cp's OEIS Frontend

Conjectured formulas: (Start)

R(2n, x) = R(n, x) + R(n - 2^f(n), x) + R(2n - 2^f(n), x) where f(n) = A007814(n) (see A329369).

b(2^m*n + q) = Sum_{i=A001511(n+1)..A000120(n)+1} T(n, i)*b(2^m*(2^(i-1)-1) + q) for n >= 0, m >= 0, q >= 0 where b(n) = A329369(n). Note that this formula is recursive for n != 2^k - 1.

R(n, x) = c(n, x)

where c(2^k - 1, x) = x^(k+1) for k >= 0,

c(n, x) = Sum_{i=0..s(n)} p(n, s(n)-i)*Sum_{j=0..i} (s(n)-j+1)^A279209(n)*binomial(i, j)*(-1)^j,

p(n, k) = Sum_{i=0..k} c(t(n) + (2^i - 1)*A062383(t(n)), x)*L(s(n), k, i) for 0 <= k < s(n) with p(n, s(n)) = c(t(n) + (2^s(n) - 1)*A062383(t(n)), x),

s(n) = A090996(n), t(n) = A087734(n),

L(n, k, m) are some integer coefficients defined for n > 0, 0 <= k < n, 0 <= m <= k that can be represented as W(n-m, k-m, m+1)

and where W(n, k, m) = (k+m)*W(n-1, k, m) + (n-k)*W(n-1, k-1, m) + [m > 1]*W(n, k, m-1) for 0 <= k < n, m > 0 with W(0, 0, m) = 1, W(n, k, m) = 0 for n < 0 or k < 0.

In particular, W(n, k, 1) = A173018(n, k), W(n, k, 2) = A062253(n, k), W(n, k, 3) = A062254(n, k) and W(n, k, 4) = A062255(n, k).

Here s(n), t(n) and A279209(n) are unique integer sequences such that n can be represented as t(n) + (2^s(n) - 1)*A062383(t(n))*2^A279209(n) where t(n) is minimal. (End)

Conjectures from Mikhail Kurkov, Jun 19 2024: (Start)

T(n, k) = d(n, 1, A000120(n) - k + 2) where d(n, m, k) = (m+1)^g(n)*d(h(n), m+1, k) - m^(g(n)+1)*d(h(n), m, k-1) for n > 0, m > 0, k > 0 with d(n, m, 0) = 0 for n >= 0, m > 0, d(0, m, k) = [k <= m]*abs(Stirling1(m, m-k+1)) for m > 0, k > 0, g(n) = A290255(n) and where h(n) = A053645(n). In particular, d(n, 1, 1) = A341392(n).

Sum_{i=A001511(n+1)..wt(n)+k} d(n, k, wt(n)-i+k+1)*A329369(2^m*(2^(i-1)-1) + q) = k!*A357990(2^m*n + q, k) for n >= 0, k > 0, m >= 0, q >= 0 where wt(n) = A000120(n).

If we change R(0, x) to Product_{i=0..m-1} (x+i), then for resulting irregular table U(n, k, m) we have U(n, k, m) = d(n, m, A000120(n) - k + m + 1).

T(n, k) = (-1)^(wt(n)-k+1)*Sum_{i=1..wt(n)-k+3} Stirling1(wt(n)-i+3, k+1)*A358612(n, wt(n)-i+3) for n >= 0, k > 0 where wt(n) = A000120(n). (End)

Conjecture: T(2^m*(2k+1), q) = (-1)^(wt(k)-q)*Sum_{i=q..wt(k)+2} Stirling1(i,q)*A358612(k,i)*i^m for m >= 0, k >= 0, q > 0 where wt(n) = A000120(n). - Mikhail Kurkov, Jan 17 2025

a(2^n-1) = A376697(n).

a(n) = integer part of 1/(2^b(n)-1) where b(n) = digit-reversal of binary of (positive integer) n, preceded by a decimal point and converted (from base 2) to base 10; initial term, a(0), is defined as 0.

a(n) = floor(1/(2^(A030101(n)/A062383(n))-1)) for n>0. - Andrey Zabolotskiy, Feb 20 2017

a(n) = n * (2^(A070939(n)*n) - 1) / (2^(A070939(n)) - 1). - Maxim Skorohodov, Oct 26 2020

a(2*n) = a(n) for any n >= 0.

a(2^k-1) = k for any k >= 0.

a(n) = 1 iff n = 2^k for some k >= 0.

a(n) = 2 iff n belongs to A173195.

a(Sum_{k=1..n} 2^(k^2)) = A016726(n) for any n >= 1.

a(Sum_{k=1..n} 2^A000069(k)) = A062383(n) for any n >= 1.

a(Sum_{k=0..n} 2^(2^k)) = A270097(n) for any n >= 0.

a(Sum_{k=1..n} 2^A000045(k+1)) = A270151(n) for any n >= 1.

a(Sum_{k=1..n} 2^A000041(k)) = A270176(n) for any n >= 1.

a(A076793(n)) = A272633(n) for any n >= 0.

a(Sum_{k=1..n} 2^A001969(k)) = A272881(n) for any n >= 1.

a(Sum_{k=1..n} 2^A005823(k)) = A272882(n) for any n >= 1.

a(Sum_{k=1..n} 2^A000215(k-1)) = A273037(n) for any n >= 1.

a(Sum_{k=1..n} 2^A000108(k)) = A273041(n) for any n >= 1.

a(Sum_{k=1..n} 2^A001566(k)) = A273043(n) for any n >= 1.

a(Sum_{k=1..n} 2^A003095(k)) = A273044(n) for any n >= 1.

a(Sum_{k=1..n} 2^A000058(k-1)) = A273056(n) for any n >= 1.

a(Sum_{k=1..n} 2^A002808(k)) = A273062(n) for any n >= 1.

a(Sum_{k=1..n} 2^(k!)) = A273064(n) for any n >= 1.

a(Sum_{k=1..n} 2^(k^k)) = A273068(n) for any n >= 1.

a(Sum_{k=1..n} 2^A000110(k)) = A273237(n) for any n >= 1.

a(Sum_{k=1..n} 2^A001147(k)) = A273377(n) for any n >= 1.

a(n) = 0 iff n = 0 or n > 1 and n belongs to A116882.

a(n) = 1 iff n belongs to A086341.

a(2^k + m) = a(2^(k+1)-m) for any k > 0 and m = 0..2^k.

a(n) = 2^ceiling(log_2(n)) * Product_{k=1..n} (k-th cyclotomic polynomial evaluated at 2).

a(n) = A034268(n)*A062383(n+1). - Michel Marcus, Jul 29 2022