A347241 -id:A347241 - cp's OEIS frontend

A336361 Number of iterations of A000593 (sum of divisors of odd part of n) needed to reach a power of 2, or -1 if never reached.

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

Offset: 1

Antti Karttunen, Jul 30 2020

nonn

Also, for n > 1, one less than the number of iterations of A000593 to reach 1.

If there exists any hypothetical odd perfect numbers w, then the iteration will get stuck into a fixed point after encountering them, and we will have a(w) = a(2^k * w) = -1 by the escape clause.

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

Cf. A000265, A000593, A161942, A209229, A336362, A336363, A347249, A347250.
Cf. A054784 (positions of 0's and 1's in this sequence).
Cf. also A347240, A347241, A347242, A347243, A347244, A347245.

A336361(n) = if(!bitand(n,n-1),0,1+A336361(sigma(n>>valuation(n,2))));

If A209229(n) = 1 [when n is a power of 2], a(n) = 0, otherwise a(n) = 1+a(A000593(n)).

a(n) = a(2n) = a(A000265(n)).

A347240 a(n) is the largest prime factor (A006530) of all terms encountered when iterating the map x -> A000593(x), when starting from x = n, but excluding the n itself. If n is a power of 2, then a(n) = 1. If 1 is never reached, then a(n) = -1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 2, 1, 13, 3, 3, 2, 7, 2, 3, 1, 13, 13, 5, 3, 2, 3, 3, 2, 31, 7, 5, 2, 5, 3, 2, 1, 3, 13, 3, 13, 19, 5, 7, 3, 7, 2, 11, 3, 13, 3, 3, 2, 19, 31, 13, 7, 5, 5, 13, 2, 5, 5, 5, 3, 31, 2, 13, 1, 7, 3, 17, 13, 3, 3, 13, 13, 37, 19, 31, 5, 3, 7, 5, 3, 19, 7, 7, 2, 5, 11, 5, 3, 13, 13, 7, 3, 2, 3, 5, 2, 19

Offset: 1

Antti Karttunen, Aug 28 2021

nonn

			For n = 17, the iteration proceeds as follows 17 -> 18 (= 2*3*3), 18 -> 13 (13 is a prime), 13 -> 14 (= 2*7), 14 -> 8 (= 2*2*2), 8 -> 1. The largest prime factor present after the initial step is 13, thus a(17) = 13.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000593, A006530, A336361, A347241, A347242, A347243, A347244.

Cf. also A161942.

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A000265(n) = (n >> valuation(n, 2));
A000593(n) = sigma(A000265(n));
A347240(n) = { my(m=1); while(n>1, n = A000593(n); m = max(m, A006530(n))); (m); };

a(n) = A347241(A000593(n)). - Antti Karttunen, Feb 10 2022

A347242 Numbers k such that when iterating the map x -> A000593(x), at some point before 1 is reached (after starting from x=k), a term is encountered whose largest prime factor is at least as large as A006530(k).

Original entry on oeis.org

9, 18, 25, 27, 36, 45, 49, 50, 54, 55, 63, 72, 75, 81, 90, 98, 99, 100, 108, 110, 117, 121, 125, 126, 135, 144, 147, 150, 162, 165, 169, 175, 180, 196, 198, 200, 216, 220, 225, 234, 242, 243, 245, 250, 252, 270, 275, 288, 289, 294, 300, 315, 324, 325, 330, 338, 343, 350, 360, 361, 363, 375, 385, 392, 396, 400

Offset: 1

Antti Karttunen, Aug 28 2021

nonn

Provided there do not exist any odd perfect numbers, these are numbers k for which A347240(k) >= A006530(k), as for any odd perfect number x, A347240(x) = -1 by its escape clause.
If k is included as a term, then 2*k is also present.
Not all odd squares of primes are present. For example, 67^2 and 79^2 are not included. See also A091490, which seems to be a subsequence of those exceptions.
Conjecture: There are no primes in this sequence. Checked up to the 2^20-th prime, 16290047.

			For n = 55 = 5*11, on the first iteration we get A000593(55) = 72 = 2^3 * 3^2, but both 2 and 3 are less than 11; therefore we iterate a second time to get A000593(72) = 13, which is the first value whose largest prime factor is larger than that of 55 (13 > 11), thus 55 is included in the sequence.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000593, A006530, A091490, A161942, A336353, A336361, A347240, A347241, A347243 (complement), A347244 (characteristic function).

Positions of nonzero terms in A347245.

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A000265(n) = (n >> valuation(n, 2));
A000593(n) = sigma(A000265(n));
A347244(n) = { my(gpf=A006530(n)); while(n>1, n = A000593(n); if(A006530(n)>=gpf,return(1))); (0); };
isA347242(n) = A347244(n);

A347243 Numbers k such that when iterating the map x -> A000593(x), we will not encounter a term x (after the starting point x=k) whose largest prime factor would be at least as large as A006530(k), before 1 is eventually reached.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 51, 52, 53, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 101

Offset: 1

Antti Karttunen, Aug 28 2021

nonn

The initial 1 is included by a convention.

			For n = 17, the iteration proceeds as follows 17 -> 18 (= 2*3*3), 18 -> 13 (13 is a prime), 13 -> 14 (= 2*7), 14 -> 8 (= 2*2*2), 8 -> 1. The largest prime factor present after the initial step is 13, which is less than the largest prime factor of 17 (which is 17 itself), thus 17 is included in this sequence.

Cf. A000593, A006530, A161942, A336361, A347240, A347241, A347242 (complement).
Positions of zeros in A347244 and in A347245.
Subsequences: A000040 (conjectured), A000079.

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A000265(n) = (n >> valuation(n, 2));
A000593(n) = sigma(A000265(n));
A347244(n) = { my(gpf=A006530(n)); while(n>1, n = A000593(n); if(A006530(n)>=gpf,return(1))); (0); };
isA347243(n) = !A347244(n);

A351460 Lexicographically earliest infinite sequence such that a(i) = a(j) => A006530(i) = A006530(j), A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 11 2022

nonn

Restricted growth sequence transform of the ordered triplet [A006530(n), A206787(n), A336651(n)].
For all i, j >= 1:
  A324400(i) = A324400(j) => a(i) = a(j),
  a(i) = a(j) => A347241(i) = A347241(j),
  a(i) = a(j) => A351461(i) = A351461(j) => A347240(i) = A347240(j).

			a(429) = a(455) because 429 = 3*11*13 and 455 = 5*7*13, so they have equal largest prime factor (A006530), and they also agree on A206787(429) = A206787(455) = 672 and on A336651(429) = A336651(455) = 1 (because both are squarefree), therefore they get equal value (which is 216) allotted to them by the restricted growth sequence transform. - _Antti Karttunen_, Feb 14 2022

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

Cf. A006530, A206787, A336651, A347241, A351461.
Cf. also A324400, A351452.
Differs from A351454 for the first time at n=121, where a(121) = 62, while A351454(121) = 51.
Differs from A103391(1+n) for the first time after n=1 at n=455, where a(455) = 216, while A103391(456) = 229.

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; };
A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A206787(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ From A206787
A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };
Aux351460(n) = [A006530(n), A206787(n), A336651(n)];
v351460 = rgs_transform(vector(up_to, n, Aux351460(n)));
A351460(n) = v351460[n];

cp's OEIS Frontend

A336361 Number of iterations of A000593 (sum of divisors of odd part of n) needed to reach a power of 2, or -1 if never reached.

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A347240 a(n) is the largest prime factor (A006530) of all terms encountered when iterating the map x -> A000593(x), when starting from x = n, but excluding the n itself. If n is a power of 2, then a(n) = 1. If 1 is never reached, then a(n) = -1.

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Formula

A347242 Numbers k such that when iterating the map x -> A000593(x), at some point before 1 is reached (after starting from x=k), a term is encountered whose largest prime factor is at least as large as A006530(k).

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A347243 Numbers k such that when iterating the map x -> A000593(x), we will not encounter a term x (after the starting point x=k) whose largest prime factor would be at least as large as A006530(k), before 1 is eventually reached.

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A351460 Lexicographically earliest infinite sequence such that a(i) = a(j) => A006530(i) = A006530(j), A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

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