A324201 -id:A324201 - cp's OEIS frontend

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

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)).

A332225 Numbers k > 1 for which A048675(A332223(k)) is equal to 2*A048675(k).

Original entry on oeis.org

4, 9, 12, 20, 44, 52, 60, 108, 124, 125, 132, 140, 156, 172, 188, 204, 236, 300, 308, 396, 412, 436, 476, 492, 612, 644, 700, 836, 876, 884, 891, 924, 972, 980, 1004, 1044, 1092, 1100, 1116, 1148, 1188, 1196, 1236, 1260, 1268, 1292, 1300, 1308, 1372, 1380, 1476, 1620, 1628, 1724, 1860, 1900, 2140, 2244, 2324, 2356, 2444, 2460, 2652, 2660, 2700

Offset: 1

Antti Karttunen, Feb 12 2020

nonn

Numbers k > 1 such that A332224(A156552(k)) = A087808(sigma(A156552(k))) is equal to 2*A048675(k) = A048675(k^2).
Notably, of the first 150 terms (4 .. 9996), 156 = 2^2 * 3 * 13 is the only even term that does not map to a prime, as A156552(156) = 267 = 3*89 (and sigma(267) = 360 = 4*90).
Although sigma(A156552(k)) = A323243(k) is a multiple of 4 for most of the terms k present in this sequence, there are exceptions, for example 840350 = A005940(1+A332445(1)) = 2^1 * 5^2 * 7^5 is one, as A048675(A332223(840350)) = 98 = 2*A048675(840350) and A323243(840350) = 2394 == 2 (mod 4).

Antti Karttunen, Table of n, a(n) for n = 1..150 (all terms < 10000, computed using Hans Havermann's factorization of A156552)

Cf. A000203, A048675, A156552, A323243, A332223, A332224, A332229, A332445, A332446.

Cf. A324201 (a subsequence).

for(n=2,2048,if(A048675(A332223(n))==2*A048675(n),print1(n,", ")))

\\ To find all terms < 10000:
v156552sigs = readvec("a156552.txt"); \\ Use the factorization file for A156552 prepared by Hans Havermann, available at https://oeis.org/A156552/a156552.txt
A323243(n) = if(n<=2,n-1,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,((ps[i]^(1+es[i]))-1)/(ps[i]-1)));
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA322225(n) = (A087808(A323243(n)) == 2*A048675(n));
for(n=2,10000,if(isA322225(n),print1(n,", ")));

A324051 a(n) = A106315(A156552(n)).

Original entry on oeis.org

0, 1, 2, 5, 4, 2, 6, 0, 1, 18, 10, 3, 16, 4, 12, 67, 12, 4, 18, 30, 36, 260, 22, 16, 8, 8, 44, 5, 20, 1029, 30, 28, 164, 36, 28, 6, 256, 96, 44, 4102, 36, 7, 66, 16, 104, 16391, 46, 12, 13, 32, 130, 8, 28, 51, 70, 480, 942, 65544, 42, 9, 2724, 32, 66, 30, 84, 262153, 124, 508, 40, 10, 4, 1048586, 3320, 20, 188, 50, 52, 11, 78, 24

Offset: 2

Antti Karttunen, Feb 19 2019

nonn

Positions of zeros, which is sequence A005940(1+A001599(n)) sorted into ascending order: 2, 9, 125, 325, 351, 4199, ..., has A324201 as its subsequence.

Antti Karttunen, Table of n, a(n) for n = 2..4473
Index entries for sequences computed from indices in prime factorization

Cf. A001599, A005940, A106315, A156552, A158879, A323243, A323244, A324049, A324105, A324201.

Cf. also A324057, A324187.

A106315(n) = (n*numdiv(n) % sigma(n));
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))));
A324051(n) = A106315(A156552(n));

a(n) = A106315(A156552(n)).

a(n) = (A156552(n)*A324105(n)) mod A323243(n).

A324115 a(n) = A002487(A323244(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 0, 3, 1, 3, 1, 1, 1, 2, 1, 2, 1, 4, -2, 4, 1, 5, -1, 7, 1, 5, 1, 3, 1, 4, 3, 11, -1, 3, 1, 1, -2, 5, 1, 4, 1, 6, 1, 13, 1, 7, -2, 7, 1, 7, 1, 3, -7, 9, -2, 25, 1, 8, 1, 76, 1, 5, 3, 8, 1, 21, 7, 3, 1, 7, 1, 31, 3, 31, -3, 13, 1, 10, -2, 199, 1, 5, -4, 101, -18, 4, 1, 2, -12, 43, 11, 266, -5, 9, 1, 11, -1, 4, 1, 6, 1, 13

Offset: 1

Antti Karttunen, Feb 20 2019

sign

If there are no odd perfect numbers then A324201 gives the positions of all zeros after the initial a(1) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..4473

Cf. A002487, A033879, A156552, A323244, A323902, A324201, A324116, A324117.

A002487(n) = if(abs(n)<=1, n, A002487(n\2) + if( n%2, A002487(n\2 + 1))); \\ This version works consistently also with negative arguments, so that a(-n) = -a(n). Except that it is very slow on large n.
A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); }; \\ So we use this one, modified from the one given in A002487
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))));
A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k));
A324115(n) = A002487(A323244(n));

a(n) = A002487(A323244(n)), with the definition of A002487 extended to the negative arguments so that A002487(-n) = -A002487(n).

a(A324201(n)) = 0.

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).

A324551 Positions of negative terms in A323244.

Original entry on oeis.org

21, 25, 35, 39, 49, 55, 57, 77, 81, 85, 87, 91, 95, 99, 105, 111, 115, 121, 129, 133, 143, 155, 159, 161, 169, 183, 185, 187, 189, 195, 201, 203, 205, 209, 213, 221, 225, 235, 237, 247, 253, 259, 265, 267, 285, 287, 289, 295, 299, 301, 303, 319, 321, 323, 325, 335, 339, 341, 343, 351, 355, 361, 365, 371, 377, 381, 385, 391, 393, 403

Offset: 1

Antti Karttunen, Mar 07 2019

nonn

Sequence A005940(1+A005101(n)) sorted into ascending order.

The first two even terms are 3710 and 4096, where A323244(3710) = -942 and A323244(4096) = -546.

Antti Karttunen, Table of n, a(n) for n = 1..986

Cf. A005101, A005231, A005940, A323244, A324201.

A329641 a(n) = gcd(A329638(n), A329639(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 6, 1, 5, 1, 10, 1, 16, 2, 6, 1, 1, 1, 18, 1, 18, 1, 22, 1, 46, 1, 22, 1, 10, 1, 30, 14, 82, 2, 1, 1, 256, 2, 22, 1, 1, 1, 66, 1, 226, 1, 46, 1, 1, 8, 130, 1, 1, 1, 70, 2, 748, 1, 42, 1, 1362, 2, 2, 10, 42, 1, 214, 254, 4, 1, 1, 1, 3838, 5, 406, 2, 2, 1, 78, 1, 5458, 1, 26, 2, 12250, 2, 10, 1, 2, 1, 934

Offset: 1

Antti Karttunen, Nov 22 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, A323243, A324201, A329610, A329638, A329639, A329640, A329644.

A323243(n) = if(1==n,0,sigma(A156552(n)));
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
A329644(n) = sumdiv(n,d,moebius(n/d)*((2*A156552(d))-A323243(d)));
A329641(n) = { my(t=0,u=0); fordiv(n, d, if((d=A329644(d))>0, t +=d, u -= d)); gcd(u,t); };

a(n) = gcd(A329638(n), A329639(n)).

a(A324201(n)) = A329610(n).

A332463 Möbius transform of A332223.

Original entry on oeis.org

1, 1, 3, 3, 7, 4, 15, 2, 21, 9, 31, 13, 63, 4, 10, 42, 127, -3, 255, 14, 21, 88, 511, 22, 117, 320, 24, 97, 1023, -22, 2047, -36, 190, 1444, 82, 34, 4095, -200, 120, 306, 8191, 14, 16383, -59, 13, 4180, 32767, 30, 609, -103, 15494, -303, 65535, 30, 141, -32, -6, 8920, 131071, 132, 262143, 506030, 564, 834, 658, -149

Offset: 1

Antti Karttunen, Feb 22 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000 (computed using 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. A008683, A332223.

Cf. also A003972, A324201, A324543.

A332463(n) = sumdiv(n,d,moebius(n/d)*A332223(d)); \\ Needs also code from A332223.

a(n) = Sum_{d|n} A008683(n/d) * A332223(d).

A324721 Positions of positive terms in A323244; numbers n for which 2*A156552(n) > A323243(n).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 82, 83, 84, 86, 88, 89, 90, 92, 93, 94, 96, 97, 98, 100

Offset: 1

Antti Karttunen, Mar 12 2019

nonn

These correspond to deficient numbers as this is sequence A005940(1+A005100(n)) sorted into ascending order. Subsequence of A324720.

Antti Karttunen, Table of n, a(n) for n = 1..7676 (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. A005100, A005940, A156552, A323243, A323244, A324732 (characteristic function).

Cf. A324551, A324201, A324720.

A324720 Positions of nonnegative terms in A323244; numbers n for which 2*A156552(n) >= A323243(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 82, 83, 84, 86, 88, 89, 90, 92, 93, 94, 96, 97, 98, 100

Offset: 1

Antti Karttunen, Mar 12 2019

nonn

These correspond to nonabundant numbers as the terms after the initial 1 are obtained by sorting the sequence A005940(1+A263837(n)) into ascending order.

Antti Karttunen, Table of n, a(n) for n = 1..7679 (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. A005940, A156552, A263837, A323243, A323244.

Cf. A324551 (complement), A324201, A324721 (subsequences).

cp's OEIS Frontend

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

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A332225 Numbers k > 1 for which A048675(A332223(k)) is equal to 2*A048675(k).

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A324051 a(n) = A106315(A156552(n)).

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A324115 a(n) = A002487(A323244(n)).

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

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A324551 Positions of negative terms in A323244.

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A329641 a(n) = gcd(A329638(n), A329639(n)).

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A332463 Möbius transform of A332223.

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A324721 Positions of positive terms in A323244; numbers n for which 2*A156552(n) > A323243(n).

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