A119733 -id:A119733 - cp's OEIS frontend

A213539 Variant of numbers for which there is at least one 3-smooth representation that is special of level k.

1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 19, 20, 22, 23, 28, 29, 31, 32, 35, 37, 38, 40, 44, 46, 47, 49, 53, 56, 58, 62, 64, 65, 67, 70, 73, 74, 76, 79, 80, 85, 88, 89, 92, 94, 97, 98, 101, 103, 106, 112, 116, 119, 121, 124, 125, 128, 130, 131, 133, 134, 140, 143, 146

Offset: 0

Kenneth Vollmar, Mar 03 2013

nonn

These numbers are of the form 3^k*2^{a_0} + 3^{k-1}*2^{a_1} + ... + 3^1*2^{a_{k-1}} + 3^0*2^{a_k} in which every power 3^i appears, 0 <= i <= k, and where a_i satisfies 0 <= a_0 < a_1 < ... < a_k.

These values are those of sequence A116640 in addition to any multiple of two of elements of this sequence. - Kenneth Vollmar, Jun 05 2013

			n=19 has two 3-smooth representations that are special of level k. At k=1, 19 = 3^1*2^0 + 3^0*2^4. At k=2, 19 = 3^2*2^0 + 3^1*2^1 + 3^0*2^2.

Kenneth Vollmar, Recursive calculation of 3-smooth representations special of level k, To be submitted mid-2013.

R. Blecksmith, M. McCallum and J. L. Selfridge, 3-smooth representations of integers, Amer. Math. Monthly, 105 (1998), 529-543.

Cf. A116623, A116640, A116641, A119733, A226383, A003586.

Corrected a reference to another sequence and added cross references - Joe Slater, Dec 19 2016

A348175 Irregular table T(n,k) read by rows: T(n,k) = T(n-1,k/2) when k is even and 3*T(n-1,(k-1)/2) + 2^(n-1) when k is odd. T(0,0) = 0 and 0 <= k <= 2^n-1.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 5, 0, 4, 2, 10, 1, 7, 5, 19, 0, 8, 4, 20, 2, 14, 10, 38, 1, 11, 7, 29, 5, 23, 19, 65, 0, 16, 8, 40, 4, 28, 20, 76, 2, 22, 14, 58, 10, 46, 38, 130, 1, 19, 11, 49, 7, 37, 29, 103, 5, 31, 23, 85, 19, 73, 65, 211

Offset: 0

Ryan Brooks, Oct 04 2021

			n\k 0  1  2  3  4  5  6  7
0   0
1   0  1
2   0  2  1  5
3   0  4  2 10  1  7  5 19

Cf. A001047 (right diagonal), A002697 (row sums), A119733.

Cf. A133457 (binary exponents).

T[0, 0] = 0; T[n_, k_] := T[n, k] = If[EvenQ[k], T[n - 1, k/2], 3*T[n - 1, (k - 1)/2] + 2^(n - 1)]; Table[T[n, k], {n, 0, 5}, {k, 0, 2^n - 1}] // Flatten (* Amiram Eldar, Oct 11 2021 *)

T(n, k) = if ((n==0) && (k==0), 0, if (k%2, 3*T(n-1,(k-1)/2) + 2^(n-1), T(n-1,k/2)));
tabf(nn) = for (n=0, nn, for (k=0, 2^n-1, print1(T(n,k), ", ")); print); \\ Michel Marcus, Oct 18 2021

T(n,k) = my(ret=0); for(i=0,n-1, if(bittest(k,n-1-i), ret=3*ret+1<Kevin Ryde, Oct 22 2021

T(n,k) = T(n-1,k/2) for k being even.
T(n,k) = 3*T(n-1,(k-1)/2) + 2^(n-1) for k being odd.
T(n,k) = 2*T(n-1,k) for 0 <= k <= 2^(n-1) - 1.
T(n,k) = Sum_{i=0..r} 2^(n-1-e[i]) * 3^i where binary expansion k = 2^e[0] + 2^e[1] + ... + 2^e[r] with ascending e[0] < e[1] < ... < e[r] (A133457). - Kevin Ryde, Oct 22 2021

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A213539 Variant of numbers for which there is at least one 3-smooth representation that is special of level k.

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A348175 Irregular table T(n,k) read by rows: T(n,k) = T(n-1,k/2) when k is even and 3*T(n-1,(k-1)/2) + 2^(n-1) when k is odd. T(0,0) = 0 and 0 <= k <= 2^n-1.

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