A336361 -id:A336361 - cp's OEIS frontend

A347249 a(n) = A331410(n) - A336361(n).

0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1, -1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 1, -1, 1, -1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 1, -1, -1, 1, 1, -1, -1, 0, 0, 3, 0, 1, 1, 0, 0, -1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 1, 0, 2, 0, -2, -2, 1, 2

Offset: 1

Antti Karttunen, Aug 28 2021

sign

Terms 0 .. 9 occur for the first time at n = 1, 15, 25, 75, 275, 725, 2175, 3725, 9025, 27075.

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

Cf. A000265, A331410, A336361, A347250 (positions of negative terms).

Cf. also A347374.

A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); };
A336361(n) = if(!bitand(n,n-1),0,1+A336361(sigma(n>>valuation(n,2))));
A347249(n) = (A331410(n)-A336361(n));

a(n) = A331410(n) - A336361(n).

For all n >= 1, a(n) = a(2*n) = a(A000265(n)).

A347250 Numbers k for which A331410(k) < A336361(k).

Original entry on oeis.org

9, 17, 18, 34, 36, 49, 67, 68, 71, 72, 81, 97, 98, 134, 136, 142, 144, 147, 162, 193, 194, 196, 268, 271, 272, 283, 284, 288, 291, 293, 294, 324, 386, 388, 392, 536, 541, 542, 544, 566, 568, 576, 579, 582, 586, 587, 588, 647, 648, 679, 772, 776, 784, 961, 1072, 1082, 1084, 1087, 1088, 1132, 1136, 1151, 1152, 1158, 1163

Offset: 1

Antti Karttunen, Aug 28 2021

nonn

If k is a term, then also 2*k is present in this sequence.

Cf. A331410, A336361.

Positions of negative terms in A347249.

A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); };
A336361(n) = if(!bitand(n,n-1),0,1+A336361(sigma(n>>valuation(n,2))));
A347249(n) = (A331410(n)-A336361(n));
isA347250(n) = (A347249(n)<0);

A054784 Integers n such that sigma(2n) - sigma(n) is a power of 2, where sigma is the sum of the divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 31, 32, 42, 48, 56, 62, 64, 84, 93, 96, 112, 124, 127, 128, 168, 186, 192, 217, 224, 248, 254, 256, 336, 372, 381, 384, 434, 448, 496, 508, 512, 651, 672, 744, 762, 768, 868, 889, 896, 992, 1016, 1024, 1302, 1344, 1488

Offset: 1

Labos Elemer, May 22 2000

nonn

If n is a squarefree product of Mersenne primes multiplied by a power of 2, then sigma(2n) - sigma(n) is a power of 2.
The reverse is also true. All numbers in this sequence have this form. - Ivan Neretin, Aug 12 2016
From Antti Karttunen, Sep 01 2021: (Start)
Numbers k such that the sum of their odd divisors [A000593(k)] is a power of 2.
Numbers k whose odd part [A000265(k)] is in A046528.
(End)

			For n=12, sigma(2n) = sigma(24) = 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 and sigma(n) = sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. So sigma(2n) - sigma(n) = 60 - 28 = 32 = 2^5 is a power of 2, and therefore 12 is in the sequence. - _Michael B. Porter_, Aug 15 2016

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000203, A000265, A000396 (even terms form a subsequence), A000593, A000668, A046528, A063883, A209229, A306204, A331410, A336923 (characteristic function).
Positions of zeros in A336922. Positions of 0's and 1's in A336361.
Cf. also A003401.

N:= 10^6: # to get all terms <= N
M:= select(isprime, [seq(2^i-1, i=select(isprime, [$2..ilog2(N+1)]))]):
R:= map(t -> seq(2^i*t, i=0..floor(log[2](N/t))), map(convert,combinat:-powerset(M),`*`)):
sort(convert(R,list)); # Robert Israel, Aug 12 2016

Sort@Select[Flatten@Outer[Times, p2 = 2^Range[0, 11], Times @@ # & /@ Subsets@Select[p2 - 1, PrimeQ]], # <= Max@p2 &] (* Ivan Neretin, Aug 12 2016 *)
Select[Range[1500],IntegerQ[Log2[DivisorSigma[1,2#]-DivisorSigma[1,#]]]&] (* Harvey P. Dale, Apr 23 2019 *)

A209229(n) = (n && !bitand(n,n-1));
isA054784(n) = A209229(sigma(n>>valuation(n,2))); \\ Antti Karttunen, Aug 28 2021

Numbers n such that A000203(2*n) - A000203(n) = 2^w for some w.

Sum_{n>=1} 1/a(n) = 2 * Product_{p in A000668} (1 + 1/p) = 2 * A306204 = 3.1711177758... . - Amiram Eldar, Jan 11 2023

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);

A347241 a(1) = 1, and for n > 1, a(n) is the largest prime factor (A006530) of all terms encountered when iterating the map x -> A000593(x), when starting from x = n, and including the n itself. If 1 is never reached when starting from n, then a(n) = -1.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 13, 5, 11, 3, 13, 7, 5, 2, 17, 13, 19, 5, 7, 11, 23, 3, 31, 13, 5, 7, 29, 5, 31, 2, 11, 17, 7, 13, 37, 19, 13, 5, 41, 7, 43, 11, 13, 23, 47, 3, 19, 31, 17, 13, 53, 5, 13, 7, 19, 29, 59, 5, 61, 31, 13, 2, 13, 11, 67, 17, 23, 7, 71, 13, 73, 37, 31, 19, 11, 13, 79, 5, 19, 41, 83, 7, 17, 43, 29

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 (when including the starting term also) is 17, thus a(17) = 17.

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

Cf. A000593, A006530, A161942, A336361, A347240, A347242.

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A000265(n) = (n >> valuation(n, 2));
A000593(n) = sigma(A000265(n));
A347241(n) = { my(m=1); while(n>1, m = max(m, A006530(n)); n = A000593(n)); (m); };

a(n) = max(A006530(n), A347240(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);

A347245 Number of iterations of the map x -> A000593(x), when starting from x = n, needed to reach a number whose largest prime factor is at least as large as that of n itself (= A006530(n)). If 1 is reached without encountering such a number, then a(n) = 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Aug 28 2021

nonn

For any hypothetical odd perfect number x, a(x) = 1.

The first occurrence of k = 0 .. 5 is at n = 0, 9, 55, 935, 102753, 262205.

			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, thus on the second iteration, we get A000593(72) = 13, which is the first time when the largest prime factor is larger than that of 55 (13 > 11), thus a(55) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..102753
Index entries for sequences related to sigma(n)

Cf. A000593, A006530, A336361, A347240, A347242 (positions of terms > 0), A347243 (positions of zeros), A347244, A347246.

A000265(n) = (n >> valuation(n, 2));
A000593(n) = sigma(A000265(n));
A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A347245(n) = { my(gpf=A006530(n), k=0); while(n>1, n = A000593(n); k++; if(A006530(n)>=gpf,return(k))); (0); };

For n >= 1, a(n) = 0 iff A347244(n) = 0.

For n > 1, a(n) = 1 iff A347246(n) = 1.

A336362 Number of iterations of map k -> k*sigma(p^e)/p^e needed to reach a power of 2, where p is the smallest odd prime factor of k and e is its exponent, when starting from k = n. a(n) = -1 if number of the form 2^k is never reached.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 0, 3, 2, 2, 1, 2, 1, 3, 0, 4, 3, 3, 2, 2, 2, 2, 1, 2, 2, 3, 1, 4, 3, 1, 0, 3, 4, 3, 3, 4, 3, 3, 2, 3, 2, 3, 2, 5, 2, 2, 1, 5, 2, 5, 2, 4, 3, 4, 1, 4, 4, 4, 3, 2, 1, 4, 0, 4, 3, 5, 4, 3, 3, 4, 3, 5, 4, 3, 3, 3, 3, 3, 2, 6, 3, 3, 2, 6, 3, 5, 2, 6, 5, 3, 2, 2, 2, 5, 1, 6, 5, 5, 2, 6, 5, 3, 2, 4

Offset: 1

Antti Karttunen, Jul 30 2020

nonn

Informally: starting from k=n, keep on replacing p^e, the maximal power of the smallest odd prime factor in k, with (1 + p + p^2 + ... + p^e), until a power of 2 is reached. Sequence counts the steps needed.

			For n = 15 = 3*5, we obtain the following path, when starting from k = n, and when we always replace the maximal power of the lowest odd prime factor, p^e of k with sigma(p^e) = (1 + p + p^2 + ... + p^e) in the prime factorization k: 3^1 * 5^1 -> (1+3)*5 = 20 = 2^2 * 5 -> 4 * (1+5) = 24 = 2^3 * 3^1 -> 2^3 * 2^2 = 2^5, thus it took three iterations to reach a power of two, and a(15) = 3.

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

Cf. A000265, A209229, A336650.

Cf. also A331410, A336361, A336363.

A336650(n) = if(!bitand(n,n-1),1,my(f=factor(n>>valuation(n,2))); f[1, 1]^f[1, 2]);
A336362(n) = if(!bitand(n,n-1),0,my(pe=A336650(n)); 1+A336362((n/pe)*sigma(pe)));

If A209229(n) = 1 [when n is a power of 2], a(n) = 0, otherwise a(n) = 1 + a(sigma(A336650(n))*(n/A336650(n))).

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

A336363 Number of iterations of map k -> k*sigma(p^e)/p^e needed to reach a power of 2, where p is the largest prime factor of k and e is its exponent, when starting from k = n. a(n) = -1 if number of the form 2^k is never reached.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 0, 3, 2, 2, 1, 2, 1, 4, 0, 4, 3, 3, 2, 2, 2, 2, 1, 2, 2, 3, 1, 5, 4, 1, 0, 4, 4, 3, 3, 4, 3, 3, 2, 3, 2, 3, 2, 4, 2, 2, 1, 6, 2, 4, 2, 4, 3, 5, 1, 5, 5, 5, 4, 2, 1, 4, 0, 4, 4, 5, 4, 4, 3, 4, 3, 5, 4, 3, 3, 3, 3, 3, 2, 6, 3, 3, 2, 5, 3, 5, 2, 5, 4, 7, 2, 2, 2, 3, 1, 7, 6, 4, 2, 5, 4, 3, 2, 5

Offset: 1

Antti Karttunen, Jul 30 2020

nonn

Informally: starting from k=n, keep on replacing p^e, the maximal power of the largest prime factor in k, with (1 + p + p^2 + ... + p^e), until a power of 2 is reached. Sequence counts the steps needed.

			For n = 15 = 3*5, we obtain the following path, when starting from k = n, and when we always replace the maximal power of the largest prime factor, p^e of k with sigma(p^e) = (1 + p + p^2 + ... + p^e) in the prime factorization k: 3^1 * 5^1 -> 3*(5+1) = 18 = 2^1 * 3^2 -> 2 * (1+3+9) = 26 = 2 * 13 -> 2 * (13+1) = 28 = 2^2 * 7 -> 4*(7+1) = 2^5, thus it took four iterations to reach a power of two, and a(15) = 4.

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

Cf. A000203, A000265, A000593, A209229, A053585.

Cf. also A331410, A336361, A336362.

A053585(n) = if(1==n, 1, my(f=factor(n)); f[#f~, 1]^f[#f~, 2]);
A336363(n) = if(!bitand(n,n-1),0,my(pe=A053585(n)); 1+A336363((n/pe)*sigma(pe)));

If A209229(n) = 1 [when n is a power of 2], a(n) = 0, otherwise a(n) = 1 + a(sigma(A053585(n))*(n/A053585(n))).

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

cp's OEIS Frontend

A347249 a(n) = A331410(n) - A336361(n).

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A347250 Numbers k for which A331410(k) < A336361(k).

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A054784 Integers n such that sigma(2n) - sigma(n) is a power of 2, where sigma is the sum of the divisors of n.

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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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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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A347241 a(1) = 1, and for n > 1, a(n) is the largest prime factor (A006530) of all terms encountered when iterating the map x -> A000593(x), when starting from x = n, and including the n itself. If 1 is never reached when starting from n, then a(n) = -1.

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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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A347245 Number of iterations of the map x -> A000593(x), when starting from x = n, needed to reach a number whose largest prime factor is at least as large as that of n itself (= A006530(n)). If 1 is reached without encountering such a number, then a(n) = 0.

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