A005187 -id:A005187 - cp's OEIS frontend

A324377 a(0) = 0; for n > 0, a(n) = A000265(A005187(n)).

0, 1, 3, 1, 7, 1, 5, 11, 15, 1, 9, 19, 11, 23, 25, 13, 31, 1, 17, 35, 19, 39, 41, 21, 23, 47, 49, 25, 53, 27, 7, 57, 63, 1, 33, 67, 35, 71, 73, 37, 39, 79, 81, 41, 85, 43, 11, 89, 47, 95, 97, 49, 101, 51, 13, 105, 109, 55, 7, 113, 29, 117, 119, 15, 127, 1, 65, 131, 67, 135, 137, 69, 71, 143, 145, 73, 149, 75, 19

Offset: 0

Antti Karttunen, Feb 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16385
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000265, A005187, A283477, A322821, A324287, A324378, A324379.

A000265(n) = (n/2^valuation(n, 2));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A324377(n) = if(!n,n,A000265(A005187(n)));

a(0) = 0; for n > 0, a(n) = A000265(A005187(n)) = A005187(n) / 2^A324379(n).

a(n) = A322821(A283477(n)).

A346237 Dirichlet inverse of A005187.

Original entry on oeis.org

1, -3, -4, 2, -8, 14, -11, 0, 0, 30, -19, -14, -23, 41, 38, 0, -32, -2, -35, -34, 49, 73, -42, 4, 17, 89, 14, -46, -54, -172, -57, 0, 88, 126, 109, 10, -71, 137, 110, 12, -79, -219, -82, -86, -6, 164, -89, 0, 26, -103, 158, -106, -102, -76, 199, 16, 170, 212, -113, 274, -117, 223, 16, 0, 240, -406, -131, -154, 201

Offset: 1

Antti Karttunen, Jul 13 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005187, A346238.

Cf. also A297111, A317927, A317928.

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
v346237 = DirInverseCorrect(vector(up_to,n,A005187(n)));
A346237(n) = v346237[n];

a(n) = A346238(n) - A005187(n).

A379008 Square array A(n, k) = A294898(A246278(n, k)), read by falling antidiagonals; Difference A005187(n)-A000203(n) applied to the prime shift array.

Original entry on oeis.org

0, 0, 0, -2, 3, 2, 0, 2, 16, 3, 0, 10, 19, 38, 7, -6, 7, 88, 54, 104, 9, 1, 8, 33, 280, 113, 151, 14, 0, 16, 96, 65, 1192, 184, 268, 15, -5, 38, 44, 389, 152, 2009, 282, 336, 18, -4, 22, 464, 88, 1279, 207, 4600, 388, 502, 24, 5, 16, 142, 1996, 174, 2445, 345, 6470, 608, 806, 25, -14, 18, 174, 623, 13170, 257, 4834, 497, 11605, 833, 924, 33

Offset: 1

Antti Karttunen, Dec 14 2024

Question: Are all columns increasing, and strictly increasing after the leftmost column?

			The top left corner of the array:
k=  |  1    2    3      4    5      6    7       8      9     10   11      12
2k= |  2    4    6      8   10     12   14      16     18     20   22      24
----+-------------------------------------------------------------------------
1   |  0,   0,  -2,     0,   0,    -6,   1,      0,    -5,    -4,   5,    -14,
2   |  0,   3,   2,    10,   7,     8,  16,     38,    22,    16,  18,     26,
3   |  2,  16,  19,    88,  33,    96,  44,    464,   142,   174,  58,    495,
4   |  3,  38,  54,   280,  65,   389,  88,   1996,   623,   469, 103,   2737,
5   |  7, 104, 113,  1192, 152,  1279, 174,  13170,  1516,  1717, 211,  14102,
6   |  9, 151, 184,  2009, 207,  2445, 257,  26172,  3208,  2756, 328,  31850,
7   | 14, 268, 282,  4600, 345,  4834, 439,  78295,  5406,  5916, 473,  82285,
8   | 15, 336, 388,  6470, 497,  7455, 533, 123071,  9035,  9501, 638, 141745,
9   | 18, 502, 608, 11605, 653, 14081, 784, 267115, 17773, 15097, 870, 324077,
Here 0's occur also after the first row. For example column 30, which corresponds with numbers 60, 315, 1925, 7007, 26741, ..., begins as -52, 0, 868, 4428, 19958, etc. See also A295296.

Cf. A000203, A005187, A246278, A294898, A295296.
Cf. A080085 (column 1, incremented by one).
Cf. also array A378979, and A324348 (another permutation of A294898).

up_to = 11325; \\ = binomial(150+1,2)
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A379008sq(row,col) = A294898(A246278sq(row,col));
A379008list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A379008sq(col,(a-(col-1))))); (v); };
v379008 = A379008list(up_to);
A379008(n) = v379008[n];

A279337 a(1) = 1, for n > 1, if n is even, a(n) = A055938(n/2), otherwise a(n) = A005187(a(A064989(n))).

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 7, 9, 8, 12, 11, 13, 19, 14, 10, 17, 35, 20, 67, 21, 22, 24, 131, 27, 15, 28, 16, 29, 259, 30, 515, 33, 25, 36, 18, 37, 1027, 40, 46, 43, 2051, 44, 4099, 45, 23, 48, 8195, 51, 26, 52, 53, 55, 16387, 58, 41, 59, 70, 60, 32771, 61, 65539, 62, 39, 65, 47, 68, 131075, 69, 78, 72, 262147, 75, 524291, 76, 38, 77, 34, 80

Offset: 1

Antti Karttunen, Dec 10 2016

nonn

For n > 1, a(n) = the number which is in the same position of array A256997 where n is located in array A246278.

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences that are permutations of the natural numbers

Inverse: A279336.
Cf. A005187, A055938, A064989, A246278, A252752, A256997.
Cf. also A279339 (a variant).

(definec (A279337 n) (cond ((= 1 n) n) ((even? n) (A055938 (/ n 2))) (else (A005187 (A279337 (A064989 n))))))

a(1) = 1; for n > 1, if n is even, a(n) = A055938(n/2), otherwise a(n) = A005187(a(A064989(n))).

Other identities: For all n >= 2, a(n) = A256997(A252752(n)).

A283996 a(n) = n OR A005187(floor(n/2)), where OR is bitwise-or (A003986).

Original entry on oeis.org

0, 1, 3, 3, 7, 7, 6, 7, 15, 15, 10, 11, 14, 15, 15, 15, 31, 31, 18, 19, 22, 23, 23, 23, 30, 31, 31, 31, 29, 29, 30, 31, 63, 63, 34, 35, 38, 39, 39, 39, 46, 47, 47, 47, 45, 45, 46, 47, 62, 63, 63, 63, 53, 53, 54, 55, 61, 61, 62, 63, 60, 61, 63, 63, 127, 127, 66, 67, 70, 71, 71, 71, 78, 79, 79, 79, 77, 77, 78, 79, 94, 95, 95, 95, 85, 85, 86, 87, 93, 93, 94, 95

Offset: 0

Antti Karttunen, Mar 19 2017

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A003986, A005187, A283997, A283998.

A[n_]:=2*n - DigitCount[2*n, 2, 1]; Table[BitOr[n, A[Floor[n/2]]], {n,0,100}] (* Indranil Ghosh, Mar 25 2017 *)

b(n) = if(n<1, 0, b(n\2) + n%2);
A(n) = 2*n - b(2*n);
for(n=0, 100, print1(bitor(n, A(floor(n/2))),", ")) \\ Indranil Ghosh, Mar 25 2017

def A(n): return 2*n - bin(2*n)[2:].count("1")
print([n|A(n//2) for n in range(101)]) # Indranil Ghosh, Mar 25 2017

(define (A283996 n) (A003986bi n (A005187 (floor->exact (/ n 2))))) ;; Where A003986bi implements bitwise-OR (A003986).

a(n) = n OR A005187(floor(n/2)), where OR is bitwise-or (A003986).
a(n) = A283997(n) + A283998(n).
a(n) = A005187(n) - A283998(n).

A283998 a(n) = n AND A005187(floor(n/2)), where AND is bitwise-and (A004198).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 4, 4, 0, 1, 8, 8, 8, 8, 10, 11, 0, 1, 16, 16, 16, 16, 18, 19, 16, 16, 18, 19, 24, 25, 26, 26, 0, 1, 32, 32, 32, 32, 34, 35, 32, 32, 34, 35, 40, 41, 42, 42, 32, 32, 34, 35, 48, 49, 50, 50, 48, 49, 50, 50, 56, 56, 56, 57, 0, 1, 64, 64, 64, 64, 66, 67, 64, 64, 66, 67, 72, 73, 74, 74, 64, 64, 66, 67, 80, 81, 82, 82, 80, 81, 82, 82, 88, 88, 88, 89

Offset: 0

Antti Karttunen, Mar 19 2017

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A004198, A005187, A283996, A283997.

A[n_]:=2*n - DigitCount[2*n, 2, 1];Table[BitAnd[n, A[Floor[n/2]]], {n, 0, 100}] (* Indranil Ghosh, Mar 25 2017 *)

b(n) = if(n<1, 0, b(n\2) + n%2);
A(n) = 2*n - b(2*n);
for(n=0, 100, print1(bitand(n, A(floor(n/2))),", ")) \\ Indranil Ghosh, Mar 25 2017

def A(n): return 2*n - bin(2*n)[2:].count("1")
print([n&A(n//2) for n in range(101)]) # Indranil Ghosh, Mar 25 2017

(define (A283998 n) (A004198bi n (A005187 (floor->exact (/ n 2))))) ;; Where A004198bi implements bitwise-AND (A004198).

a(n) = n AND A005187(floor(n/2)), where AND is bitwise-and (A004198).
a(n) = A283996(n) - A283997(n).
a(n) = A005187(n) - A283996(n) = (A005187(n) - A283997(n))/2.

A283999 a(n) = A005187(n) XOR A006068(n), where XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 0, 0, 6, 0, 14, 14, 14, 0, 30, 30, 30, 30, 30, 18, 16, 0, 62, 62, 62, 62, 62, 50, 48, 62, 62, 34, 32, 34, 32, 44, 44, 0, 126, 126, 126, 126, 126, 114, 112, 126, 126, 98, 96, 98, 96, 108, 108, 126, 126, 66, 64, 66, 64, 76, 76, 66, 64, 92, 92, 92, 92, 92, 82, 0, 254, 254, 254, 254, 254, 242, 240, 254, 254, 226, 224, 226, 224, 236, 236, 254, 254, 194, 192, 194

Offset: 0

Antti Karttunen, Mar 20 2017

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A003987, A005187, A006068, A283997.

Table[BitXor[Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]], 2 n - DigitCount[2 n, 2, 1]], {n, 0, 84}] (* Michael De Vlieger, Mar 20 2017, after Jan Mangaldan at A006068 *)

b(n) = if(n<1, 0, b(n\2) + n%2);
A(n) = 2*n - b(2*n);
a(n) = if(n<2, n, 2*a(floor(n/2)) + (n%2 + a(floor(n/2))%2)%2);
for(n=0, 110, print1(bitxor(A(n),a(n)),", ")) \\ Indranil Ghosh, Mar 25 2017

def A(n): return 2*n - bin(2*n)[2:].count("1")
def a(n): return n if n<2 else 2*a(n//2) + (n%2 + a(n//2)%2)%2
print([A(n)^a(n) for n in range(111)]) # Indranil Ghosh, Mar 25 2017

(define (A283999 n) (A003987bi (A005187 n) (A006068 n))) ;; Where A003987bi implements bitwise-XOR (A003987).

a(n) = A005187(n) XOR A006068(n), where XOR is bitwise-xor (A003987).

a(n) = A006068(2*n) XOR A283997(2*n).

A295655 a(n) = A000203(n) / A294896(n) = A000203(n) / gcd(A000203(n), A005187(n)).

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 8, 1, 13, 1, 12, 14, 14, 24, 12, 1, 9, 39, 4, 21, 32, 36, 4, 30, 31, 6, 4, 56, 5, 9, 32, 1, 3, 9, 48, 13, 38, 60, 28, 15, 42, 32, 22, 84, 39, 9, 48, 62, 3, 93, 36, 98, 9, 15, 24, 120, 8, 45, 60, 42, 62, 96, 13, 1, 21, 72, 68, 63, 32, 144, 12, 195, 74, 114, 62, 140, 16, 21, 80, 93, 121, 18, 14

Offset: 1

Antti Karttunen, Nov 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n)

Cf. A000203, A005187, A294896, A295656.

Array[#1/GCD[#1, #2] & @@ {DivisorSigma[1, #], IntegerExponent[(2 #)!, 2]} &, 83] (* Michael De Vlieger, Nov 26 2017 *)

a(n) = A000203(n) / A294896(n) = A000203(n) / gcd(A000203(n), A005187(n)).

A295656 a(n) = A005187(n) / A294896(n) = A005187(n) / gcd(A000203(n), A005187(n)).

Original entry on oeis.org

1, 1, 1, 1, 4, 5, 11, 1, 16, 1, 19, 11, 23, 25, 13, 1, 16, 34, 7, 19, 39, 41, 7, 23, 47, 7, 5, 53, 9, 7, 57, 1, 4, 11, 67, 10, 71, 73, 37, 13, 79, 27, 41, 85, 43, 11, 89, 47, 5, 97, 49, 101, 17, 13, 35, 109, 11, 56, 113, 29, 117, 119, 15, 1, 32, 65, 131, 67, 45, 137, 23, 142, 143, 145, 73, 149, 25, 19, 153, 79, 159, 23, 27

Offset: 1

Antti Karttunen, Nov 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A005187, A294896, A295655.

a(n) = A005187(n) / A294896(n) = A005187(n) / gcd(A000203(n), A005187(n)).

A318445 a(n) = Sum_{d|n, dA005187(d).

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 11, 5, 12, 1, 25, 1, 15, 13, 26, 1, 34, 1, 37, 16, 23, 1, 62, 9, 27, 21, 47, 1, 70, 1, 57, 24, 36, 20, 97, 1, 39, 28, 90, 1, 93, 1, 71, 55, 46, 1, 139, 12, 77, 37, 83, 1, 118, 28, 115, 40, 58, 1, 193, 1, 61, 71, 120, 32, 142, 1, 109, 47, 133, 1, 228, 1, 75, 86, 119, 31, 164, 1, 199, 71, 83, 1, 256, 41

Offset: 1

Antti Karttunen, Aug 26 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A005187, A293214, A293447, A318446, A318447.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A318445(n) = sumdiv(n,d,(dA005187(d));

a(n) = Sum_{d|n, dA005187(d).
a(n) = A318446(n) - A005187(n).
a(n) = A211779(n) + A318447(n).
a(n) = A293447(A293214(n)).

cp's OEIS Frontend

A324377 a(0) = 0; for n > 0, a(n) = A000265(A005187(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A346237 Dirichlet inverse of A005187.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A379008 Square array A(n, k) = A294898(A246278(n, k)), read by falling antidiagonals; Difference A005187(n)-A000203(n) applied to the prime shift array.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A279337 a(1) = 1, for n > 1, if n is even, a(n) = A055938(n/2), otherwise a(n) = A005187(a(A064989(n))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A283996 a(n) = n OR A005187(floor(n/2)), where OR is bitwise-or (A003986).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

Formula

A283998 a(n) = n AND A005187(floor(n/2)), where AND is bitwise-and (A004198).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

Formula

A283999 a(n) = A005187(n) XOR A006068(n), where XOR is bitwise-xor (A003987).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

Formula

A295655 a(n) = A000203(n) / A294896(n) = A000203(n) / gcd(A000203(n), A005187(n)).

Views

Author

Keywords

Links