A006513 -id:A006513 - cp's OEIS frontend

A076057 Numbers k such that Sum_{j=1..k} A006513(j) = 3*k/2.

2, 4, 8, 16, 18, 32, 38, 46, 48, 62, 66, 80, 90, 102, 120, 138, 158, 160, 180, 212, 242, 278, 324, 364, 436, 482, 574, 576, 658, 736, 738, 882, 892, 900, 972, 974, 976, 1162, 1164, 1318, 1320, 1524, 1526, 1528, 1762, 1788, 1796, 1982, 2342, 2344, 2346, 2386, 2392, 2634, 3110

Offset: 1

Benoit Cloitre, Oct 30 2002

nonn

Michel Marcus, Table of n, a(n) for n = 1..858

Cf. A006513, A014682.

f(n) = if (n%2, (3*n+1)/2, n/2); \\ A014682
g(n) = my(last = n); while (1, my(new = f(f(last))); if (new == last, return(new)); last = new;); \\ A006513
isok(m) = !(m%2) && (sum(k=1, m, g(k)) == 3*m/2); \\ Michel Marcus, Feb 03 2022

f(n) = if (n%2, (3*n+1)/2, n/2); \\ A014682
g(n) = my(last = n); while (1, my(new = f(f(last))); if (new == last, return(new)); last = new;); \\ A006513
lista(nn) = {my(v = vector(nn, k, g(k)), w = vector(nn)); w[1] = v[1]; for (i=2, nn, w[i] = w[i-1] + v[i];); forstep (i=2, nn, 2, if (w[i] == 3*i/2, print1(i, ", ")););} \\ Michel Marcus, Feb 03 2022

a(n) seems to be asymptotic to c*n^2 with 1/2 < c < 1.

More terms from Michel Marcus, Feb 03 2022

A076054 a(n) = Sum_{k=1..n} A006513(k).

Original entry on oeis.org

1, 3, 5, 6, 7, 8, 10, 12, 14, 16, 17, 19, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 34, 35, 36, 37, 38, 40, 42, 44, 46, 48, 49, 50, 51, 53, 55, 57, 59, 61, 63, 65, 66, 67, 68, 69, 70, 72, 74, 76, 78, 80, 82, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 99, 101, 103, 105, 107

Offset: 1

Benoit Cloitre, Oct 30 2002

nonn

Cf. A006513, A014682.

f(n) = if (n%2, (3*n+1)/2, n/2); \\ A014682
g(n) = my(last = n); while (1, my(new = f(f(last))); if (new == last, return(new)); last = new;); \\ A006513
a(n) = sum(k=1, n, g(k)); \\ Michel Marcus, Feb 03 2022

a(n) seems to be asymptotic to (3/2)*n.

A076182 a(n) = A006666(n) mod 2.

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Benoit Cloitre, Nov 01 2002

nonn

Cf. A014682, A006513, A006666, A076092.

a(n)=if(n<0,0,s=n; c=0; while(s>1,s=(s%2)*(3*s+1)/2+(1-s%2)*s/2; c++); c)%2

(define (A076182 n) (modulo (A006666 n) 2))
(definec (A006666 n) (if (= 1 n) 0 (+ 1 (A006666 (A014682 n))))) ;; With memoization-macro definec
(define (A014682 n) (if (even? n) (/ n 2) (/ (+ n n n 1) 2)))
;; Antti Karttunen, Aug 13 2017

a(n) = A006513(n) - 1.

Apparently also the antiparity of A064433. - Ralf Stephan, Nov 16 2004

A076138 a(1)=1; a(n+1) is the smallest integer > a(n) such that C_2k(a(n+1))=C_2k(a(n)) for k large enough, where C_m(n) is the modified Collatz map iterated m times on n ( x->x/2 if x is even x->(3x+1)/2 if x is odd).

Original entry on oeis.org

1, 4, 5, 6, 11, 14, 15, 16, 18, 19, 20, 21, 24, 25, 26, 27, 33, 34, 35, 43, 44, 45, 46, 47, 56, 57, 58, 59, 60, 61, 62, 63, 64, 72, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 96, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 113, 130, 131, 132, 133, 134, 135

Offset: 1

Benoit Cloitre, Oct 31 2002

nonn

a(n) seems to be asymptotic to 2*n and a(n)=2*n for some n (5, 14, 43, 54, 72, 93, ...)

Positions of 1's in A006513. - Sean A. Irvine, Mar 21 2025

Cf. A006513, A076057.

a(n+1) = min (k>a(n) : A076057(k)=A076057(a(n))).

Missing 16 and 18 inserted by Sean A. Irvine, Mar 21 2025

cp's OEIS Frontend

A076057 Numbers k such that Sum_{j=1..k} A006513(j) = 3*k/2.

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A076054 a(n) = Sum_{k=1..n} A006513(k).

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A076182 a(n) = A006666(n) mod 2.

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A076138 a(1)=1; a(n+1) is the smallest integer > a(n) such that C_2k(a(n+1))=C_2k(a(n)) for k large enough, where C_m(n) is the modified Collatz map iterated m times on n ( x->x/2 if x is even x->(3x+1)/2 if x is odd).

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