A364564 -id:A364564 - cp's OEIS frontend

A364551 Odd numbers k such that k is a multiple of A005941(k).

1, 3, 5, 3125, 7875, 12005, 13365, 22869, 23595, 46475, 703395, 985439, 2084775, 2675673, 13619125, 19144125

Offset: 1

Antti Karttunen, Jul 28 2023

Odd numbers k such that k is a multiple of 1+A156552(k).
Sequence A005940(A364545(n)) sorted into ascending order.
This is a subsequence of A364561, so the comments given in A364564 apply also here (see also the example section).

			In all these cases, the right hand side is a divisor of the left hand side:
      Term   (and its factorization)             A005941(term)
         1   (unity)                         ->    1
         3   (prime)                         ->    3
         5   (prime)                         ->    5
      3125 = 5^5                             ->    125 = 5^3
      7875 = 3^2 * 5^3 * 7                   ->    375 = 3 * 5^3
     12005 = 5 * 7^4                         ->    245 = 5 * 7^2
     13365 = 3^5 * 5 * 11                    ->    1215 = 3^5 * 5
     22869 = 3^3 * 7 * 11^2                  ->    847 = 7 * 11^2
     23595 = 3 * 5 * 11^2 * 13               ->    715 = 5 * 11 * 13
     46475 = 5^2 * 11 * 13^2                 ->    845 = 5 * 13^2
    703395 = 3^2 * 5 * 7^2 * 11 * 29         ->    33495 = 3 * 5 * 7 * 11 * 29
    985439 = 7^3 * 13^2 * 17                 ->    2873 = 13^2 * 17
   2084775 = 3 * 5^2 * 7 * 11 * 19^2         ->    12635 = 5 * 7 * 19^2
   2675673 = 3^5 * 7 * 11^2 * 13             ->    11583 = 3^4 * 11 * 13
  13619125 = 5^3 * 13 * 17^2 * 29            ->    36125 = 5^3 * 17^2
  19144125 = 3^2 * 5^3 * 7 * 11 * 13 * 17    ->    21879 = 3^2 * 11 * 13 * 17.

Cf. A005940, A005941, A156552, A364545, A364549, A364564.

Subsequence of A364561, which is a subsequence of A364560.

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)
isA364551(n) = ((n%2)&&!(n%A005941(n)));

A364561 Odd numbers k for which A156552(k) < k.

Original entry on oeis.org

1, 3, 5, 9, 15, 21, 25, 27, 35, 45, 49, 55, 63, 75, 77, 81, 91, 99, 105, 121, 125, 135, 143, 147, 165, 169, 175, 187, 189, 195, 221, 225, 231, 243, 245, 273, 275, 289, 297, 315, 323, 325, 343, 351, 357, 363, 375, 385, 405, 425, 429, 441, 455, 495, 507, 525, 539, 561, 567, 585, 595, 605, 625, 627, 637, 663, 665

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Odd numbers k such that A005941(k) <= k.

Antti Karttunen, Table of n, a(n) for n = 1..13739; terms less than 2^21

Odd terms in A364560.
Cf. A005940, A005941, A156552, A364545, A364564 (largest prime factor).
Cf. also A364551, A364576 (subsequences).

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 };
isA364561(n) = ((n%2)&&(A156552(n) < n));

A364576 Starting from k=1, each subsequent term is the next larger odd k such that A156552(k) < k and the ratio A156552(k)/k is nearer to 1.0 than for any previous k in the sequence.

Original entry on oeis.org

1, 3, 5, 21, 323, 66297, 139965, 263375, 264845, 528581

Offset: 1

Antti Karttunen, Aug 06 2023

All the odd fixed points of map n -> A005940(n) [and its inverse, map n -> A005941(n)] are included in this sequence. This includes both the known odd fixed points, 1, 3 and 5 (see A029747), and any additional hypothetical odd composites that would satisfy the condition n == A005940(n).

This is a subsequence of A364561, so the comments given in A364564 apply also here.

			       k  A156552(k)    A156552(k)/k  k-(1+A156552(k)) factorization of k
       1:       0         0                0
       3:       2         0.6666667        0
       5:       4         0.8              0
      21:      18         0.8571429        2           (3 * 7)
     323:     320         0.9907121        2           (17 * 19)
   66297:   65714         0.9912062      582           (3 * 7^2 * 11 * 41)
  139965:  139306         0.9952917      658           (3 * 5 * 7 * 31 * 43)
  263375:  262364         0.9961614     1010           (5^3 * 7^2 * 43)
  264845:  264244         0.9977307      600           (5 * 7^2 * 23 * 47)
  528581:  528576         0.9999905        4           (17^2 * 31 * 59).

Subsequence of A364561.
Cf. A005940, A005941, A029747, A156552.
Cf. also A364551, A364564, A364572.

cp's OEIS Frontend

A364551 Odd numbers k such that k is a multiple of A005941(k).

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A364561 Odd numbers k for which A156552(k) < k.

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A364576 Starting from k=1, each subsequent term is the next larger odd k such that A156552(k) < k and the ratio A156552(k)/k is nearer to 1.0 than for any previous k in the sequence.

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