A005187 -id:A005187 - cp's OEIS frontend

A289618 a(n) = A289617(n) - A046645(n) = A005187(A001222(n)) - A046645(n).

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 2, 1, 1, 0, 2, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 2, 1, 1, 1, 2, 0, 2, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2

Offset: 1

Antti Karttunen, Jul 08 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A046645, A289617.

Cf. A268375 (positions of zeros), A289619 (of ones).

(define (A289618 n) (- (A289617 n) (A046645 n)))

a(n) = A289617(n) - A046645(n) = A005187(A001222(n)) - A046645(n).

A324348 a(n) = A294898(A005940(1+n)), where A294898(k) = A005187(k) - A000203(k).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 0, 2, -6, 16, -5, 10, 0, 7, 1, 7, -4, 19, -16, 8, -14, 38, 4, 22, -21, 88, -16, 38, 0, 9, 5, 16, -3, 33, -15, 16, -12, 54, -7, 14, -52, 96, -58, 26, -30, 104, 22, 62, -20, 142, -76, 43, -53, 280, 26, 119, -68, 464, -42, 116, 0, 14, 7, 18, 1, 44, -14, 38, -11, 65, -1, 38, -59, 174, -66, 52, -28, 113, 16, 72, -59, 191, -160, 0, -124

Offset: 0

Antti Karttunen, Feb 24 2019

sign

Positions of zeros is given by the sequence A156552(A295296(n)), n >= 1, sorted into ascending order: 0, 1, 2, 3, 7, 9, 15, 31, 63, 86, 127, 255, 511, 519, 1023, 2047, 4095, 8191, 16383, 32767, ...

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Cf. A000120, A000203, A005187, A005940, A294898, A324055.

Cf. also A156552, A295296, A323248.

A324348(n) = { my(m1=2, m2=1, p=2, mp=p*p); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, m1 *= p; if(3==(n%4), mp *= p, m2 *= (mp-1)/(p-1))); n>>=1); ((m1-m2)-hammingweight(m1)); };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n) - sigma(n));
A324348(n) = A294898(A005940(1+n));

a(n) = A294898(A005940(1+n)).

a(n) = A324055(n) - A000120(A005940(1+n)).

A326131 Positive numbers n for which A000120(n) = k*A294898(n), with k < 0; numbers for which A326130(n) = sigma(n) - A005187(n).

Original entry on oeis.org

6, 28, 110, 496, 884, 8128, 18632, 85936, 116624, 15370304, 33550336, 73995392, 815634435, 3915380170, 5556840416, 6800695312, 8589869056

Offset: 1

Antti Karttunen, Jun 09 2019

No further terms below 2^31.
See also comments in A326133.
The quotients A000120(k)/(sigma(k)-A005187(k)) for these terms are: 1, 1, 5, 1, 3, 1, 5, 9, 2, 2, 1, 2, 2. Ones occur at the positions of perfect numbers.
a(18) > 10^11. - Amiram Eldar, Jan 03 2021

			110 is "1101110" in binary, thus A000120(110) = 5. Sigma(110) = 216, while A005187(110) = 215, thus as 5 = 5*(216-215), 110 is included in this sequence.

Intersection of A326132 and A326133, also of A326132 and A326138.
Cf. A000120, A000396 (subsequence), A005187, A294898, A295296, A295297, A295298, A295299, A326130.
Cf. also A325981, A326141.

q[n_] := Module[{bw = DigitCount[n, 2, 1], ab = DivisorSigma[1, n] - 2*n, sum}, (sum = ab + bw) > 0 && Divisible[bw, sum]]; Select[Range[10^5], q] (* Amiram Eldar, Jan 03 2021 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
isA326131(n) = { my(t=sigma(n)-A005187(n)); (gcd(hammingweight(n), t) == t); };

a(14)-a(17) from Amiram Eldar, Jan 03 2021

A122247 Partial sums of A005187.

Original entry on oeis.org

0, 1, 4, 8, 15, 23, 33, 44, 59, 75, 93, 112, 134, 157, 182, 208, 239, 271, 305, 340, 378, 417, 458, 500, 546, 593, 642, 692, 745, 799, 855, 912, 975, 1039, 1105, 1172, 1242, 1313, 1386, 1460, 1538, 1617, 1698, 1780, 1865, 1951, 2039, 2128, 2222, 2317, 2414

Offset: 0

Paul Barry, Aug 27 2006

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Hsien-Kuei Hwang, Svante Janson and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47. Also first authors' copy, 2016. See example 6.2.

Cf. A005187 (first differences).

b:= proc(n) option remember; `if`(n=0, 0, b(iquo(n, 2))+n) end:
a:= proc(n) option remember; `if`(n=0, 0, b(n)+a(n-1)) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 25 2022

Accumulate[Table[2n-Count[IntegerDigits[2 n,2],1],{n,0,70}]] (* Harvey P. Dale, Oct 22 2011 *)

a(n) = Sum_{k=1..n} Sum_{j=0..n} floor(k/2^j).

G.f.: (1/(1-x))*Sum_{k>=0} x^(2^k)/((1-x)*(1-x^(2^k))).

A213715 a(n) = position of A179016(n) in A005187.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 26, 28, 31, 32, 33, 35, 37, 39, 40, 42, 44, 47, 48, 50, 52, 54, 56, 58, 60, 63, 64, 65, 67, 69, 71, 72, 74, 76, 79, 80, 82, 84, 86, 88, 90, 92, 95, 96, 98, 100, 102, 104, 107, 110, 112, 115, 118, 120, 123, 127, 128, 129, 131, 133, 135, 136, 138, 140, 143, 144, 146, 148, 150, 152, 154, 156, 159, 160, 162, 164, 166, 168

Offset: 0

Antti Karttunen, Oct 26 2012

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8727

Cf. A213716, A213717.

: (define (A213715 n) (A213714 (A179016 n)))

a(n) = A213714(A179016(n)).

Offset changed from 1 to 0 by Antti Karttunen, Nov 05 2012

A256994 a(n) = n + 1 when n <= 3, otherwise a(n) = 2^(n-2) + 3; also iterates of A005187 starting from a(1) = 2.

Original entry on oeis.org

2, 3, 4, 7, 11, 19, 35, 67, 131, 259, 515, 1027, 2051, 4099, 8195, 16387, 32771, 65539, 131075, 262147, 524291, 1048579, 2097155, 4194307, 8388611, 16777219, 33554435, 67108867, 134217731, 268435459, 536870915, 1073741827, 2147483651, 4294967299, 8589934595, 17179869187, 34359738371, 68719476739, 137438953475, 274877906947

Offset: 1

Antti Karttunen, Apr 15 2015

nonn

Note that if we instead iterated function b(n) = 1+A005187(n), from b(1) onward, we would get the powers of two, A000079.

Antti Karttunen, Table of n, a(n) for n = 1..128

Topmost row of A256995, leftmost column of A256997.

Cf. A000079, A005187, A062709, A068156.

Table[If[n<4,n+1,2^(n-2)+3],{n,40}] (* Harvey P. Dale, May 14 2019 *)

A256994(n) = if(n < 4, n+1, 2^(n-2) + 3);

\\ By iterating A005187:
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
i=1; k=2; print1(k); while(i <= 40, k = A005187(k); print1(", ", k); i++);

(define (A256994 n) (if (< n 4) (+ n 1) (+ (A000079 (- n 2)) 3)))

;; The following uses memoization-macro definec:
(definec (A256994 n) (if (= 1 n) 2 (A005187 (A256994 (- n 1)))))

If n < 4, a(n) = n + 1, otherwise a(n) = 2^(n-2) + 3 = A062709(n-2).

a(1) = 2; for n > 1, a(n) = A005187(a(n-1)).

A276343 Permutation of natural numbers: a(1) = 1, a(A087686(1+n)) = A005187(1+a(n)), a(A088359(n)) = A055938(a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 3, 2, 7, 6, 5, 4, 15, 14, 13, 12, 11, 9, 10, 8, 31, 30, 29, 28, 27, 26, 24, 20, 25, 21, 23, 22, 17, 18, 19, 16, 63, 62, 61, 60, 59, 58, 57, 55, 51, 43, 56, 52, 44, 54, 48, 53, 50, 45, 36, 47, 37, 39, 49, 40, 41, 46, 42, 33, 34, 35, 38, 32, 127, 126, 125, 124, 123, 122, 121, 120, 118, 114, 106, 90, 119, 115, 107, 91, 117, 111, 99

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276344.
Cf. A004001, A005187, A055938, A080677, A087686, A088359, A093879.
Similar or related permutations: A233276, A233278, A267111, A276345, A276441.
Compare also to the scatter-plots of A276443 and A276445.

(definec (A276343 n) (cond ((< n 2) n) ((zero? (A093879 (- n 1))) (A005187 (+ 1 (A276343 (+ -1 (A080677 n)))))) (else (A055938 (A276343 (+ -1 (A004001 n)))))))

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = A005187(1+a(A080677(n)-1)), otherwise [when n is in A088359], a(n) = A055938(a(A004001(n)-1)).
As a composition of other permutations:
a(n) = A233276(A267111(n)).
a(n) = A233278(A276441(n)).

A276344 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A087686(1+a(n)), a(A055938(n)) = A088359(a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 3, 2, 7, 6, 5, 4, 15, 13, 14, 12, 11, 10, 9, 8, 31, 28, 29, 30, 23, 25, 27, 26, 22, 24, 21, 20, 19, 18, 17, 16, 63, 59, 60, 61, 50, 52, 62, 53, 55, 56, 58, 41, 44, 49, 57, 51, 46, 54, 48, 40, 43, 47, 45, 39, 42, 38, 37, 36, 35, 34, 33, 32, 127, 122, 123, 124, 108, 110, 125, 111, 113, 114, 126, 89, 92, 117, 115, 118, 94, 119, 121

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276343.
Cf. A004001, A005187, A055938, A079559, A087686, A088359, A213714, A234017.
Similar or related permutations: A233275, A233277, A267112, A276346, A276442.

(definec (A276344 n) (cond ((< n 2) n) ((not (zero? (A079559 n))) (A087686 (+ 1 (A276344 (- (A213714 n) 1))))) (else (A088359 (A276344 (A234017 n))))))

a(1)=1; for n > 1, if A079559(n)=1 [when n is in A005187], a(n) = A087686(1+a(A213714(n)-1)), otherwise a(n) = A088359(a(A234017(n))).
As a composition of other permutations:
a(n) = A267112(A233275(n)).
a(n) = A276442(A233277(n)).

A276345 Permutation of natural numbers: a(1) = 1, a(A087686(1+n)) = A055938(a(n)), a(A088359(n)) = A005187(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 12, 10, 8, 15, 9, 11, 14, 13, 27, 23, 19, 16, 31, 21, 18, 22, 17, 26, 30, 20, 25, 24, 29, 28, 58, 53, 46, 38, 32, 63, 48, 41, 35, 42, 40, 34, 50, 33, 57, 62, 44, 39, 49, 37, 47, 45, 36, 56, 55, 61, 43, 54, 52, 51, 60, 59, 121, 113, 104, 89, 74, 64, 127, 108, 95, 81, 70, 82, 93, 79, 67, 98, 77, 66, 112, 65, 120

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276346.
Cf. A004001, A005187, A055938, A080677, A087686, A088359, A093879.
Similar or related permutations: A233276, A233278, A267111, A276343, A276441.

(definec (A276345 n) (cond ((< n 2) n) ((zero? (A093879 (- n 1))) (A055938 (A276345 (+ -1 (A080677 n))))) (else (A005187 (+ 1 (A276345 (+ -1 (A004001 n))))))))

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = A055938(a(A080677(n)-1)), otherwise [when n is in A088359], a(n) = A005187(1+a(A004001(n)-1)).
As a composition of other permutations:
a(n) = A233276(A276441(n)).
a(n) = A233278(A267111(n)).

A276346 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A088359(a(n)), a(A055938(n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 10, 12, 9, 13, 8, 15, 14, 11, 19, 24, 22, 18, 27, 21, 23, 17, 29, 28, 25, 16, 31, 30, 26, 20, 36, 45, 43, 40, 54, 51, 35, 49, 42, 39, 41, 58, 48, 53, 34, 52, 38, 50, 44, 61, 60, 33, 59, 56, 55, 46, 32, 63, 62, 57, 47, 37, 69, 83, 81, 78, 102, 99, 74, 97, 93, 91, 68, 116, 112, 80, 88, 77, 109, 73, 75, 96, 90

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276345.
Cf. A004001, A005187, A055938, A079559, A087686, A088359, A213714, A234017.
Similar or related permutations: A233275, A233277, A267112, A276344, A276442.

(definec (A276346 n) (cond ((< n 2) n) ((not (zero? (A079559 n))) (A088359 (A276346 (- (A213714 n) 1)))) (else (A087686 (+ 1 (A276346 (A234017 n)))))))

a(1)=1; for n > 1, if A079559(n)=1 [when n is in A005187], a(n) = A088359(a(A213714(n)-1)), otherwise a(n) = A087686(1+a(A234017(n))).
As a composition of other permutations:
a(n) = A276442(A233275(n)).
a(n) = A267112(A233277(n)).

cp's OEIS Frontend

A289618 a(n) = A289617(n) - A046645(n) = A005187(A001222(n)) - A046645(n).

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A324348 a(n) = A294898(A005940(1+n)), where A294898(k) = A005187(k) - A000203(k).

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A326131 Positive numbers n for which A000120(n) = k*A294898(n), with k < 0; numbers for which A326130(n) = sigma(n) - A005187(n).

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A122247 Partial sums of A005187.

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Maple

Mathematica

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A213715 a(n) = position of A179016(n) in A005187.

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A256994 a(n) = n + 1 when n <= 3, otherwise a(n) = 2^(n-2) + 3; also iterates of A005187 starting from a(1) = 2.

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A276343 Permutation of natural numbers: a(1) = 1, a(A087686(1+n)) = A005187(1+a(n)), a(A088359(n)) = A055938(a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

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Scheme

Formula

A276344 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A087686(1+a(n)), a(A055938(n)) = A088359(a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

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