable functions: A008836, A064179, A096268, A332814.
A000035, A003961, A028234, A055396, A067029, A097248, A225546, A297845, A331590 are used to express relationship between terms of this sequence.
The formula section also details how the sequence maps the terms of A000040, A332461, A332462.

A332823(n) = { my(f = factor(n),u=(sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); if(2==u,-1,u); };

a(n) = A102283(A048675(n)) = -1 + (1 + A048675(n)) mod 3.
a(1) = 0; for n > 1, a(n) = A102283[(A067029(n) * (2-(A000035(A055396(n))))) + a(A028234(n))].
For all n >= 1, k >= 1: (Start)
a(n * k) == a(n) + a(k) (mod 3).
a(A331590(n,k)) == a(n) + a(k) (mod 3).
a(n^2) = -a(n).
a(A003961(n)) = -a(n).
a(A297845(n,k)) = a(n) * a(k).
(End)
For all n >= 1: (Start)
a(A000040(n)) = (-1)^(n-1).
a(A225546(n)) = a(n).
a(A097248(n)) = a(n).
a(A332461(n)) = a(A332462(n)) = A332814(n).
(End)
a(n) = A332814(A332462(n)). [Compare to the formula above. For a proof, see A353350.] - Antti Karttunen, Apr 16 2022

A277905 Irregular table: Each row n (n >= 0) lists in ascending order all A018819(n) numbers k for which A048675(k) = n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 9, 12, 16, 10, 18, 24, 32, 15, 20, 27, 36, 48, 64, 30, 40, 54, 72, 96, 128, 7, 25, 45, 60, 80, 81, 108, 144, 192, 256, 14, 50, 90, 120, 160, 162, 216, 288, 384, 512, 21, 28, 75, 100, 135, 180, 240, 243, 320, 324, 432, 576, 768, 1024, 42, 56, 150, 200, 270, 360, 480, 486, 640, 648, 864, 1152, 1536, 2048, 35, 63, 84, 112, 125, 225, 300, 400

Offset: 1

Antti Karttunen, Nov 14 2016

Each row beginning with an odd number (rows with even index) is followed by a row of the same length, with the same terms, but multiplied by 2. See also comments in the Formula section of A018819.
Note that although the indexing of rows start from zero, the indexing of this sequence starts from 1, with a(1) = 1.
Also Heinz numbers of integer partitions whose binary rank is n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). For example, row n = 6 is 15, 20, 27, 36, 48, 64, corresponding to the partitions (3,2), (3,1,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1). - Gus Wiseman, May 25 2024
Also, row n lists in ascending order all A018819(n) numbers k for which A097248(k) = A019565(n). - Flávio V. Fernandes, Jul 19 2025

			The irregular table begins as:
  row terms
   0   1;
   1   2;
   2   3,  4;
   3   6,  8;
   4   5,  9,  12,  16;
   5  10, 18,  24,  32;
   6  15, 20,  27,  36,  48,  64;
   7  30, 40,  54,  72,  96, 128;
   8   7, 25,  45,  60,  80,  81, 108, 144, 192, 256;
   9  14, 50,  90, 120, 160, 162, 216, 288, 384, 512;
  10  21, 28,  75, 100, 135, 180, 240, 243, 320, 324, 432,  576,  768, 1024;
  11  42, 56, 150, 200, 270, 360, 480, 486, 640, 648, 864, 1152, 1536, 2048;
...

Antti Karttunen, Table of n, a(n) for n = 1..166
Index entries for sequences that are permutations of the natural numbers

Cf. A019565 (the left edge, the only terms that are squarefree).
Cf. A000079 (the trailing edge).
Cf. A048675, A260443, A277886, A277896, A277903, A277904.
Row lengths are A018819 (number of partitions of binary rank n).
A000009 counts strict partitions, ranks A005117.
A029837 stc_sum or A070939 bin_len, opposite A070940 binexp_lastpos_1.
A048675 gives binary rank of prime indices, distinct A087207.
A048793 lists binary indices, product A096111, reverse A272020.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, cf. A001222, A003963, A056239, A296150.
A372890 adds up binary ranks of partitions, strict A372888.
Cf. A000040, A000041, A001511, A005940, A118462, A215366, A225620, A246868.
Cf. A097248, A048675.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Select[Range[0,2^k],Total[2^(prix[#]-1)]==k&],{k,0,10}] (* Gus Wiseman, May 25 2024 *)

(definec (A277905 n) (A277905bi (A277903 n) (A277904 n)))
(define (A277905bi row col) (let outloop ((k (A019565 row)) (col col)) (if (zero? col) k (let inloop ((j (+ 1 k))) (if (= (A048675 j) row) (outloop j (- col 1)) (inloop (+ 1 j))))))) ;; Very slow implementation.
;; Implementation based on a naive recurrence:
(definec (A277905 n) (if (= 1 n) n (let ((maybe_next (A277896 (A277905 (- n 1))))) (if (not (zero? maybe_next)) maybe_next (A019565 (A277903 n))))))

a(1) = 1; for n > 1, if A277896(a(n-1)) > 0, then a(n) = A277896(a(n-1)), otherwise a(n) = A019565(A277903(n)). [A naive recurrence for a one-dimensional version.]
Other identities. For all n >= 1:
A048675(a(n)) = A277903(n).

A283475 a(n) = A019565(A005187(n)).

Original entry on oeis.org

1, 2, 6, 5, 30, 7, 21, 42, 210, 11, 33, 66, 165, 330, 154, 231, 2310, 13, 39, 78, 195, 390, 182, 273, 1365, 2730, 286, 429, 1430, 2145, 1001, 2002, 30030, 17, 51, 102, 255, 510, 238, 357, 1785, 3570, 374, 561, 1870, 2805, 1309, 2618, 19635, 39270, 442, 663, 2210, 3315, 1547, 3094, 15470, 23205, 2431, 4862, 12155

Offset: 0

Antti Karttunen, Mar 15 2017

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A000040, A000051, A001221, A001222, A004198, A005117, A005187, A019565, A046523, A097248, A108951, A280700, A280705, A283477, A283478.

Cf. A283476 (same sequence sorted into ascending order).

Map[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[#, 2] &, Table[2 n - DigitCount[2 n, 2, 1], {n, 0, 60}]] (* Michael De Vlieger, Mar 16 2017 *)

(define (A283475 n) (A019565 (A005187 n)))

a(n) = A019565(A005187(n)).
a(n) = A097248(A283477(n)) = A097248(A108951(A019565(n))) = A283478(A019565(n)).
Other identities:
If A004198(x,y) = 0, then a(x+y) = A097248(a(x)*a(y)).
For all n >= 1, a(A000051(n)) = A000040(n+2).
For all n >= 0, A001221(a(n)) = A001222(a(n)) = A280700(n).
For all n >= 0, A046523(a(n)) = A280705(n).

A097246 Replace factors of n that are squares of a prime with the prime succeeding this prime.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 6, 5, 10, 11, 9, 13, 14, 15, 9, 17, 10, 19, 15, 21, 22, 23, 18, 7, 26, 15, 21, 29, 30, 31, 18, 33, 34, 35, 15, 37, 38, 39, 30, 41, 42, 43, 33, 25, 46, 47, 27, 11, 14, 51, 39, 53, 30, 55, 42, 57, 58, 59, 45, 61, 62, 35, 27, 65, 66, 67, 51, 69, 70, 71, 30, 73

Offset: 1

Reinhard Zumkeller, Aug 03 2004

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

Cf. A000035, A000040, A002110, A049084, A097250, A000079, A000188, A000244, A003961, A004526, A005117, A007814, A007913, A048675, A064989, A151800.

Cf. A097247, A097248 (fixed points of iteration), A097249 (number of iterations needed to reach them for each n), A277886, A277899.

Table[Times @@ Map[#1^#2 & @@ # &, Partition[#, 2, 2] &@ Flatten[ FactorInteger[n] /. {p_, e_} /; e >= 2 :> {If[OddQ@ e, {p, 1}, {1, 1}], {NextPrime@ p, Floor[e/2]}}]], {n, 73}] (* 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))); }; \\ Antti Karttunen, Mar 18 2017

from sympy import factorint, nextprime
from operator import mul
def a(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]) # Indranil Ghosh, May 15 2017

(definec (A097246 n) (if (= 1 n) 1 (* (A000244 (A004526 (A007814 n))) (A000079 (A000035 (A007814 n))) (A003961 (A097246 (A064989 n))))))
(define (A097246 n) (* (A003961 (A000188 n)) (A007913 n)))
;; Antti Karttunen, Nov 15 2016

Multiplicative with p^e -> NextPrime(p)^floor(e/2) * p^(e mod 2), where p prime and NextPrime(p)=A000040(A049084(p)+1).
a(n) <= n; a(n) = n iff n is squarefree: a(A005117(n)) = A005117(n);
a(m*n) <= a(m)*a(n); a(m*n) = a(m)*a(n) iff m and n are coprime;
a(A000040(k)^n) = A000040(k+1)^floor(n/2)*A000040(k)^(n mod 2); a(2^n) = 3^floor(n/2) * (1 + n mod 2);
a(A000040(k)*A002110(n)/A002110(k-1)) = A000040(k+1)*A002110(n)/A002110(k) for k <= n, see also A097250.
From Antti Karttunen, Nov 15 2016: (Start)
a(1) = 1; for n > 1, a(n) = 2^A000035(A007814(n)) * 3^A004526(A007814(n)) * A003961(a(A064989(n))).
a(n) = A003961(A000188(n)) * A007913(n).
A048675(a(n)) = A048675(n).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (p^4-p^2)/(p^4-nextprime(p)) = 0.4059779303..., where nextprime is A151800. - Amiram Eldar, Nov 29 2022

cp's OEIS Frontend

A283478 a(n) = A097248(A108951(n)).

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A019565 The squarefree numbers ordered lexicographically by their prime factorization (with factors written in decreasing order). a(n) = Product_{k in I} prime(k+1), where I is the set of indices of nonzero binary digits in n = Sum_{k in I} 2^k.

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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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A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

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A225546 Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).

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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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