A285332 -id:A285332 - cp's OEIS frontend

A285320 If n == 0 or A008683(n) == 0, then a(n) = 0, otherwise a(n) = 1+a(A048675(n)); number of iterations of A048675 needed before the result is either zero or nonsquarefree number (A013929).

0, 1, 2, 3, 0, 1, 4, 1, 0, 0, 2, 1, 0, 1, 1, 5, 0, 1, 0, 1, 0, 3, 2, 1, 0, 0, 2, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 3, 0, 1, 2, 1, 0, 0, 2, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 3, 1, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 3, 2, 1, 0, 1

Offset: 0

Antti Karttunen, Apr 18 2017

Conjecture: all terms are well-defined (finite). This implies also the conjecture I have made in A019565.

			a(38) = 3 because 38 = 2*19 (thus squarefree), A048675(38) = 129 (= 3*43), A048675(129) = 8194 (= 2*17*241) and A048675(8194) = 4503599627370561 (= 3^2 * 37 * 71 * 190483425427), so three steps were needed before nonsquarefree number was reached.
a(74) >= 3 as A048675(74) = 2049 (squarefree), A048675(2049) =  10633823966279326983230456482242756610 (squarefree), A048675(10633823966279326983230456482242756610) = ???

A left inverse of A109162.
Cf. A008683, A005117, A013929, A048675.
Cf. also A285319, A285331, A285332.

A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016
A285320(n) = if(!n || !moebius(n),0,1+A285320(A048675(n)));

(define (A285320 n) (if (or (zero? n) (zero? (A008683 n))) 0 (+ 1 (A285320 (A048675 n)))))

If n == 0 or A008683(n) == 0, then a(n) = 0, otherwise a(n) = 1+a(A048675(n)).

a(A109162(n)) = n.

A322807 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = A285330(n) for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

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

Cf. A048675, A285328, A285330, A285332, A322806.

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; };
A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n,(n-1))), (n/2), r=A007947(n); if(r==n,1,n = n-r; while(A007947(n) <> r, n = n-r); n)); };
A285330(n) = if(issquarefree(n),A048675(n),A285328(n));
A322807aux(n) = if((n%2)&&isprime(n),-1,A285330(n));
v322807 = rgs_transform(vector(up_to,n,A322807aux(n)));
A322807(n) = v322807[n];

A286543 Restricted growth sequence of A286542.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 2, 5, 4, 6, 4, 5, 4, 3, 2, 7, 8, 6, 4, 6, 9, 6, 8, 7, 4, 6, 9, 5, 9, 3, 2, 10, 8, 11, 8, 6, 9, 6, 4, 12, 8, 13, 9, 12, 14, 11, 9, 10, 9, 6, 4, 12, 8, 13, 14, 7, 8, 13, 9, 5, 9, 3, 2, 15, 16, 11, 8, 11, 14, 11, 8, 12, 8, 13, 14, 6, 4, 6, 4, 17, 18, 11, 14, 13, 9, 13, 9, 6, 19, 20, 14, 11, 14, 13, 14, 15, 18, 13, 14, 6, 18, 6, 4, 6, 18, 11, 8

Offset: 0

Antti Karttunen, May 18 2017

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

Cf. A046523, A285332, A286542, A286544.

allocatemem(2^30);
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A000120(n) = hammingweight(n);
A002110(n) = prod(i=1,n,prime(i));
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
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); };
A285332(n) = { if(n<=1,n+1,if(!(n%2),A019565(A285332(n/2)),A065642(A285332((n-1)/2)))); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286542(n) = if(!n,1,if(!(n%2),A002110(A000120(A285332(n/2))),A046523(A285332(n))));
write_to_bfile(0,rgs_transform(vector(8192,n,A286542(n-1))),"b286543.txt");

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

Original entry on oeis.org

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)).

A322806 Lexicographically earliest such sequence a that a(i) = a(j) => A285330(i) = A285330(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 6, 4, 5, 7, 8, 9, 10, 11, 9, 6, 12, 13, 14, 15, 15, 16, 17, 18, 7, 19, 11, 20, 21, 22, 23, 8, 18, 24, 13, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 22, 38, 39, 40, 41, 42, 29, 43, 44, 45, 46, 47, 48, 49, 50, 10, 37, 51, 52, 28, 53, 54, 55, 56, 57, 58, 59, 60, 25, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 38

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

Restricted growth sequence transform of A285330.

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

Cf. A048675, A285328, A285330, A285332, A285333, A286543, A322807.

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; };
A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n,(n-1))), (n/2), r=A007947(n); if(r==n,1,n = n-r; while(A007947(n) <> r, n = n-r); n)); };
A285330(n) = if(issquarefree(n),A048675(n),A285328(n));
v322806 = rgs_transform(vector(up_to,n,A285330(n)));
A322806(n) = v322806[n];

A285318 a(n) = A048675(A285317(n)).

Original entry on oeis.org

18, 36, 19, 8194, 37, 2147483648, 18014398509481984, 8195, 2056, 18014398509481985, 67108868, 16400, 2057, 158456325028528675187087900672, 2097184, 67108869, 5986310706507378352962293074805895248510699696029696, 11972621413014756705924586149611790497021399392059392, 16401, 295147905179352825858, 158456325028528675187087900673, 34359738376

Offset: 1

Antti Karttunen, Apr 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..130

Cf. A048675, A285317, A285319, A285332.

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));
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016
n=0; k=1; while(k <= 130, n=n+1; if(isA285317(n),write("b285318.txt", k, " ", A048675(n));k=k+1));

(define (A285318 n) (A048675 (A285317 n)))

a(n) = A048675(A285317(n)).

A286544 Restricted growth sequence of A285333.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 18 2017

nonn

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

Cf. A285332, A285333, A286543.

allocatemem(2^30);
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
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); };
A285332(n) = { if(n<=1,n+1,if(!(n%2),A019565(A285332(n/2)),A065642(A285332((n-1)/2)))); };
A285333(n) = if(!n,n,if(!(n%2),A285332(n/2),A048675(A285332(n))));
write_to_bfile(0,rgs_transform(vector(8192,n,A285333(n-1))),"b286544.txt");

cp's OEIS Frontend

A285320 If n == 0 or A008683(n) == 0, then a(n) = 0, otherwise a(n) = 1+a(A048675(n)); number of iterations of A048675 needed before the result is either zero or nonsquarefree number (A013929).

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A322807 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = A285330(n) for all other numbers.

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A286543 Restricted growth sequence of A286542.

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

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A322806 Lexicographically earliest such sequence a that a(i) = a(j) => A285330(i) = A285330(j) for all i, j.

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A285318 a(n) = A048675(A285317(n)).

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A286544 Restricted growth sequence of A285333.

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