A019565 -id:A019565 - cp's OEIS frontend

A319992 a(n) = Product_{d|n, dA019565(d)^[2 == d mod 3].

1, 1, 1, 3, 1, 3, 1, 3, 1, 30, 1, 3, 1, 3, 10, 21, 1, 3, 1, 30, 1, 126, 1, 21, 10, 3, 1, 315, 1, 30, 1, 21, 42, 66, 10, 3, 1, 3, 1, 11550, 1, 315, 1, 126, 10, 990, 1, 21, 1, 30, 22, 693, 1, 3, 420, 2205, 1, 2310, 1, 1650, 1, 3, 1, 273, 10, 126, 1, 66, 330, 245700, 1, 21, 1, 3, 10, 585, 42, 693, 1, 11550, 1, 546, 1, 315, 220, 3, 770

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

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

Cf. A019565, A293214, A293896, A293898, A319990, A319991, A320005, A320012 (rgs-transform).

Cf. also A293222.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319992(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };

a(n) = Product_{d|n, dA019565(d)^[2 == d mod 3].
a(n) = A293214(n) / (A319990(n)*A319991(n)).
For all n >= 1:
A007814(a(n)) = A320005(n).
A048675(a(n)) = A293898(n).
A195017(a(n)) = -A293896(n) mod 3.

A109162 a(1) = 1; for n > 1, a(n) = A019565(a(n-1)).

Original entry on oeis.org

1, 2, 3, 6, 15, 210, 10659, 54230826, 249853434654335387610276087

Offset: 1

Leroy Quet, Aug 18 2005

nonn

After the initial 1, even-indexed terms are of the form 4k+2 (members of A016825) and odd-indexed terms are of the form 6k+3 (members of A016945). However, not all even terms after 2 are multiples of three, because not all odd-indexed terms are of the form 4k+3. For example, because a(11) is of the form 4k+1, a(12) cannot be a multiple of three. - Antti Karttunen, Jun 18 2017

			a(4) = 6, which is 110 in binary. So a(5) is the product of the primes corresponding to the 1's of 110, 3*5 = 15.

Frank Adams-Watters, Table of n, a(n) for n = 1..11

Cf. A019565, A285320 (a left inverse).
The left edge of A285332 and A285333.
Cf. A153013, A328316 for similar iteration sequences, and also A376406, A376407, A376408.

NestList[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[#, 2] &, 1, 11] (* Michael De Vlieger, Aug 20 2017 *)

More terms from Franklin T. Adams-Watters, Aug 29 2006

A285331 Inverse for A285332: a(1) = 0, a(2) = 1, a(A019565(n)) = 2a(n), a(A065642(n)) = 1 + 2a(n).

Original entry on oeis.org

0, 1, 2, 3, 6, 4, 14, 7, 5, 12, 30, 9, 62, 10, 8, 15, 126, 19, 254, 25, 24, 252, 510, 39, 13, 76, 11, 21, 1022, 28, 2046, 31, 38, 316, 18, 79, 4094

Offset: 1

Antti Karttunen, Apr 17 2017, comments edited Apr 19 2017

Note the indexing: the domain starts from 1, while the range includes also zero.
For the question whether this sequence and A285332 are permutations of natural numbers, see comments in A285332 and the conjecture stated in A019565.
As a practical problem, it seems next-to-impossible to compute even the value of a(38). Even though we know that 38 certainly is not in a finite cycle of A019565, because A048675(38) = 129, A048675(129) = 8194 and A048675(8194) = 4503599627370561 which factorizes as 3^2 * 37 * 71 * 190483425427 (thus is not squarefree and A285320(38) = 3), the value of a(38) is most likely so huge that it will not fit into the data section or even into a b-file. The same problem applies to all numbers that share prime factors with 38, namely 76, 152, 304, 608, 722, ...
Terms a(39) .. a(61) are [632, 51, 8190, 60, 16382, 505, 17, 72057594037927932, 32766, 159, 29, 103, 1016, 153, 65534, 319, 50, 43, 16376, 131014, 131070, 57, 262142].
The name is slightly misleading. The given definition of a(n) is not always very helpful to compute the terms (cf. example of n = 38), it is actually not clear whether the sequence is well defined. - M. F. Hasler, Mar 01 2018

			a(1) = 0 and a(2) = 1 by definition.
a(3) = a(prime(2)) = a(A019565(2^1)) = 2*a(2) = 2.
a(4) = a(2^2) = a(A065642(2)) = 1 + 2*a(2) = 3.
a(5) = a(prime(3)) = a(A019565(2^2)) = 2*a(4) = 6.
a(9) = a(3^2) = a(A065642(3)) = 1 + 2*a(3) = 5.
a(10) = a(2*5) = a(prime(1)*prime(3)) = a(A019565(2^0+2^2)) = 2*a(1+4) = 12.
To compute a(38), write 38 = prime(1)*prime(8) = A019565(2^7+2^0), so a(38) = 2*a(129). To compute this, use 129 = prime(2)*prime(14) = A019565(2^13+2^1), so a(129) = 2*a(8194). But 8194 = prime(1)*prime(7)*prime(53) = A019565(2^0+2^6+2^52), so a(8194) = 2*a(4503599627370561)...

Inverse: A285332.
Cf. A005117, A008683, A019565, A048675, A065642, A087207, A285319, A285320, A285328.
Compare also to permutation A285111.

A285331(n)={ if(n<=2,n-1,if(moebius(n)<>0, 2*A285331(A048675(n)), 1+2*A285331(A285328(n))))} \\ See A048675 & A285328 for respective PARI code. (We avoid content duplication leading to obsolete code.)

;; With memoization-macro definec.
(definec (A285331 n) (cond ((<= n 2) (- n 1)) ((not (zero? (A008683 n))) (* 2 (A285331 (A048675 n)))) (else (+ 1 (* 2 (A285331 (A285328 n)))))))

a(1) = 0, a(2) = 1, and for n > 2, if A008683(n) <> 0 [when n is squarefree], a(n) = 2*a(A048675(n)), otherwise a(n) = 1 + 2*a(A285328(n)).

For all n >= 0, a(A285332(n)) = n.

A293443 Multiplicative with a(p^e) = A019565(A193231(e)).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 3, 6, 4, 2, 12, 2, 4, 4, 10, 2, 12, 2, 12, 4, 4, 2, 6, 6, 4, 3, 12, 2, 8, 2, 5, 4, 4, 4, 36, 2, 4, 4, 6, 2, 8, 2, 12, 12, 4, 2, 20, 6, 12, 4, 12, 2, 6, 4, 6, 4, 4, 2, 24, 2, 4, 12, 15, 4, 8, 2, 12, 4, 8, 2, 18, 2, 4, 12, 12, 4, 8, 2, 20, 10, 4, 2, 24, 4, 4, 4, 6, 2, 24, 4, 12, 4, 4, 4, 10, 2, 12, 12, 36, 2, 8, 2, 6, 8

Offset: 1

Antti Karttunen, Oct 31 2017

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

Cf. A019565, A028234, A067029, A193231.

Cf. also A270428, A293442, A293231, A293439.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ And this from Franklin T. Adams-Watters
vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
A293443(n) = vecproduct(apply(e -> A019565(A193231(e)), factorint(n)[, 2]));

;; With memoization-macro definec.
(definec (A293443 n) (if (= 1 n) n (* (A019565 (A193231 (A067029 n))) (A293443 (A028234 n)))))

a(1) = 1; for n > 1, a(n) = A019565(A193231(A067029(n))) * a(A028234(n)).
For all n >= 1, A007814(a(n)) = A293439(n).
For all k in A270428, A007814(a(k)) = A001221(k).

A300831 a(n) = Product_{d|n, dA019565(d)^[moebius(n/d) = +1].

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 6, 1, 3, 2, 2, 1, 5, 1, 2, 1, 3, 1, 180, 1, 1, 2, 2, 2, 15, 1, 2, 2, 5, 1, 540, 1, 3, 6, 2, 1, 7, 1, 10, 2, 3, 1, 14, 2, 5, 2, 2, 1, 1575, 1, 2, 6, 1, 2, 756, 1, 3, 2, 900, 1, 35, 1, 2, 10, 3, 2, 1260, 1, 7, 1, 2, 1, 7875, 2, 2, 2, 5, 1, 44100, 2, 3, 2, 2, 2, 11, 1, 30, 6, 21, 1, 396, 1, 5, 1800

Offset: 1

Antti Karttunen, Mar 16 2018

nonn

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

Cf. A008683, A008966, A019565.

Cf. also A293214, A300830, A300832, A300833.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300831(n) = { my(m=1); fordiv(n,d,if((d < n)&&(1==moebius(n/d)), m *= A019565(d))); m; };

a(n) = A293214(n) / (A300830(n)*A300832(n)).

A300834 a(n) = Product_{d|n, dA019565(A003714(d)), where A003714(n) is the n-th Fibbinary number.

Original entry on oeis.org

1, 2, 2, 6, 2, 30, 2, 60, 10, 42, 2, 4200, 2, 126, 70, 660, 2, 9240, 2, 13860, 210, 330, 2, 5082000, 14, 78, 220, 32760, 2, 3783780, 2, 42900, 550, 780, 294, 924924000, 2, 1092, 130, 41621580, 2, 3898440, 2, 112200, 60060, 306, 2, 28078050000, 42, 235620, 1300, 92820, 2, 200119920, 770, 128648520, 1820, 1122, 2, 424964656116000, 2, 3366

Offset: 1

Antti Karttunen, Mar 18 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3194

Cf. A003714, A019565, A300835 (rgs-transform of this sequence), A300836.

Cf. also A293214, A293216, A293221, A293222, A293231.

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A003714(d)))); m; };

a(n) = Product_{d|n, dA019565(A003714(d)).

For n >= 1, A001222(a(n)) = A300836(n).

A318834 a(n) = Product_{d|n, dA019565(phi(d)), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 12, 6, 20, 2, 108, 2, 60, 30, 60, 2, 540, 2, 300, 90, 84, 2, 2700, 10, 140, 90, 2700, 2, 6300, 2, 420, 126, 44, 150, 121500, 2, 132, 210, 10500, 2, 283500, 2, 5292, 3150, 660, 2, 132300, 30, 5500, 66, 14700, 2, 267300, 210, 472500, 198, 1540, 2, 4630500, 2, 4620, 47250, 4620, 350, 873180, 2, 1452, 990

Offset: 1

Antti Karttunen, Sep 04 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A000010, A019565, A318835 (rgs-transform).

Cf. also A293214, A293231, A300834.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(eulerphi(d)))); m; };

a(n) = Product_{d|n, dA019565(A000010(d)).

A048675(a(n)) = A051953(n).

A064273 Permutation of nonnegative integers: a(n) = A013928(A019565(n)).

Original entry on oeis.org

0, 1, 2, 4, 3, 6, 10, 18, 5, 9, 13, 27, 22, 43, 64, 128, 7, 14, 20, 40, 33, 68, 100, 202, 47, 93, 143, 282, 232, 469, 702, 1404, 8, 16, 25, 48, 39, 79, 119, 235, 56, 110, 167, 333, 278, 553, 832, 1660, 88, 175, 260, 520, 437, 872, 1303, 2609, 608, 1216, 1826, 3649

Offset: 0

Howard A. Landman, Sep 23 2001

From Antti Karttunen, Aug 24 2014: (Start)
The original name of the sequence was: "Inverse of sequence A048672 considered as a permutation of the nonnegative integers".
However, the real inverse to A048672 is A246353(n) (= a(n)+1), satisfying A246353(A048672(n)) = n for all n. This sequence subtracts one from the terms of A246353 so as to obtain a permutation of nonnegative integers (bijection [0..] --> [0..]).
Sequence is obtained when the range of A019565 is compacted so that it becomes surjective, thus the logarithmic scatter plots look very similar. (Same applies to A246353.) Compare also to the plot of A005940.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..478
Index entries for sequences that are permutations of the natural numbers

One less than A246353.

Cf. A000040, A005940, A013928, A019565, A048672.

allocatemem(234567890);
default(primelimit, 2^22)
uplim_for_13928 = 13123111;
v013928 = vector(uplim_for_13928); A013928(n) = v013928[n];
v013928[1]=0; n=1; while((n < uplim_for_13928), if(issquarefree(n), v013928[n+1] = v013928[n]+1, v013928[n+1] = v013928[n]); n++);
A019565(n) = {factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ M. F. Hasler
A064273(n) = A013928(A019565(n));
for(n=0, 478, write("b064273.txt", n, " ", A064273(n))); \\ Antti Karttunen, Aug 23 2014

from math import prod, isqrt
from sympy import prime, mobius
def A064273(n):
    m = prod(prime(i) for i,j in enumerate(bin(n)[-1:1:-1],1) if j=='1')
    return int(sum(mobius(k)*(m//k**2) for k in range(1, isqrt(m)+1))-1) # Chai Wah Wu, Feb 23 2025

(define (A064273 n) (let loop ((n n) (i 1) (p 1)) (cond ((zero? n) (- (A013928 (+ 1 p)) 1)) ((odd? n) (loop (/ (- n 1) 2) (+ 1 i) (* p (A000040 i)))) (else (loop (/ n 2) (+ 1 i) p))))) ;; Antti Karttunen, Aug 23 2014

From Antti Karttunen, Aug 24 2014: (Start)
a(n) = A013928(A019565(n)).
a(n) = A246353(n) - 1.
(End)

More terms from Carl R. White, Apr 19 2006

Name changed by Antti Karttunen, Aug 23 2014

A285316 Numbers n for which A019565(n) > n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 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, 60, 61, 62, 63, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Antti Karttunen, Apr 18 2017

nonn

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

Complement: A285315.

Cf. A019565.

a019565[n_]:=Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2] ; Select[Range[0, 100], a019565[#]># &] (* Indranil Ghosh, Apr 18 2017, after Michael De Vlieger *)

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
isA285316(n) = (A019565(n) > n);
n=0; k=1; while(k <= 10000, if(isA285316(n),write("b285316.txt", k, " ", n);k=k+1); n=n+1);

from operator import mul
from sympy import prime
from functools import reduce
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
print([n for n in range(101) if a019565(n)>n]) # Indranil Ghosh, Apr 18 2017, after Chai Wah Wu

;; with Antti Karttunen's IntSeq-library
(define A285316 (MATCHING-POS 1 0 (lambda (n) (> (A019565 n) n))))

A300835 Restricted growth sequence transform of A300834, product_{d|n, dA019565(A003714(d)); Filter sequence related to Zeckendorf-representations of proper divisors of n.

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, 7, 35, 36, 37, 2, 38, 39, 40, 41, 42, 2, 43, 2, 44, 45, 46, 47, 48, 2, 49, 50, 51, 2, 52, 2, 53, 54, 55, 56, 57, 2, 58, 59, 60, 2, 61, 41, 62, 63, 64, 2, 65, 66, 67, 68, 69

Offset: 1

Antti Karttunen, Mar 18 2018

nonn

For all i, j: a(i) = a(j) => A001065(i) = A001065(j).

For all i, j: a(i) = a(j) => A300836(i) = A300836(j).

			For cases n=10 and 49, we see that 10 has proper divisors 1, 2 and 5 and these have Zeckendorf-representations (A014417) 1, 10 and 1000, while 49 has proper divisors 1 and 7 and these have Zeckendorf-representations 1 and 1010. When these Zeckendorf-representations are summed (columnwise without carries), result in both cases is 1011, thus a(10) = a(49).

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

Cf. A003714, A014417, A019565, A300834, A300836.

Cf. also A293215, A293217, A293223, A293224, A293232, A300833 for similar filtering sequences.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A003714(d)))); m; };
write_to_bfile(1,rgs_transform(vector(up_to,n,A300834(n))),"b300835.txt");

cp's OEIS Frontend

A319992 a(n) = Product_{d|n, dA019565(d)^[2 == d mod 3].

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A109162 a(1) = 1; for n > 1, a(n) = A019565(a(n-1)).

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A285331 Inverse for A285332: a(1) = 0, a(2) = 1, a(A019565(n)) = 2*a(n), a(A065642(n)) = 1 + 2*a(n).

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A293443 Multiplicative with a(p^e) = A019565(A193231(e)).

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A300831 a(n) = Product_{d|n, dA019565(d)^[moebius(n/d) = +1].

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A300834 a(n) = Product_{d|n, dA019565(A003714(d)), where A003714(n) is the n-th Fibbinary number.

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A318834 a(n) = Product_{d|n, dA019565(phi(d)), where phi is the Euler totient function A000010.

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A064273 Permutation of nonnegative integers: a(n) = A013928(A019565(n)).

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A285331 Inverse for A285332: a(1) = 0, a(2) = 1, a(A019565(n)) = 2a(n), a(A065642(n)) = 1 + 2a(n).