A099874 -id:A099874 - cp's OEIS frontend

A365747 Decimal expansion of Trinv(1 + Trinv(2 + Trinv(3 + Trinv(4 + ... )))) where Trinv(n) = (sqrt(8*n+1)-1)/2.

2, 2, 8, 1, 4, 3, 7, 4, 1, 4, 0, 5, 3, 4, 2, 4, 4, 1, 5, 2, 7, 1, 7, 7, 2, 7, 2, 8, 5, 9, 1, 6, 5, 0, 7, 6, 0, 7, 3, 3, 3, 8, 8, 4, 5, 2, 6, 6, 1, 0, 1, 1, 8, 2, 7, 9, 7, 2, 9, 3, 1, 7, 6, 7, 1, 7, 2, 4, 0, 7, 2, 3, 2, 2, 1, 9, 0, 5, 9, 5, 0, 8, 9, 7, 1, 0, 3, 6, 8, 1, 7, 8, 9, 2, 2, 8, 7, 9, 3, 3, 4, 8, 2, 7, 7, 7, 3, 0, 1, 7, 7

Offset: 1

Kelvin Voskuijl, Sep 17 2023

Trinv(n) = (sqrt(8*n+1)-1)/2 is the inverse of A000217.

			2.2814374140534244152717727285916507607333884526610...

Cf. A072449 (analog for square root), A099874 (analog for cube root).

Cf. A000217 (triangular numbers), A003056.

TriangleRoot[n_] =(-1 + Sqrt[1 + 8 n])/2; RealDigits[ Fold[ TriangleRoot[ #1 + #2] &, 0, Reverse[ Range[200]]], 10,111][[1]]

A347185 Decimal expansion of (2 + (3 + (5 + (7 + ...)^(1/3))^(1/3))^(1/3))^(1/3), continued cube root map applied to primes.

Original entry on oeis.org

1, 5, 4, 6, 8, 3, 7, 9, 8, 4, 5, 2, 9, 7, 3, 1, 0, 8, 1, 7, 6, 3, 5, 1, 8, 4, 3, 7, 9, 0, 6, 9, 7, 5, 2, 2, 2, 1, 6, 5, 4, 8, 9, 4, 5, 1, 2, 6, 2, 9, 3, 2, 0, 9, 8, 9, 1, 3, 1, 5, 1, 7, 1, 5, 3, 5, 0, 7, 8, 1, 0, 4, 5, 6, 0, 1, 1, 7, 7, 6, 5, 4, 3, 9, 6, 4, 5, 9, 1, 0, 8, 2, 7, 5, 9, 3, 4, 9, 1, 0, 9, 0, 6, 1, 8, 7, 5, 0, 8, 8

Offset: 1

Ilya Gutkovskiy, Aug 21 2021

			1.546837984529731081763518437906975222165489451262932...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A000040, A099874, A105546, A239349, A347074.

A373009 Decimal expansion of PentInv(1 + PentInv(2 + PentInv(3 + PentInv(4 + ... )))) where PentInv(n) = (1 + sqrt(1 + 24*n))/6.

Original entry on oeis.org

1, 5, 4, 7, 1, 5, 7, 5, 8, 7, 2, 9, 7, 8, 7, 6, 6, 6, 1, 2, 5, 8, 8, 0, 7, 9, 4, 0, 0, 8, 6, 5, 1, 8, 4, 8, 3, 6, 2, 0, 6, 4, 1, 3, 1, 3, 4, 6, 8, 5, 0, 5, 1, 3, 4, 9, 8, 1, 6, 5, 8, 8, 4, 7, 3, 3, 8, 7, 5, 9, 7, 9, 8, 0, 1, 1, 1, 2, 3, 0, 3, 5, 2, 6, 4, 5, 7, 3, 6, 5, 5, 3, 1, 5, 8, 0, 7, 3, 4, 9, 4, 2, 9, 6, 0, 1, 6, 4, 7, 7, 7

Offset: 1

Kelvin Voskuijl, May 22 2024

The inverse of the pentagonal numbers is defined here as (1 + sqrt(1 + 24*n))/6.

			1.54715758729787666125880794008651848362064131346850513...

Cf. A365747 (triangular inverse), A072449 (square root), A099874 (cubes).

Cf. A000326 (pentagonal numbers).

cp's OEIS Frontend

A365747 Decimal expansion of Trinv(1 + Trinv(2 + Trinv(3 + Trinv(4 + ... )))) where Trinv(n) = (sqrt(8*n+1)-1)/2.

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A347185 Decimal expansion of (2 + (3 + (5 + (7 + ...)^(1/3))^(1/3))^(1/3))^(1/3), continued cube root map applied to primes.

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A373009 Decimal expansion of PentInv(1 + PentInv(2 + PentInv(3 + PentInv(4 + ... )))) where PentInv(n) = (1 + sqrt(1 + 24*n))/6.

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