A324876 -id:A324876 - cp's OEIS frontend

A324861 a(n) is the binary length of A324876(n).

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

Offset: 1

Antti Karttunen, Mar 21 2019

nonn

Differs from A324863 [binary length of A324866(n)] for the first time at n=50.

			For n = 50, A324876(50) = 9, in binary "1001" with length 4, thus a(50) = 4.

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. A000040, A000523, A070939, A156552, A324876, A324862, A324863, A324872.

A324861(n) = #binary(A324876(n)); \\ Needs also code from A324876.

a(1) = 0; for n > 1, a(n) = A070939(A324876(n)) = 1 + A000523(A324876(n)).

a(A000040(n)) = n.

A324866 a(n) = A156552(n) OR A324865(n), where OR is bitwise-OR, A003986.

Original entry on oeis.org

0, 1, 3, 3, 7, 5, 15, 7, 6, 13, 31, 11, 63, 17, 10, 15, 127, 13, 255, 19, 23, 47, 511, 23, 28, 83, 14, 47, 1023, 31, 2047, 31, 54, 175, 22, 31, 4095, 257, 78, 55, 8191, 37, 16383, 67, 30, 799, 32767, 47, 60, 31, 250, 131, 65535, 29, 55, 71, 270, 1301, 131071, 43, 262143, 2735, 54, 63, 126, 95, 524287, 303, 774, 41, 1048575, 55

Offset: 1

Antti Karttunen, Mar 18 2019

nonn

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. A001065, A003986, A156552, A318456, A323243, A324865, A324867, A324398, A324876.

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
A318456(n) = bitor(n,sigma(n)-n);
A324866(n) = if(1==n,0,A318456(A156552(n)));

A324866(n) = { my(k=A156552(n)); bitor(k,(A323243(n)-k)); }; \\ Needs also code from A323243.

a(1) = 0; for n > 1, a(n) = A318456(A156552(n)).

a(n) = A156552(n) OR (A323243(n) - A156552(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].

A324821 Xor-Moebius transform of A324819, where A324819(n) = bitor(2*A156552(n), A323243(n)), where bitor is A003986.

Original entry on oeis.org

0, 3, 7, 5, 15, 10, 31, 8, 11, 19, 63, 21, 127, 46, 30, 16, 255, 27, 511, 44, 63, 78, 1023, 40, 19, 170, 16, 65, 2047, 38, 4095, 32, 78, 334, 58, 48, 8191, 766, 236, 64, 16383, 110, 32767, 177, 49, 1246, 65535, 80, 35, 51, 260, 341, 131071, 48, 107, 176, 1004, 2794, 262143, 104, 524287, 5454, 80, 64, 142, 219, 1048575, 641, 1278, 110

Offset: 1

Antti Karttunen, Mar 18 2019

nonn

It seems that the records, which are A000225(1+n) = 2^(1+n) - 1 occur at primes, as occur also the records for the width of terms, A000523(a(n)), and the records for the binary weight of terms, A000120(a(n)).

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. A003986, A156552, A323243, A324815, A324819, A324820, A324876, A324878.

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
A324819(n) = bitor(2*A156552(n),A323243(n)); \\ Needs code also from A323243.
A324821(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A324819(d)))); (v); };

a(A000040(n)) = A000225(1+n).

A324878 Xor-Moebius transform of A324398, where A324398(n) = A156552(n) AND (A323243(n) - A156552(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 0, 6, 0, 0, 1, 0, 1, 8, 8, 0, 6, 0, 0, 16, 1, 0, 0, 0, 1, 12, 1, 0, 8, 0, 8, 0, 1, 20, 8, 0, 1, 66, 0, 0, 17, 0, 1, 8, 1, 0, 8, 0, 0, 2, 1, 0, 12, 36, 0, 258, 1, 0, 0, 0, 1, 16, 40, 0, 1, 0, 1, 0, 20, 0, 24, 0, 1, 24, 1, 32, 67, 0, 8, 0, 1, 0, 1, 132, 1, 1026, 0, 0, 40, 72, 1, 0, 1, 256, 16, 0, 1, 68, 16, 0, 3, 0, 0, 46

Offset: 1

Antti Karttunen, Mar 18 2019

nonn

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. A156552, A318458, A324820, A324821, A324876, A324877.

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
A318458(n) = bitand(n, sigma(n)-n);
A324398(n) = if(1==n,0,A318458(A156552(n)));
\\ Or, equivalently:
A324398(n) = { my(k=A156552(n)); bitand(k,(A323243(n)-k)); }; \\ Needs also code from A323243.
A324878(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A324398(d)))); (v); };

a(p) = 0 for all primes p.

A324872 a(n) = A001222(n) + A061395(n) - A324861(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Mar 21 2019

nonn

Records 0, 1, 2, 6, 7, 10, 14, etc., occur at n = 1, 2, 50, 125, 243, 1729, 8192, etc.

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. A001222, A061395, A252464, A324861, A324870, A324876.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A324861(n) = #binary(A324876(n)); \\ Needs also code from A324876.
A324872(n) = (bigomega(n)+A061395(n)-A324861(n));

a(n) = A001222(n) + A061395(n) - A324861(n).
a(n) = 1 + A252464(n) - A324861(n).
a(p) = 1 for all primes p.

cp's OEIS Frontend

A324861 a(n) is the binary length of A324876(n).

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A324866 a(n) = A156552(n) OR A324865(n), where OR is bitwise-OR, A003986.

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

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A324821 Xor-Moebius transform of A324819, where A324819(n) = bitor(2*A156552(n), A323243(n)), where bitor is A003986.

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A324878 Xor-Moebius transform of A324398, where A324398(n) = A156552(n) AND (A323243(n) - A156552(n)).

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A324872 a(n) = A001222(n) + A061395(n) - A324861(n).

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