A003261 -id:A003261 - cp's OEIS frontend

A131812 Sum of all n-digit Woodall numbers.

8, 86, 1437, 6654, 81917, 827389, 17956860, 157286397, 1434451965, 12884901885, 114353504253, 1005022347261, 8761733283837, 166026255794172, 1337006139375613, 11434920928870397, 97390341941886973, 1799188051134513148, 14231374822490767357, 119903836479112085501

Offset: 1

Parthasarathy Nambi, Oct 23 2007

			Sum of all 1-digit Woodall numbers is 1 + 7 = 8.
Sum of all 2-digit Woodall numbers is 23 + 63 = 86.
Sum of all 3-digit Woodall numbers is 159 + 383 + 895 = 1437.

Eric Weisstein's World of Mathematics, Woodall Number.

Cf. A003261.

digNum[n_] := Length @ IntegerDigits[n]; woodall[n_] := n * 2^n - 1; digCount = 0; sum = 0; cumsum = {}; Do[w = woodall[n]; If[digNum[w] > digCount, digCount++; AppendTo[cumsum, sum]]; sum += w, {n, 1, 65}]; Differences[cumsum] (* Amiram Eldar, Nov 30 2019 *)

More terms from Amiram Eldar, Nov 30 2019

A137810 a(n) = 2^(2^n+n) - 1.

Original entry on oeis.org

1, 7, 63, 2047, 1048575, 137438953471, 1180591620717411303423, 43556142965880123323311949751266331066367, 29642774844752946028434172162224104410437116074403984394101141506025761187823615

Offset: 0

Ant King, Feb 12 2008

An integer is simultaneously a Mersenne number and a Woodall number if and only if it is a member of this sequence. Hence this sequence is the intersection of A000225 and A003261.

			The fourth integer which is both a Mersenne number and a Woodall number is 2047. Hence a(3)=2047 (as the offset is zero).

Wilfrid Keller, New Cullen Primes, Mathematics of Computation, Vol. 64, No. 212 (Ocober 1995), pp. 1733-1741.

Cf. A000225, A003261, A006127.

Mathematica
```
2^(2^#+#)-1 &/@Range[0,8]
```

a(n) = 2^(2^n+n)-1 = A000225(2^n+n) = A003261(2^n).

A382447 Number of positive k <= n such that k*2^n - 1 is prime.

Original entry on oeis.org

0, 2, 2, 2, 2, 3, 2, 1, 1, 3, 3, 2, 3, 2, 2, 4, 6, 3, 1, 3, 3, 0, 1, 0, 1, 1, 2, 3, 2, 3, 4, 2, 2, 1, 5, 2, 4, 2, 1, 3, 4, 3, 4, 2, 2, 3, 2, 3, 2, 3, 3, 3, 4, 5, 2, 2, 3, 1, 3, 3, 3, 4, 3, 1, 0, 1, 2, 1, 4, 3, 3, 5, 3, 3, 6, 2, 3, 3, 3, 2, 3, 1, 1, 1, 3, 1, 2, 2, 2, 2, 3, 3, 2, 3, 2, 3, 2, 3, 3, 2

Offset: 1

Juri-Stepan Gerasimov, Mar 26 2025

nonn

Cf. A002234, A003261, A382119.

Cf. A061411 (indices of 0s), A061414 (indices of 1s).

[#[k: k in [1..n] | IsPrime(k*2^n-1)]: n in [1..100]];

a[n_]:=Length[Select[Range[n],PrimeQ[#*2^n-1] &]]; Array[a,100] (* Stefano Spezia, Mar 26 2025 *)

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A131812 Sum of all n-digit Woodall numbers.

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A137810 a(n) = 2^(2^n+n) - 1.

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A382447 Number of positive k <= n such that k*2^n - 1 is prime.

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