A323243 -id:A323243 - cp's OEIS frontend

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

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

Antti Karttunen, Mar 12 2019

nonn

Properties of Xor-Moebius transform are explained in A295901.

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

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

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

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

A324543 Möbius transform of A323243, where A323243(n) = sigma(A156552(n)).

Original entry on oeis.org

0, 1, 3, 3, 7, 2, 15, 4, 9, 5, 31, 3, 63, 2, 8, 16, 127, -1, 255, 4, 21, 16, 511, 8, 21, 20, 12, 27, 1023, 6, 2047, 8, 20, 48, 20, 20, 4095, 2, 78, 32, 8191, -6, 16383, 17, 9, 288, 32767, 8, 45, -3, 122, 45, 65535, 4, 53, 20, 270, 278, 131071, 2, 262143, 688, 12, 72, 56, 23, 524287, 125, 260, -8, 1048575, 20, 2097151, 260, 3, 363, 44, -7, 4194303

Offset: 1

Antti Karttunen, Mar 07 2019

sign

The first four zeros after a(1) occur at n = 192, 288, 3645, 6075.
There are 1562 negative terms among the first 10000 terms.
Applying this function to the divisors of the first four terms of A324201 reveals the following pattern:
----------------------------------------------------------------------------------
  A324201  divisors                             a(n) applied to each:         Sum
        9: [1, 3, 9]                         -> [0, 3, 9]                      12 = 2*6
      125: [1, 5, 25, 125]                   -> [0, 7, 21, 28]                 56 = 2*28
   161051: [1, 11, 121, 1331, 14641, 161051] -> [0, 31, 93, 124, 496, 248]    992 = 2*496
410338673: [1, 17, 289, 4913, 83521, 1419857, 24137569, 410338673]
                             -> [0, 127, 381, 508, 2032, 1016, 9144, 3048]  16256 = 2*8128
The second term (the first nonzero) of the latter list = A000668(n), and the sum is always twice the corresponding perfect number, which forces either it or at least many of its divisors to be present. For example, in the fourth case, although 8128 = A000396(4) itself is not present, we still have 127, 508, 1016 and 2032 in the list. See also A329644.

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

Cf. A000040, A000043, A000668, A000203, A000225, A000396, A008683, A068156, A156552, A173033, A297112, A297113, A323243, A323244, A324201, A324542, A324547, A324548, A324549, A324712, A329644.

Table[DivisorSum[n, MoebiusMu[n/#] If[# == 1, 0, DivisorSigma[1, Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]]]] &], {n, 79}] (* Michael De Vlieger, Mar 11 2019 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
memoA323243 = Map();
A323243(n) = if(1==n, 0, my(v); if(mapisdefined(memoA323243,n,&v),v, v=sigma(A156552(n)); mapput(memoA323243,n,v); (v)));
A324543(n) = sumdiv(n,d,moebius(n/d)*A323243(d));

a(n) = Sum_{d|n} A008683(n/d) * A323243(d).
a(A000040(n)) = A000225(n).
a(A001248(n)) = A173033(n) - A000225(n) = A068156(n) = 3*(2^n - 1).
a(2*A000040(n)) = A324549(n).
a(A002110(n)) = A324547(n).
a(n) = 2*A297112(n) - A329644(n), and for n > 1, a(n) = 2^A297113(n) - A329644(n). - Antti Karttunen, Dec 08 2019

A323248 a(n) = A323247(n) - A323243(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 3, -2, 3, 0, 7, 0, 14, 0, 2, 0, 9, 0, 15, -5, 16, 0, 18, -6, 44, 1, 19, 0, 7, 0, 25, 12, 80, -4, 10, 0, 254, -14, 18, 0, 33, 0, 63, 5, 224, 0, 41, -14, 16, 6, 127, 0, 24, -21, 66, -14, 746, 0, 38, 0, 1360, 13, 16, 8, 39, 0, 211, 252, 37, 0, 33, 0, 3836, 7, 403, -12, 103, 0, 73, -16, 5456, 0, 22, -74, 12248, -350, 26, 0, 8

Offset: 1

Antti Karttunen, Jan 10 2019

sign

Cf. A005187, A156552, A323243, A323246, A323247.

Cf. also A323167.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A323247(n) = A005187(A156552(n));
A323248(n) = (A323247(n) - A323243(n));

a(n) = A323247(n) - A323243(n).
a(n) = A323244(n) - A001222(n).
For n > 1, a(n) = A294898(A156552(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

Antti Karttunen, Mar 13 2019

nonn

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.

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

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

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

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

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].

A324876 Xor-Moebius transform of A324866, where A324866(n) = A156552(n) OR (A323243(n) - A156552(n)).

Original entry on oeis.org

0, 1, 3, 2, 7, 7, 15, 4, 5, 11, 31, 12, 63, 31, 14, 8, 127, 13, 255, 28, 27, 49, 511, 24, 27, 109, 8, 60, 1023, 24, 2047, 16, 42, 209, 30, 28, 4095, 511, 114, 32, 8191, 43, 16383, 110, 17, 737, 32767, 48, 51, 9, 134, 210, 65535, 24, 47, 108, 498, 1771, 131071, 38, 262143, 3409, 36, 32, 70, 94, 524287, 386, 762, 42, 1048575, 52, 2097151, 7933, 11

Offset: 1

Antti Karttunen, Mar 18 2019

nonn

It seems that the records, which are A000225(n) = 2^n - 1 occur at primes, as occur also the records for the width of terms, A000523(a(n)), and the records for the binary weights 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. A156552, A323243, A324820, A324821, A324866, A324877, 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
A318456(n) = bitor(n,sigma(n)-n);
A324866(n) = if(1==n,0,A318456(A156552(n)));
A324876(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A324866(d)))); (v); };

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

A324815 a(n) = 2*A156552(n) AND A323243(n), where AND is bitwise-and, A004198.

Original entry on oeis.org

0, 0, 0, 4, 0, 2, 0, 8, 12, 0, 0, 4, 0, 2, 16, 24, 0, 10, 0, 4, 36, 0, 0, 8, 24, 0, 24, 0, 0, 32, 0, 32, 4, 0, 40, 32, 0, 2, 128, 8, 0, 2, 0, 4, 36, 0, 0, 16, 48, 18, 4, 4, 0, 26, 72, 8, 512, 2, 0, 4, 0, 0, 12, 104, 8, 0, 0, 0, 4, 2, 0, 72, 0, 0, 32, 0, 80, 0, 0, 16, 8, 0, 0, 20, 256, 0, 2048, 0, 0, 74, 128, 0, 0, 0, 520, 56, 0, 32, 128, 64, 0, 2, 0, 8, 64

Offset: 1

Antti Karttunen, Mar 17 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. A003987, A004198, A139256, A156552, A318468, A323243, A324201, A324716, A324816.

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
A324815(n) = bitand(2*A156552(n),A323243(n)); \\ Needs code also from A323243.

a(n) = 2*A156552(n) AND A323243(n), where AND is A004198.
a(n) = 2*A156552(n) - A324716(n) = 2*A156552(n) XOR A324716(n), where XOR is A003987.
For n > 1, a(n) = A318468(A156552(n)).
a(p) = 0 for all primes p.
a(A324201(n)) = A139256(n).
A000120(a(n)) = A324816(n).

A324825 Number of divisors d of n such that A323243(d) is odd; number of terms of A324813 larger than 1 that divide n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 2, 1, 3, 3, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 2, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 1, 2, 1, 4, 2, 2, 1, 2, 3, 2, 2, 2, 1, 4, 1, 2, 3, 1, 2, 3, 1, 2, 2, 4, 1, 2, 1, 2, 2, 2, 2, 3, 1, 3, 1, 2, 1, 4, 2, 2, 2, 2, 1, 4, 3, 2, 2, 2, 2, 2, 1, 3, 2, 4, 1, 3, 1, 2, 4

Offset: 1

Antti Karttunen, Mar 16 2019

nonn

Inverse Möbius transform of A324823.

Cf. A000961, A156552, A324813, A324823, A324826, A324827, A324830, A324831, A324832.

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
A324823(n) = if(1==n,0, n=A156552(n); (issquare(n) || (!(n%2) && issquare(n/2))));
A324825(n) = sumdiv(n,d,A324823(d));

A324825(n) = sumdiv(n,d,A323243(d)%2); \\ This needs code also from A323243.

a(n) = Sum_{d|n} A324823(d).

a(p^k) = 1, for all primes p and exponents k >= 1.

A324829 a(n) = A000120(A324712(n)); binary weight of Xor-Moebius transform of A323243.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 16 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. A000120, A324712, A323243, A324828.

A324712(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A323243(d)))); (v); }; \\ Needs also code from A323243.
A324829(n) = hammingweight(A324712(n));

a(n) = A000120(A324712(n)).

a(A000040(n)) = n.

A324830 Number of divisors d of n such that A323243(d) is a multiple of 3.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 16 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. A000005, A156552, A323243, A324831, A324832.

A324830(n) = sumdiv(n,d,(0==(A323243(d))%3));

a(n) = A000005(n) - (A324831(n) + A324832(n)).

For all n >= 1, a(A000040(n)) = 2-A000035(n).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A324543 Möbius transform of A323243, where A323243(n) = sigma(A156552(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A323248 a(n) = A323247(n) - A323243(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A324876 Xor-Moebius transform of A324866, where A324866(n) = A156552(n) OR (A323243(n) - A156552(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A324815 a(n) = 2*A156552(n) AND A323243(n), where AND is bitwise-and, A004198.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A324825 Number of divisors d of n such that A323243(d) is odd; number of terms of A324813 larger than 1 that divide n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs