cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A324712 Xor-Moebius transform of A323243, where A323243(n) = sigma(A156552(n)).

Original entry on oeis.org

0, 1, 3, 5, 7, 4, 15, 12, 15, 11, 31, 15, 63, 28, 22, 16, 127, 7, 255, 28, 43, 46, 511, 24, 27, 106, 20, 39, 1023, 60, 2047, 56, 42, 206, 34, 44, 4095, 508, 172, 32, 8191, 24, 16383, 113, 57, 734, 32767, 56, 51, 9, 128, 213, 65535, 4, 67, 116, 748, 1768, 131071, 26, 262143, 3406, 20, 72, 70, 125, 524287, 389, 762, 22
Offset: 1

Views

Author

Antti Karttunen, Mar 12 2019

Keywords

Comments

Properties of Xor-Moebius transform are explained in A295901.

Links

Crossrefs

Cf. A156552, A295901, A297106, A323243, A324543, A324713, A324714.

Programs

  • PARI
    A324712(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A323243(d)))); (v); } \\ For A323243, see under that entry.

Formula

a(n) = A297106(n) XOR A324714(n).
a(A000040(n)) = A000225(n).

A324713 a(n) = 2*A156552(n) XOR A323243(n).

Original entry on oeis.org

0, 3, 7, 2, 15, 12, 31, 6, 0, 31, 63, 26, 127, 48, 6, 6, 255, 20, 511, 50, 3, 114, 1023, 54, 4, 214, 4, 118, 2047, 10, 4095, 30, 114, 434, 2, 30, 8191, 768, 20, 118, 16383, 108, 32767, 194, 8, 1826, 65535, 110, 12, 45, 504, 386, 131071, 36, 19, 198, 20, 3348, 262143, 122, 524287, 6834, 112, 22, 246, 234, 1048575, 822, 1794, 120
Offset: 1

Views

Author

Antti Karttunen, Mar 13 2019

Keywords

Comments

a(n) is also the cumulative XOR of (2*A297106(d) XOR A324712(d)) over the divisors d of n.
It is conjectured that a(n) may obtain value zero only when n is a power of prime, and especially for n > 1, it must be a prime power present in A324201.

Links

Crossrefs

Cf. A003987, A156552, A297106, A323243, A323244, A324201, A324712, A324714.

Programs

  • PARI
    A324713(n) = { my(x=0,s=0); fordiv(n,d,x = bitxor(x,A324712(d)); s = bitxor(s,A297106(d))); bitxor(x,2*s); };

Formula

a(n) = 2*A156552(n) XOR A323243(n).
a(n) = XORsum_{d|n} (2*A297106(d) XOR A324712(d)).

A324715 Xor-Moebius transform of A324713.

Original entry on oeis.org

0, 3, 7, 1, 15, 8, 31, 4, 7, 19, 63, 23, 127, 44, 14, 0, 255, 31, 511, 44, 27, 78, 1023, 40, 11, 170, 4, 71, 2047, 20, 4095, 24, 74, 334, 18, 28, 8191, 764, 108, 64, 16383, 72, 32767, 177, 9, 1246, 65535, 88, 19, 57, 256, 341, 131071, 52, 35, 180, 492, 2792, 262143, 74, 524287, 5454, 116, 8, 134, 221, 1048575, 645, 1274, 70, 2097151, 20
Offset: 1

Views

Author

Antti Karttunen, Mar 15 2019

Keywords

Links

Crossrefs

Cf. A003987, A297106, A324712, A324713, A324714, A324717.

Programs

  • PARI
    A324715(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A324713(d)))); (v); };

Formula

a(n) = A324712(n) XOR 2*A297106(n).

A324875 Xor-Moebius transform of A324865, where A324865(n) = A323243(n) - A156552(n).

Original entry on oeis.org

0, 0, 1, 1, 3, 0, 7, 0, 7, 7, 15, 1, 31, 6, 10, 8, 63, 7, 127, 4, 19, 0, 255, 0, 19, 12, 12, 13, 511, 4, 1023, 8, 26, 16, 18, 12, 2047, 126, 80, 16, 4095, 18, 8191, 15, 1, 480, 16383, 8, 35, 17, 68, 19, 32767, 12, 59, 12, 368, 234, 65535, 14, 131071, 336, 4, 40, 38, 15, 262143, 3, 506, 22, 524287, 28, 1048575, 1788, 11, 109, 58, 89
Offset: 1

Views

Author

Antti Karttunen, Mar 18 2019

Keywords

Links

Crossrefs

Cf. A156552, A323243, A324714, A324865, A324867, A324877.

Programs

Showing 1-4 of 4 results.

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