A364498 -id:A364498 - cp's OEIS frontend

A364496 Numbers k such that k is a multiple of A163511(k).

0, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 16383, 24576, 32766, 49152, 65532, 98304, 131064, 196608, 262128, 393216, 524256, 786432, 1048512, 1572864, 2097024, 3145728, 4194048, 6291456, 8388096, 12582912, 16776192, 25165824, 33552384, 50331648, 67104768, 100663296, 134209536, 201326592

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).
Sequence A243071(A364497(.)) sorted into ascending order.

			16383 is present, because A163511(16383) = 43, as 16383 = 2^14 - 1 and A000040(14) = 43, and 43 is a factor of 16383 = 3*43*127.
536870895 is present, because A163511(536870895) = 1177 (11*107), which divides 536870895 (3*5*11*47*107*647). See also example in A364498.

Antti Karttunen, Table of n, a(n) for n = 1..53

Positions of 1's in A364492.
Subsequence of A364292.
Cf. A007283 (subsequence), A163511, A364963 (odd terms).
Cf. also A364295, A364494, A364497, A364498.

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)))
isA364496(n) = !(n%A163511(n));

A364495 Odd numbers k such that k divides A163511(k).

Original entry on oeis.org

1, 3, 9, 105, 429, 1365, 1617, 3887, 4235, 10829, 14025, 17745, 21125, 22627, 38025, 54587, 70805, 100555, 115159, 147875, 168751, 169065, 175769, 181447, 181545, 291525, 297297, 303875, 338675, 350987, 501787, 513825, 518035, 549081, 560947, 566865, 594473, 624169, 676039, 735875, 745147, 831875, 869193, 957125

Offset: 1

Antti Karttunen, Jul 27 2023

nonn

			For n = 513825 = 3 * 5^2 * 13 * 17 * 31, A163511(n) = 13873275 = 3^4 * 5^2 * 13 * 17 * 31, so A163511(n)/n = 27 (which is an integer), and therefore 513825 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..801

Odd terms in A364494.
After 1, sequence A243071(A364965(n)), for n>=1, sorted into ascending order.
Cf. A163511.
Cf. also A364296, A364498, A364545, A364963.

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)))
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));
isA364495(n) = ((n%2)&&!(A163511(n)%n));

A364497 Numbers k such that k divides A243071(k).

Original entry on oeis.org

1, 3, 6, 12, 24, 43, 48, 86, 96, 172, 192, 344, 384, 688, 768, 1177, 1376, 1536, 2354, 2752, 3072, 3503, 4708, 5504, 6144, 7006, 9416, 11008, 12288, 14012, 18832, 22016, 24576, 28024, 37664, 44032, 49152, 49477, 56048, 75328, 88064, 98304, 98954, 112096, 150656, 169413, 176128, 196608, 197908, 224192, 301312, 338826

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).
Sequence A163511(A364496(.)) sorted into ascending order.

Index entries for sequences computed from indices in prime factorization

Cf. A163511, A243071.
Cf. A007283 (subsequence), A364498 (odd terms).
Cf. also A364295, A364494, A364496.

A054429(n) = ((3<<#binary(n\2))-n-1); \\ From A054429
A156552(n) = { 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]); res };
A243071(n) = if(n<=2, n-1, A054429(A156552(n)));
isA364497(n) = !(A243071(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.

A364549 Odd numbers k that divide A005941(k).

Original entry on oeis.org

1, 3, 5, 97, 345, 549, 1093, 64621, 671515, 3280317, 8957089

Offset: 1

Antti Karttunen, Jul 28 2023

Sequence A005940(A364547(.)) sorted into ascending order.
Odd numbers k such that k divides 1+A156552(k).
The first ten terms factored:
        1   (unity)
        3   (prime)
        5   (prime)
       97   (prime)
      345 = 3*5*23
      549 = 3^2 * 61
     1093   (prime)
    64621   (prime)
   671515 = 5*13*10331
  3280317 = 3*79*13841.
Primes p present are those that occur as factors of 1 + 2^(A000720(p)-1).

Odd terms in A364548.
Cf. A000720, A005940, A005941, A156552.
Cf. also A364498, A364547, A364551.

A005941(n) = { 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]); (1+res) }; \\ (After David A. Corneth's program for A156552)
isA364549(n) = ((n%2)&&!(A005941(n)%n));

from itertools import count, islice
from sympy import primepi, factorint
def A364549_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue+(startvalue&1^1),1),2):
        if not (sum(pow(2,i+int(primepi(p))-1,n) for i, p in enumerate(factorint(n, multiple=True)))+1) % n:
            yield n
A364549_list = list(islice(A364549_gen(),8)) # Chai Wah Wu, Jul 28 2023

a(11) from Chai Wah Wu, Jul 28 2023

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)));

cp's OEIS Frontend

A364496 Numbers k such that k is a multiple of A163511(k).

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A364495 Odd numbers k such that k divides A163511(k).

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A364497 Numbers k such that k divides A243071(k).

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

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A364549 Odd numbers k that divide A005941(k).

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

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