A285317 -id:A285317 - cp's OEIS frontend

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

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)).

A285319 Squarefree numbers n for which A019565(n) < n and A048675(n) is also squarefree.

Original entry on oeis.org

66, 129, 130, 258, 514, 1034, 1041, 1042, 2049, 2054, 2055, 2066, 2082, 2114, 4098, 4101, 4102, 4130, 4161, 4162, 4226, 4353, 4354, 4610, 5122, 8193, 8198, 8202, 8205, 8206, 8210, 8211, 8229, 8259, 8706, 9218, 9219, 12291

Offset: 1

Antti Karttunen, Apr 18 2017

Any finite cycle in A019565, if such cycles exist at all, must have at least one member that occurs somewhere in this sequence. Furthermore, such a number n should satisfy A019565(n) < n and that A048675(n)^k is squarefree for all k >= 0.

Subsequence of A285317.
Cf. A008683, A019565, A048675.
Cf. also A285320 and discussion in A285331 and A285332.

lim = 4000;
A019565 = Table[Times @@ Prime@Flatten@Position[#, 1] &@
   Reverse@IntegerDigits[n, 2], {n, 1, lim}]; (* From Michael De Vlieger in A019565 *)
A048675 = Table[Total[#[[2]]*2^(PrimePi[#[[1]]] - 1) & /@ FactorInteger[n]], {n, 1, lim}]; (* From Jean-François Alcover in A048675 *)
Select[Range[lim], A019565[[#]] < # && SquareFreeQ[#] &&
SquareFreeQ[A048675[[#]]] &] (* Robert Price, Apr 07 2019 *)

allocatemem(2^30);
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
isA285319(n) = (issquarefree(n) & (A019565(n) < n) && issquarefree(A048675(n)));
n=0; k=0; while(k <= 60, n=n+1; if(isA285319(n),print1(n,", ");k=k+1));

;; With Antti Karttunen's IntSeq-library.
(define A285319 (MATCHING-POS 1 0 (lambda (n) (and (< (A019565 n) n) (not (zero? (A008683 n))) (not (zero? (A008683 (A048675 n))))))))

A285315 Numbers n for which A019565(n) < n.

Original entry on oeis.org

8, 16, 32, 33, 64, 65, 66, 128, 129, 130, 131, 132, 136, 256, 257, 258, 259, 260, 261, 264, 272, 512, 513, 514, 515, 516, 517, 518, 520, 521, 528, 544, 576, 640, 768, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032, 1033, 1034, 1040, 1041, 1042, 1056, 1057, 1088, 1089, 1152, 1280, 1536, 2048, 2049, 2050, 2051

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 consecutively nested subsequences A285317, A285319, ..., and in general, it should satisfy A019565(n) < n and that A048675^{k}(n) is squarefree for all k = 0 .. ∞.

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

Complement: A285316.
Cf. A019565, A048675.
Cf. A285317, A285319 (subsequences).

a019565[n_]:=Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2] ; Select[Range[3000], 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
isA285315(n) = (A019565(n) < n);
n=0; k=1; while(k <= 10000, n=n+1; if(isA285315(n),write("b285315.txt", k, " ", n);k=k+1));

from sympy import prime, prod
def a019565(n): return prod(prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1') if n > 0 else 1
[n for n in range(1, 3001) if a019565(n)Indranil Ghosh, Apr 18 2017, after Chai Wah Wu

;; With Antti Karttunen's IntSeq-library.
(define A285315 (MATCHING-POS 1 0 (lambda (n) (< (A019565 n) n))))

cp's OEIS Frontend

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

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A285319 Squarefree numbers n for which A019565(n) < n and A048675(n) is also squarefree.

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A285315 Numbers n for which A019565(n) < n.

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