A364495 -id:A364495 - cp's OEIS frontend

A364494 Numbers k such that k divides A163511(k).

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 105, 128, 144, 192, 210, 256, 288, 384, 420, 429, 512, 576, 768, 840, 858, 1024, 1152, 1365, 1536, 1617, 1680, 1716, 2048, 2304, 2730, 3072, 3234, 3360, 3432, 3887, 4096, 4235, 4608, 5460, 6144, 6468, 6720, 6864, 7774, 8192, 8470, 9216, 10829, 10920, 12288

Offset: 1

Antti Karttunen, Jul 27 2023

nonn

If n is present, then 2*n is also present, and vice versa.

A007283 is included as a subsequence, because it gives the known fixed points of map n -> A163511(n).

Positions of 1's in A364491.
Cf. A163511.
Subsequences: A007283, A029744, A364495 (odd terms).
Cf. also A364295, A364496, A364497.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A054429(n) = ((3<<#binary(n\2))-n-1);
A163511(n) = if(!n,1,A005940(1+A054429(n)))
isA364494(n) = !(A163511(n)%n);

A364545 Odd numbers k such that k divides A005940(k).

Original entry on oeis.org

1, 3, 5, 125, 245, 375, 715, 845, 847, 1215, 2873, 11583, 12635, 21879, 24255, 31213, 33495, 36125, 42875, 48125, 48841, 71269, 100793, 102245, 104907, 157035, 173641, 191607, 206045, 240787, 244205, 251459, 302575, 313937, 351509, 359513, 375687, 384475, 388531, 417605, 419957, 444889, 468999, 521703, 586177, 635375

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Odd terms in A364544.
Cf. A005940, A364501.
Cf. also A364495, A364547.

nn = 2^20; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Select[Range[1, nn, 2], Divisible[a[#], #] &] (* Michael De Vlieger, Jul 28 2023 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
isA364545(n) = ((n%2)&&!(A005940(n)%n));

A364963 Odd numbers k such that k is a multiple of A163511(k).

Original entry on oeis.org

3, 16383, 536870895, 2147482623

Offset: 1

Antti Karttunen, Sep 02 2023

Sequence A243071(A364498(n)), for n > 1, sorted into ascending order, therefore terms 151115727451794287099901, 60708402882054033466233184588234965832575213720379360039119137804340758912662765515 (and many others that do not fit in this space) are also present.
Consider the sequence 1 + 5*2^k (with k>=1): 11, 21, 41, 81, 161, 321, etc, (A083575(n) from n>=1), and compare to the sequence A163511(1 + 5*2^k): 25, 75, 225, 675, 2025, 6075, etc (= 3^(k-1) * 25). Clearly, the first sequence does not contain any multiples of 5, while all the terms in the second one are multiples of 25, and thus of 5 also.
Then consider sequences 1 + 2*(1 + 11*2^k): 47, 91, 179, 355, 707, 1411, etc., and A163511(1 + 2*(1 + 11*2^k)): 121, 605, 3025, 15125, 75625, 378125, etc. The terms in the first one are never multiples of 11, while the terms of second one are all multiples of 121, thus of 11 also.
Consider also sequences 1 + (2^k)*(1+2*11): 47, 93, 185, 369, 737, 1473, 2945, 5889, 11777, 23553, 47105, 94209, 188417, 376833, 753665, 1507329,  etc, and 1 + (2^k)*(1+4*11): 91, 181, 361, 721, 1441, 2881, 5761, 11521, 23041, 46081, 92161, 184321, 368641, 737281, 1474561, 2949121, etc. The only time their terms are multiples of 11 is when k = 5, 15, 25, ..., 5 + 10*j, j>= 0, while for sequences A163511(1 + (2^k)*(1+2*11)): 121, 363, 1089, 3267, 9801, 29403, etc, and A163511(1 + (2^k)*(1+4*11)): 605, 1815, 5445, 16335, 49005, 147015, etc, all the terms are multiples of 121, thus of 11 also.
There are numerous other such correspondences that forbid the occurrence of factor x in n, when n is a member of a certain subset of odd numbers, while on the other hand, force the same factor x to be present in A163511(n), thus making it impossible that n were a multiple of A163511(n) in those cases. However, this sequence shows that such subsets do not completely cover all odd numbers. Similar observation applies to Doudna sequence (see A364547).

			        Term [in binary]                         Factorization         A163511(Term)
           3 [11]                                (prime)             -> 3
       16383 [11111111111111]                  = 3*43*127            -> 43
   536870895 [11111111111111111111111101111]   = 3*5*11*47*107*647   -> 1177 = 11*107
  2147482623 [1111111111111111111101111111111] = 3*11*13*31*113*1429 -> 3503 = 31*113

Odd terms in A364496.
Cf. A083575, A163511, A243071, A364498.
Cf. also A364495, A364547.

A364296 Odd numbers k for which A292943(k) == A292944(k).

Original entry on oeis.org

1, 3, 9, 45, 165, 189, 114595, 330993, 1277601

Offset: 1

Antti Karttunen, Jul 27 2023

The first six terms are binary palindromes, in A006995. Most are multiples of 3.

a(10) > 2^21 if it exists.

Odd terms in A364295.
Cf. A163511, A243071, A292943, A292944.
Cf. also A364495.

A364965 Odd numbers k such that k is a multiple of A243071(k).

Original entry on oeis.org

3, 27, 315, 3003, 42757, 72765, 195195, 799425, 13873275, 18131225

Offset: 1

Antti Karttunen, Sep 01 2023

Odd terms in A364964.
Sequence A163511(A364495(n)), for n>1, sorted into ascending order.
Cf. A243071.
Cf. also A364498, A364551.

A243071(n) = if(n<=2, n-1, my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p*p2*(2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); ((3<<#binary(res\2))-res-1)); \\ (Combining programs given in A156552 and A054429)
isA364965(n) = ((n>1)&&(n%2)&&!(n%A243071(n)));

A374469 The odd part of gcd(n, A163511(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 3, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 9, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 5, 1, 1, 1, 1, 11, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 9, 1, 1, 5, 1, 7, 1, 1, 1, 27, 1, 1, 3, 5, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 105

Offset: 0

Antti Karttunen, Aug 07 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..100000

Cf. A000265, A006519, A007814, A163511, A364255, A364495 (fixed points).

A000265(n) = (n>>valuation(n,2));
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A364255(n) = gcd(n, A163511(n));
A374469(n) = A000265(A364255(n));

a(n) = A000265(A364255(n)) = A000265(gcd(n, A163511(n))).
For n >= 1, a(n) = A364255(n) / A006519(n).
For n >= 0, a(2*n) = a(n).

cp's OEIS Frontend

A364494 Numbers k such that k divides A163511(k).

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A364545 Odd numbers k such that k divides A005940(k).

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A364963 Odd numbers k such that k is a multiple of A163511(k).

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A364296 Odd numbers k for which A292943(k) == A292944(k).

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A364965 Odd numbers k such that k is a multiple of A243071(k).

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A374469 The odd part of gcd(n, A163511(n)).

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