A019565 -id:A019565 - cp's OEIS frontend

A285317 Squarefree numbers n for which A019565(n) < n.

33, 65, 66, 129, 130, 131, 257, 258, 259, 514, 515, 517, 518, 521, 1027, 1030, 1031, 1033, 1034, 1041, 1042, 1057, 2049, 2051, 2053, 2054, 2055, 2059, 2065, 2066, 2081, 2082, 2113, 2114, 2177, 2305, 2561, 3073, 4097, 4098, 4099, 4101, 4102, 4103, 4105, 4106, 4109, 4115, 4129, 4130, 4161, 4162, 4226, 4353, 4354, 4609, 4610, 5122

Offset: 1

Antti Karttunen, Apr 18 2017

nonn

Any finite cycle in A019565, if such cycles exist at all, must have at least one member that occurs somewhere in this sequence, although certainly not all terms of this sequence could occur in a finite cycle. Specifically, such a number n must occur also in subsequence A285319, and in general, it should satisfy A019565(n) < n and that A048675^{k}(n) is squarefree for all k = 0 .. oo.

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

Intersection of A005117 and A285315.

Cf. A005117, A008683, A019565, A048675, A285318, A285319 (subsequence), A285332.

a019565[n_]:=Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2] ; Select[Range[5200], SquareFreeQ[#] && 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
isA285317(n) = (issquarefree(n) & (A019565(n) < n));
n=0; k=1; while(k <= 10000, n=n+1; if(isA285317(n),write("b285317.txt", k, " ", n);k=k+1));

from operator import mul
from functools import reduce
from sympy import prime
from sympy.ntheory.factor_ import core
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(1, 5201) if core(n) == n and a019565(n) < n]) # Indranil Ghosh, Apr 18 2017, after Chai Wah Wu

;; With Antti Karttunen's IntSeq-library.
(define A285317 (MATCHING-POS 1 0 (lambda (n) (and (< (A019565 n) n) (not (zero? (A008683 n)))))))

a(n) = A019565(A285318(n)).

A285327 Row 3 of A285325: a(n) = A048675(A065642(A065642(A019565(n)))).

Original entry on oeis.org

0, 3, 6, 5, 12, 7, 10, 9, 24, 11, 18, 13, 20, 15, 18, 17, 48, 19, 22, 21, 36, 23, 26, 25, 40, 27, 34, 29, 36, 31, 34, 33, 96, 35, 38, 37, 68, 39, 42, 41, 72, 43, 50, 45, 52, 47, 50, 49, 80, 51, 54, 53, 68, 55, 58, 57, 72, 59, 66, 61, 68, 63, 66, 65, 192, 67, 70, 69, 132, 71, 74, 73, 136, 75, 82, 77, 84, 79, 82, 81, 144, 83, 86, 85, 100, 87, 90, 89

Offset: 0

Antti Karttunen, Apr 19 2017

nonn

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

Row 3 of A285325 (after the initial zero).

Cf. A007947, A019565, A048675, A065642, A285323, A285324.

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
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016
A007947(n) = factorback(factorint(n)[, 1]); \\ From Andrew Lelechenko, May 09 2014
A065642(n) = { my(r=A007947(n)); if(1==n,n,n = n+r; while(A007947(n) <> r, n = n+r); n); };
A285327(n) = A048675(A065642(A065642(A019565(n))));

(define (A285327 n) (A048675 (A065642 (A065642 (A019565 n)))))

a(n) = A048675(A065642(A065642(A019565(n)))).

A286612 Numbers n for which A019565(n) <= A087207(n) < n.

Original entry on oeis.org

65, 513, 1026, 4107, 8200, 8256, 16400, 16450, 16512, 16544, 16641, 32800, 32900, 33024, 33282, 33920, 49184, 65600, 65800, 66048, 66080, 131200, 131334, 132096, 132160, 163968, 262400, 262668, 264192, 264320, 274432, 327936, 524342, 524610, 524800, 524832, 525826, 528384, 532500, 540736, 548864, 655872, 786467, 1048617

Offset: 1

Antti Karttunen, Jun 20 2017

Any 2-cycle of A087207 and also any 2-cycle of A019565 (in which case A019565(x) = A087207(x) for both members of the cycle), if such cycles exist at all, must have the larger one of its members included in this sequence.

Intersection of A286608 and A286611.
Subsequence of A285315.
Cf. A019565, A087207, A285319, A286609.

A294932 Multiplicative with a(p^e) = A019565(A289814(e)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1

Offset: 1

Antti Karttunen, Nov 11 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, A289814, A293442, A294931, A294933.

(define (A294932 n) (if (= 1 n) n (* (A019565 (A289814 (A067029 n))) (A294932 (A028234 n)))))

A296071 a(n) = Product_{d|n, dA019565(A289813(A295882(d))); a product obtained from the 1's present in balanced ternary representation of the deficiencies of the proper divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 8, 6, 24, 2, 24, 2, 20, 36, 16, 2, 60, 2, 144, 30, 40, 2, 48, 12, 60, 30, 240, 2, 1080, 2, 32, 60, 56, 60, 120, 2, 28, 90, 576, 2, 3600, 2, 400, 900, 168, 2, 96, 10, 1008, 84, 1200, 2, 420, 120, 480, 42, 56, 2, 4320, 2, 84, 1500, 64, 180, 4200, 2, 784, 252, 90720, 2, 1200, 2, 140, 2520, 784, 100, 75600, 2, 1152, 210, 840, 2

Offset: 1

Antti Karttunen, Dec 04 2017

nonn

Used as a part of filter A296073.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A033879, A019565, A289813, A295882, A296072, A296073.

Cf. also A293221.

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
A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A289813(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From Rémy Sigrist
A295882(n) = { my(x = (2*n)-sigma(n)); if(x >= 0,A117967(x),A117968(-x)); };
A296071(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(A295882(d))))); m; };

(define (A296071 n) (let loop ((m 1) (props (proper-divisors n))) (cond ((null? props) m) (else (loop (* m (A019565 (A289813 (A295882 (car props))))) (cdr props))))))
(define (proper-divisors n) (reverse (cdr (reverse (divisors n)))))
(define (divisors n) (let loop ((k n) (divs (list))) (cond ((zero? k) divs) ((zero? (modulo n k)) (loop (- k 1) (cons k divs))) (else (loop (- k 1) divs)))))

a(n) = Product_{d|n, dA019565(A289813(A295882(d))).

A296072 a(n) = Product_{d|n, dA019565(A289814(A295882(d))); a product obtained from the -1's present in balanced ternary representation of the deficiencies of the proper divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 12, 1, 2, 6, 1, 1, 12, 1, 1, 12, 3, 1, 12, 1, 1, 2, 15, 3, 216, 1, 5, 2, 6, 1, 6, 1, 2, 36, 5, 1, 180, 3, 10, 30, 1, 1, 1080, 1, 3, 10, 1, 1, 3240, 1, 1, 36, 1, 1, 20, 1, 450, 10, 30, 1, 45360, 1, 1, 30, 75, 3, 10, 1, 60, 360, 1, 1, 540, 15, 105, 2, 2, 1, 3240, 3, 50, 2, 35, 5, 2520, 1, 630, 60, 90, 1, 900

Offset: 1

Antti Karttunen, Dec 04 2017

nonn

Used as a part of filter A296073.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A033879, A019565, A289814, A295882, A296071, A296073.

Cf. also A293222.

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
A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A289814(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From Rémy Sigrist
A295882(n) = { my(x = (2*n)-sigma(n)); if(x >= 0,A117967(x),A117968(-x)); };
A296072(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(A295882(d))))); m; };

(define (A296072 n) (let loop ((m 1) (props (proper-divisors n))) (cond ((null? props) m) (else (loop (* m (A019565 (A289814 (A295882 (car props))))) (cdr props))))))
(define (proper-divisors n) (reverse (cdr (reverse (divisors n)))))
(define (divisors n) (let loop ((k n) (divs (list))) (cond ((zero? k) divs) ((zero? (modulo n k)) (loop (- k 1) (cons k divs))) (else (loop (- k 1) divs)))))

a(n) = Product_{d|n, dA019565(A289814(A295882(d))).

A304087 Divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A304083(n)).

Original entry on oeis.org

1, 2, 6, 3, 30, 15, 5, 10, 210, 105, 35, 7, 70, 14, 42, 21, 2310, 1155, 385, 77, 11, 770, 154, 22, 462, 231, 33, 330, 165, 55, 110, 30030, 66, 6006, 3003, 1001, 143, 13, 15015, 5005, 715, 65, 10010, 2002, 286, 26, 4290, 2145, 429, 39, 858, 78, 2730, 1365, 455, 91, 910, 182, 546, 273, 510510, 1430, 130, 390, 195, 255255, 85085, 17017, 2431, 221, 17, 170170

Offset: 0

Antti Karttunen, May 06 2018

nonn

Each a(n+1) is either a divisor or a multiple of a(n).

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

Cf. A019565, A304083, A304085.

Cf. also A303778.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A304087(n) = A019565(A304083(n)); \\ Needs also code from A304083.

a(n) = A019565(A304083(n)).

A304537 Suspected divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A304533(n)).

Original entry on oeis.org

1, 2, 6, 3, 15, 5, 65, 13, 26, 182, 7, 14, 42, 21, 105, 35, 455, 91, 910, 10, 30, 210, 70, 2730, 39, 78, 546, 273, 1365, 195, 7995, 41, 82, 246, 123, 615, 205, 2665, 533, 1066, 11726, 11, 22, 66, 33, 165, 55, 715, 143, 286, 2002, 77, 154, 462, 231, 1155, 385, 5005, 1001, 10010, 110, 330, 2310, 770, 30030, 429, 858, 6006, 3003, 15015, 2145, 87945, 451, 902

Offset: 0

Antti Karttunen, May 15 2018

nonn

Each a(n) is always either a divisor or a multiple of a(n+1).
Consider A052330. Imagine that it is an automatic piano that "plays sequences" when an appropriate punched "tape" is fed to it (as its input), i.e., when it is composed from the right with an appropriate sequence p, as A019565(p(n)). The 1-bits in the binary expansion of each p(n) are the "holes" in the tape, and they determine which "tunes" are present on beat n. The "tunes" are actually "Fermi-Dirac primes" (A050376) that are multiplied together.
If the tape is constructed in such a way that between the successive beats (when moving from p(n) to p(n+1)), either a subset of 0-bits are toggled on (changed to 1's), or a subset of 1-bits are toggled off (changed to 0's), but no both kind of changes occur simultaneously, then when fed as an input to this piano, the resulting sequence "s" (the output) is guaranteed to satisfy the condition that s(n+1) is either a multiple or a divisor of s(n). Furthermore, if the given sequence p is itself a permutation of natural numbers, then also the produced sequence is. For example, Gray code A003188 and its inverse A006068 are such sequences, and when given as an "input tape" for A052330, they produce permutations A207901 and A302783.
There is a simpler instrument, called "squarefree piano" (A019565), with which it is possible to produce similar divisor-or-multiple sequences, but that contain only squarefree numbers. Given A003188 or A006068 as an input tape for it produces correspondingly sequences A302033 and A284003.
This sequence is obtained by playing "squarefree piano" with the same tape which yields A304531 when "Fermi-Dirac piano" is played with it. However, in this case the sequence A304531 is produced by a greedy algorithm, and thus its tape (A304533) is actually a back-formation, obtained from the "music" (A304531) by applying "tape-recorder" (A052331) to it. Note that this in not a subsequence of A304531, as the terms occur in different order than the squarefree terms of A304531.
See also Peter Munn's Apr 11 2018 message on SeqFan-mailing list.

Antti Karttunen, Table of n, a(n) for n = 0..8447
Michel Marcus, Peter Munn, et al, Discussion on SeqFan-list, April 2018

Cf. A019565, A304531, A304533.

Cf. also A303760, A303771.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A304533(n) = A052331(A304531(1+n));
A304537(n) = A019565(A304533(n));

a(n) = A019565(A304533(n)) = A019565(A052331(A304531(1+n))).

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

Original entry on oeis.org

2, 2, 3, 3, 5, 3, 15, 5, 15, 5, 21, 5, 35, 15, 7, 7, 11, 15, 33, 7, 35, 21, 165, 7, 55, 35, 33, 35, 385, 7, 1155, 11, 55, 11, 77, 35, 65, 33, 77, 11, 91, 35, 273, 55, 77, 165, 1365, 11, 273, 55, 13, 77, 715, 33, 91, 77, 65, 385, 3003, 11, 5005, 1155, 65, 13, 143, 55, 51, 13, 455, 77, 255, 77, 119, 65, 91, 65, 5005, 77

Offset: 1

Antti Karttunen, Feb 25 2020

nonn

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

Cf. A000010, A003961, A019565, A318834, A332824.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A332825(n) = A019565(eulerphi(n));

a(n) = A019565(A000010(n)).
a(n) = A332824(n) / A318834(n)
a(4n) = A003961(a(2n)), a(4n+2) = a(2n+1).

A339810 a(n) = A046523(A019565(n) - 1).

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 2, 6, 2, 12, 2, 6, 6, 24, 6, 6, 6, 32, 6, 24, 2, 12, 6, 12, 12, 30, 2, 384, 2, 6, 2, 12, 4, 6, 6, 64, 6, 6, 2, 60, 2, 48, 6, 6, 12, 60, 2, 6, 30, 12, 2, 210, 2, 96, 2, 216, 30, 30, 6, 180, 2, 6, 2, 16, 6, 12, 2, 60, 4, 6, 2, 6, 6, 12, 6, 120, 6, 24, 6, 30, 2, 240, 6, 6, 30, 12, 6, 60, 2, 30, 2, 48

Offset: 1

Antti Karttunen, Dec 18 2020

nonn

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

Cf. A019565, A046523, A339809, A339811 (rgs-transform), A339812.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A339810(n) = A046523(A019565(n)-1);

a(n) = A046523(A339809(n)) = A046523(A019565(n) - 1).

cp's OEIS Frontend

A285317 Squarefree numbers n for which A019565(n) < n.

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A285327 Row 3 of A285325: a(n) = A048675(A065642(A065642(A019565(n)))).

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A286612 Numbers n for which A019565(n) <= A087207(n) < n.

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A294932 Multiplicative with a(p^e) = A019565(A289814(e)).

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A296071 a(n) = Product_{d|n, dA019565(A289813(A295882(d))); a product obtained from the 1's present in balanced ternary representation of the deficiencies of the proper divisors of n.

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A296072 a(n) = Product_{d|n, dA019565(A289814(A295882(d))); a product obtained from the -1's present in balanced ternary representation of the deficiencies of the proper divisors of n.

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A304087 Divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A304083(n)).

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A304537 Suspected divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A304533(n)).

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