A324874 -id:A324874 - cp's OEIS frontend

A324879 Numbers k such that A324863(k) is equal to A324874(k).

1, 9, 15, 16, 21, 27, 35, 39, 55, 57, 64, 75, 77, 85, 87, 90, 91, 95, 99, 105, 111, 115, 119, 125, 129, 133, 143, 147, 155, 159, 161, 175, 183, 185, 189, 195, 201, 203, 205, 209, 213, 221, 235, 237, 243, 245, 253, 256, 259, 265, 267, 275, 285, 287, 295, 299, 301, 303, 319, 321, 323, 325, 335, 339, 341, 351, 355, 363, 365

Offset: 1

Antti Karttunen, Mar 27 2019

nonn

In range 1..10000, there are only three such numbers k for which A324868(k) == A000120(A324866(k)): 1, 9, 125. See A324201.

Antti Karttunen, Table of n, a(n) for n = 1..2240 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000120, A324863, A324866, A324868, A324874.

Subsequences: A324201, A324880 (even terms).

for(n=1,oo,if(A324863(n)==A324874(n), print1(n,", ")));

A324880 Even numbers k such that A324863(k) = A324874(k).

Original entry on oeis.org

16, 64, 90, 256, 484, 490, 750, 756, 810, 988, 1000, 1024, 1296, 1440, 2116, 2622, 2662, 3630, 3710, 4004, 4116, 4624, 4896, 4900, 5880, 6426, 6724, 6760, 7290, 7744, 7840, 7920, 8100, 8924, 9604, 10000

Offset: 1

Antti Karttunen, Mar 27 2019

nonn

Of 36 such numbers found in the range 1..10000, the following ten: 90, 490, 750, 810, 2622, 2662, 3630, 3710, 6426, 7290, are such that a positive integer of the form 4m+1 is produced when A156552 is applied to them: 45, 105, 117, 189, 225, 405, 765, 2205, 2565, 262185 (when sorted into ascending order). See also A324647.

Cf. A156552, A323243, A324398, A324647, A324863, A324866, A324874.

Subsequence of A324879.

for(n=1,10000,if(!(n%2)&&A324863(n)==A324874(n), print1(n,", ")));

A324863 Binary length of A324866(n), where A324866(n) = A156552(n) OR (A323243(n) - A156552(n)).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 5, 4, 6, 5, 4, 4, 7, 4, 8, 5, 5, 6, 9, 5, 5, 7, 4, 6, 10, 5, 11, 5, 6, 8, 5, 5, 12, 9, 7, 6, 13, 6, 14, 7, 5, 10, 15, 6, 6, 5, 8, 8, 16, 5, 6, 7, 9, 11, 17, 6, 18, 12, 6, 6, 7, 7, 19, 9, 10, 6, 20, 6, 21, 13, 5, 10, 6, 8, 22, 7, 6, 14, 23, 7, 8, 15, 11, 8, 24, 6, 7, 11, 12, 16, 9, 7, 25, 6, 7, 6, 26, 9, 27, 9, 6

Offset: 1

Antti Karttunen, Mar 21 2019

nonn

Differs from A324861 [binary length of A324876(n)] for the first time at n=50.

Provided that the maximal value that A324861(d) attains among divisors d of n is attained an odd number of times, then a(n) gives that maximal value. It is conjectured that this always holds. Among n = 1..10000, there are only two such cases, where the maximal value occurs more than once among the divisors: 3675 and 7623, where it occurs three times in both (see the examples).

			For n = 50, we have A156552(50) = 25 and A323243(50) = 31. Taking bitwise-OR (A003986) of 25 and 31-25 = 6, we get 31, in binary "11111", with length 5, thus a(50) = 5.
The rest of examples pertain to the conjectured interpretation of this sequence:
Divisors of 8 are [1, 2, 4, 8]. A324861 applied to these gives values [0, 1, 2, 3], of which the largest is 3, thus a(8) = 3.
Divisors of 25 are [1, 5, 25]. A324861 applied to these gives values [0, 3, 5], of which the largest is 5, thus a(25) = 5.
Divisors of 50 are [1, 2, 5, 10, 25, 50]. A324861 applied to these gives values [0, 1, 3, 4, 5, 4], of which the largest is 5, thus a(50) = 5.
Divisors of 88 are [1, 2, 4, 8, 11, 22, 44, 88]. A324861 applied to these gives values [0, 1, 2, 3, 5, 6, 7, 8], of which the largest is 8, thus a(88) = 8.
Divisors of 3675 are [1, 3, 5, 7, 15, 21, 25, 35, 49, 75, 105, 147, 175, 245, 525, 735, 1225, 3675]. A324861 applied to these gives values [0, 2, 3, 4, 4, 5, 5, 5, 6, 4, 6, 5, 6, 5, 8, 7, 8, 8], of which the largest is 8 (occurs three times), thus a(3675) = 8.
Divisors of 7623 are [1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 121, 231, 363, 693, 847, 1089, 2541, 7623]. A324861 applied to these gives values [0, 2, 4, 3, 5, 5, 6, 6, 6, 7, 7, 7, 6, 8, 6, 9, 9, 9], of which the largest is 9 (occurs three times), thus a(7623) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000523, A003986, A070939, A156552, A252464, A323243, A324861, A324864, A324866, A324874, A324879.

Differs from A252464 for the first time at n=25, A324870 gives the differences.

A156552(n) = {my(f = factor(n), 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}; \\ From A156552 by David A. Corneth
A324866(n) = { my(k=A156552(n)); bitor(k,(A323243(n)-k)); }; \\ Needs also code from A323243.
A324863(n) = #binary(A324866(n));

A324876(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A324866(d)))); (v); };
A324861(n) = #binary(A324876(n));
A324863(n) = { my(m=0,w,c=0); fordiv(n,d,w=A324861(d); if(w>=m,if(w==m,c++,c=1;m=w))); (m); };

a(1) = 0; for n > 1, a(n) = A070939(A324866(n)) = 1 + A000523(A324866(n)).
a(A000040(n)) = n.
a(n) = Max_{d|n} A324861(d) [conjectured].

A324862 a(n) is the binary length of A324878(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 0, 3, 0, 0, 1, 0, 1, 4, 4, 0, 3, 0, 0, 5, 1, 0, 0, 0, 1, 4, 1, 0, 4, 0, 4, 0, 1, 5, 4, 0, 1, 7, 0, 0, 5, 0, 1, 4, 1, 0, 4, 0, 0, 2, 1, 0, 4, 6, 0, 9, 1, 0, 0, 0, 1, 5, 6, 0, 1, 0, 1, 0, 5, 0, 5, 0, 1, 5, 1, 6, 7, 0, 4, 0, 1, 0, 1, 8, 1, 11, 0, 0, 6, 7, 1, 0, 1, 9, 5, 0, 1, 7, 5, 0, 2, 0, 0, 6

Offset: 1

Antti Karttunen, Mar 21 2019

nonn

Cf. A000040, A000523, A070939, A324878, A324861, A324864, A324874.

A324862(n) = #binary(A324878(n)); \\ Needs also code from A324878.

If A324878(n) = 0, a(n) = 0, otherwise a(n) = A070939(A324878(n)) = 1 + A000523(A324878(n)).

a(p) = 0 for all primes p.

A324881 Number of nonleading zeros in binary representation of A324398, where A324398(n) = A156552(n) AND (A323243(n) - A156552(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 2, 0, 0, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 7, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 4, 0, 5, 0, 0, 0, 2, 0, 0, 0, 6, 0, 9, 0, 0, 4, 5, 0, 0, 0, 8, 2, 0, 0, 5, 3, 0, 0, 0, 0, 4

Offset: 1

Antti Karttunen, Mar 27 2019

nonn

			For n=4, A324398(4) = 1, in binary "1", thus a(4) = 0.
For n=9, A324398(9) = 6, in binary "110", thus a(9) = 1.
For n=16, A324398(16) = 9, in binary "1001", thus a(16) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A080791, A156552, A323243, A324398, A324868, A324874.

PARI
```
A324881(n) = (A324874(n)-A324868(n));
```

a(n) = A080791(A324398(n)) = A324874(n) - A324868(n).

a(p) = 0 for all primes p.

cp's OEIS Frontend

A324879 Numbers k such that A324863(k) is equal to A324874(k).

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A324880 Even numbers k such that A324863(k) = A324874(k).

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A324863 Binary length of A324866(n), where A324866(n) = A156552(n) OR (A323243(n) - A156552(n)).

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A324862 a(n) is the binary length of A324878(n).

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A324881 Number of nonleading zeros in binary representation of A324398, where A324398(n) = A156552(n) AND (A323243(n) - A156552(n)).

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