A364547 -id:A364547 - cp's OEIS frontend

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

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));

A364546 Numbers k such that k is a multiple of A005940(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 128, 160, 192, 256, 320, 384, 512, 640, 768, 1024, 1035, 1280, 1536, 2048, 2070, 2560, 3072, 4096, 4140, 5120, 6144, 8192, 8280, 10240, 12288, 16384, 16560, 20480, 24576, 32768, 33120, 40960, 49152, 65536, 66240, 81920, 98304, 131072, 132480, 163840

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Sequence A005941(A364548(.)) sorted into ascending order.
If k is a term, then also 2*k is present in this sequence, and vice versa.
A029747 is included as a subsequence, because it gives the known fixed points of map n -> A005940(n).

Positions of 1's in A364502.
Subsequence of A364541.
Subsequences: A029747, A364547 (odd terms).
Cf. A005940, A364544, A364548.
Cf. also A364496.

nn = 2^18; 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[nn], 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 };
isA364546(n) = !(n%A005940(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

A364543 Odd numbers k for which A005940(k) <= k.

Original entry on oeis.org

1, 3, 5, 9, 17, 33, 35, 65, 67, 69, 129, 131, 133, 135, 137, 257, 259, 261, 263, 265, 267, 273, 289, 385, 513, 515, 517, 519, 521, 523, 525, 527, 529, 531, 545, 577, 641, 769, 1025, 1027, 1029, 1031, 1033, 1035, 1037, 1039, 1041, 1043, 1045, 1047, 1057, 1059, 1089, 1091, 1153, 1281, 1537, 2049, 2051, 2053, 2055

Offset: 1

Antti Karttunen, Aug 06 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..11023; terms less than 2^23

Odd terms of A364541.
Cf. A005940, A364563 [= -A364499(a(n))].
Subsequences: A364547, A364573.
Cf. also A364293.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
isA364543(n) = ((n%2)&&(A005940(n)<=n));

cp's OEIS Frontend

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

A364546 Numbers k such that k is a multiple of A005940(k).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Crossrefs

A364549 Odd numbers k that divide A005941(k).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Python

Extensions

A364543 Odd numbers k for which A005940(k) <= k.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI