A323244 -id:A323244 - cp's OEIS frontend

A329644 Möbius transform of A323244, the deficiency of A156552(n).

0, 1, 1, 1, 1, 2, 1, 4, -1, 3, 1, 5, 1, 14, 0, 0, 1, 9, 1, 12, -5, 16, 1, 8, -5, 44, 4, 5, 1, 2, 1, 24, 12, 80, -4, -4, 1, 254, -14, 0, 1, 22, 1, 47, 7, 224, 1, 24, -13, 19, 6, 83, 1, 12, -21, 44, -14, 746, 1, 14, 1, 1360, 20, -8, 8, 9, 1, 131, 252, 24, 1, 12, 1, 3836, 13, 149, -12, 71, 1, 56, -16, 5456, 1, -21, -74, 12248, -350, -40, 1

Offset: 1

Antti Karttunen, Nov 21 2019

sign

The first eleven zeros occur at n = 1, 15, 16, 40, 96, 119, 120, 160, 893, 2464, 6731. There are 3091 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(n) divisors                             a(n) applied       Sum of positive
                                                 to each:          terms, A329610
        9: [1, 3, 9]                         -> [0, 1, -1]                     1
      125: [1, 5, 25, 125]                   -> [0, 1, -5, 4]                  5
   161051: [1, 11, 121, 1331, 14641, 161051] -> [0, 1, -29, 4, -240, 264]    269
410338673: [1, 17, 289, 4913, 83521, 1419857, 24137569, 410338673]
                             -> [0, 1, -125, 4, -1008, 1032, -5048, 5144]   6181
The positive and negative terms seem to alternate, and the fourth term (from case n=125 onward) is always 4. See also array A329637.

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. A000203, A008683, A036563, A156552, A297112, A297113, A323243, A323244, A324543, A329610, A329637, A329638, A329639, A329645, A329646.

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
A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k));
A329644(n) = sumdiv(n,d,moebius(n/d)*A323244(d));

a(n) = Sum_{d|n} A008683(n/d) * A323244(d).
a(n) = Sum_{d|n} A008683(n/d) * (2*A156552(d) - A323243(d)).
a(1) = 0; for n > 1, a(n) = 2*A297112(n) - A324543(n) = 2^A297113(n) - A324543(n).
a(n) = A329642(n) - A329643(n).
For all n >= 1, a(A000040(n)^2) = A323244(A000040(n)^2)-1 = -A036563(n).
For all primes p, a(p^3) = A323244(p^3) - A323244(p^2) = 4.

A329638 Sum of A329644(d) for all such divisors d of n for which that value is positive. Here A329644 is the Möbius transform of A323244, the deficiency of A156552(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 6, 1, 5, 1, 10, 1, 16, 2, 6, 1, 13, 1, 18, 2, 18, 1, 22, 1, 46, 5, 22, 1, 10, 1, 30, 14, 82, 2, 19, 1, 256, 2, 22, 1, 41, 1, 66, 9, 226, 1, 46, 1, 24, 8, 130, 1, 29, 2, 70, 2, 748, 1, 42, 1, 1362, 22, 30, 10, 42, 1, 214, 254, 44, 1, 43, 1, 3838, 15, 406, 2, 120, 1, 78, 5, 5458, 1, 52, 2, 12250, 2, 70, 1, 26, 2, 934

Offset: 1

Antti Karttunen, Nov 21 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, A323244, A329639, A329641, A329644.

Cf. also A318878.

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
A323243(n) = if(1==n,0,sigma(A156552(n)));
A324543(n) = sumdiv(n,d,moebius(n/d)*A323243(d));
A297113(n) = if(1==n, 0, (primepi(vecmax(factor(n)[, 1])) + (bigomega(n)-omega(n))));
A329644(n) = if(1==n,0, 2^A297113(n) - A324543(n));
A329638(n) = sumdiv(n,d,if((d=A329644(d))>0,d,0));

a(n) = Sum_{d|n} [A329644(d) > 0] * A329644(d), where [ ] is Iverson bracket.

a(n) = A323244(n) + A329639(n).

A329639 Sum of -A329644(d) for all such divisors d of n for which A329644(d) < 0. Here A329644 is the Möbius transform of A323244, the deficiency of A156552(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 5, 0, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 5, 0, 0, 14, 0, 0, 5, 0, 0, 1, 0, 0, 0, 13, 5, 0, 0, 0, 1, 21, 0, 14, 0, 0, 0, 0, 0, 6, 8, 0, 0, 0, 0, 0, 4, 0, 5, 0, 0, 5, 0, 12, 14, 0, 0, 17, 0, 0, 26, 74, 0, 350, 40, 0, 14, 53, 0, 0, 0, 70, 0, 0, 13, 18, 7, 0, 0, 0, 0, 15

Offset: 1

Antti Karttunen, Nov 21 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. A323244, A329638, A329640, A329641, A329644.

Cf. also A318879.

A329639(n) = sumdiv(n,d,if((d=A329644(d))<0,-d,0));

a(n) = -Sum_{d|n} [A329644(d) < 0] * A329644(d), where [ ] is Iverson bracket.

a(n) = A329638(n) - A323244(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.

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.

A323240 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = [A001222(n), A323244(n)] for n > 1, and f(1) = 0.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 3, 10, 2, 11, 2, 12, 13, 14, 2, 15, 16, 17, 18, 19, 2, 8, 2, 20, 21, 22, 23, 24, 2, 25, 26, 15, 2, 27, 2, 28, 29, 30, 2, 31, 26, 32, 33, 34, 2, 35, 36, 37, 26, 38, 2, 39, 2, 40, 41, 42, 43, 44, 2, 45, 46, 47, 2, 48, 2, 49, 8, 50, 51, 52, 2, 53, 54, 55, 2, 56, 57, 58, 59, 60, 2, 61, 62, 63, 64, 65, 66, 67, 2, 68, 69, 60, 2

Offset: 1

Antti Karttunen, Jan 10 2019

nonn

Restricted growth sequence transform of function f, where f(1) = 0 and f(n) = [A001222(n), A323244(n)] for n > 1.
Equally, restricted growth sequence transform of function f, where f(1) = 0 and f(n) = A318310(A156552(n)) for n > 1.
For all i, j:
  a(i) = a(j) => A323245(i) = A323245(j), [Equally: A323244(i) = A323244(j)]
  a(i) = a(j) => A323246(i) = A323246(j). [Equally: A323248(i) = A323248(j)]

Cf. A001222, A156552, A318310, A323244, A323245, A323246, A323248.

Cf. also A323168.

up_to = 1024;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
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));
A323240aux(n) = if(1==n,-1,[bigomega(n),A323244(n)]);
v323240 = rgs_transform(vector(up_to, n, A323240aux(n)));
A323240(n) = v323240[n];

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.

A329642 a(n) = Sum_{d|n} [1 == A008683(n/d)] * A323244(d), where A323244(x) gives the deficiency of A156552(x).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 6, 0, 5, 1, 11, 1, 16, 2, 6, 1, 13, 1, 19, -3, 18, 1, 24, -4, 46, 4, 23, 1, 13, 1, 30, 14, 82, -2, 18, 1, 256, -12, 24, 1, 39, 1, 67, 9, 226, 1, 52, -12, 20, 8, 131, 1, 28, -19, 72, -12, 748, 1, 53, 1, 1362, 17, 22, 10, 45, 1, 215, 254, 43, 1, 48, 1, 3838, 11, 407, -10, 109, 1, 84, -12, 5458, 1, 48, -72, 12250, -348, 32, 1, 18

Offset: 1

Antti Karttunen, Nov 21 2019

sign

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. A008683, A156552, A323243, A323244, A329643, A329644.

Cf. A329645 (inverse Möbius transform).

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
A323243(n) = if(1==n,0,sigma(A156552(n)));
A329642(n) = sumdiv(n,d,(1==moebius(n/d))*((2*A156552(d))-A323243(d)));

a(n) = Sum_{d|n} [1 == A008683(n/d)] * (2*A156552(d) - A323243(d)).

a(n) = A329643(n) + A329644(n).

A329643 a(n) = Sum_{d|n} [-1 == A008683(n/d)] * A323244(d), where A323244(x) gives the deficiency of A156552(x).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 6, 0, 2, 2, 6, 0, 4, 0, 7, 2, 2, 0, 16, 1, 2, 0, 18, 0, 11, 0, 6, 2, 2, 2, 22, 0, 2, 2, 24, 0, 17, 0, 20, 2, 2, 0, 28, 1, 1, 2, 48, 0, 16, 2, 28, 2, 2, 0, 39, 0, 2, -3, 30, 2, 36, 0, 84, 2, 19, 0, 36, 0, 2, -2, 258, 2, 38, 0, 28, 4, 2, 0, 69, 2, 2, 2, 72, 0, 31, 2, 228, 2, 2, 2, 76, 0, 4, 14, 37, 0, 94, 0, 136, -3

Offset: 1

Antti Karttunen, Nov 21 2019

sign

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. A006881, A008683, A156552, A323243, A323244, A329642, A329644.

Cf. A329646 (inverse Möbius transform).

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
A323243(n) = if(1==n,0,sigma(A156552(n)));
A329643(n) = sumdiv(n,d,(-1==moebius(n/d))*((2*A156552(d))-A323243(d)));

a(n) = Sum_{d|n} [-1 == A008683(n/d)] * (2*A156552(d) - A323243(d)).
a(n) = A329642(n) - A329644(n).
For all n, a(A000040(n)) = 0, a(A006881(n)) = 2.

A323245 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A323244(n) for n > 1, and f(1) = -1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 3, 5, 2, 10, 2, 11, 12, 11, 2, 13, 14, 15, 4, 13, 2, 8, 2, 16, 17, 18, 19, 17, 2, 20, 21, 13, 2, 22, 2, 23, 24, 25, 2, 15, 21, 26, 24, 27, 2, 28, 29, 30, 21, 31, 2, 32, 2, 33, 9, 13, 8, 32, 2, 34, 35, 36, 2, 37, 2, 38, 8, 39, 40, 41, 2, 42, 21, 43, 2, 44, 45, 46, 47, 16, 2, 10, 48, 49, 50, 51, 52, 30, 2, 53, 14, 16, 2

Offset: 1

Antti Karttunen, Jan 10 2019

nonn

Restricted growth sequence transform of function f, defined as f(1) = -1, and for n > 1, f(n) = A033879(A156552(n)).

Antti Karttunen, Table of n, a(n) for n = 1..4096

up_to = 1024;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
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));
A323245aux(n) = if(1==n,-1,A323244(n));
v323245 = rgs_transform(vector(up_to, n, A323245aux(n)));
A323245(n) = v323245[n];

cp's OEIS Frontend

A329644 Möbius transform of A323244, the deficiency of A156552(n).

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A329638 Sum of A329644(d) for all such divisors d of n for which that value is positive. Here A329644 is the Möbius transform of A323244, the deficiency of A156552(n).

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A329639 Sum of -A329644(d) for all such divisors d of n for which A329644(d) < 0. Here A329644 is the Möbius transform of A323244, the deficiency of A156552(n).

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

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A324551 Positions of negative terms in A323244.

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A323240 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = [A001222(n), A323244(n)] for n > 1, and f(1) = 0.

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

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A329642 a(n) = Sum_{d|n} [1 == A008683(n/d)] * A323244(d), where A323244(x) gives the deficiency of A156552(x).

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A329643 a(n) = Sum_{d|n} [-1 == A008683(n/d)] * A323244(d), where A323244(x) gives the deficiency of A156552(x).

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