A115590 -id:A115590 - cp's OEIS frontend

A360826 a(1) = 1, a(n) = (k+1)(2k+1), where k = Product_{i=1..n-1} a(i).

1, 6, 91, 597871, 213122969971321411, 9680343693975641657052402556458789711774336036960631

Offset: 1

Ivan N. Ianakiev, Feb 22 2023

nonn

A sequence of pairwise relatively prime triangular (and also hexagonal) numbers.
As a clarification to the problem definition by Sierpinski, here we show that only one triangular (hexagonal) seed is needed to produce such a sequence.
This sequence can be used for proving the infinitude of primes.
In general: Let m = 2*q, for any q > 0. There are infinitely many sequences of pairwise coprime m-gonal numbers, whose first term is any positive m-gonal number and whose general term is of the form a(n) = (k + 1)*((q - 1)*k + 1), where k = Product_{i=1..n-1} a(i).

W. Sierpinski, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #42.

Michel Marcus, Table of n, a(n) for n = 1..8

Cf. A000217, A000384, A004019, A034792, A115590.

a[1]=1; a[n_]:=Module[{k=Product[a[i],{i,1,n-1}]},(k+1)*(2*k+1)];
a/@Range[6]
Join[{1}, RecurrenceTable[{a[2] == 6, a[n+1] == (1 + a[n]*(Sqrt[1 + 8*a[n]] - 3)/4) * (1 + 2*a[n]*(Sqrt[1 + 8*a[n]] - 3)/4)}, a, {n, 2, 8}]] (* Vaclav Kotesovec, May 05 2023 *)

a(n) = if (n==1, 1, my(k = prod(i=1,n-1, a(i))); (k+1)*(2*k+1)); \\ Michel Marcus, Mar 25 2025

a(1) = 1, a(n) = (k+1)*(2*k+1), where k = Product_{i=1..n-1} a(i).

a(n) ~ c^(3^n), where c = 1.1784502032269064445225839284451956694752084180050932315805089054871825498... - Vaclav Kotesovec, May 05 2023

A368326 a(0) = 0; a(n) = (1+a(n-1))^4 for n > 0.

Original entry on oeis.org

0, 1, 16, 83521, 48663522406470666256, 5608079543150183734470340565498778265577622654664540468785094021747120982222401

Offset: 0

Michael De Vlieger, Dec 21 2023

Geir Agnarsson, Elie Alhajjar, and Aleyah Dawkins, On locally finite ordered rooted trees and their rooted subtrees, arXiv:2312.11379 [math.CO], 2023. See p. 6.

Cf. A004019, A115590, A368327.

Mathematica
```
{0}~Join~NestList[(# + 1)^4 &, 1, 5]
```

A368327 a(0) = 0; a(n) = (1+a(n-1))^5 for n > 0.

Original entry on oeis.org

0, 1, 32, 39135393, 91801241053644953553642221885011784224

Offset: 0

Michael De Vlieger, Dec 21 2023

Geir Agnarsson, Elie Alhajjar, and Aleyah Dawkins, On locally finite ordered rooted trees and their rooted subtrees, arXiv:2312.11379 [math.CO], 2023. See p. 6 with error at a(4).

Cf. A004019, A115590, A368326.

Mathematica
```
{0}~Join~NestList[(# + 1)^5 &, 1, 5]
```

cp's OEIS Frontend

A360826 a(1) = 1, a(n) = (k+1)(2k+1), where k = Product_{i=1..n-1} a(i).

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A368326 a(0) = 0; a(n) = (1+a(n-1))^4 for n > 0.

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Author

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Mathematica

A368327 a(0) = 0; a(n) = (1+a(n-1))^5 for n > 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

cp's OEIS Frontend

A360826 a(1) = 1, a(n) = (k+1)*(2*k+1), where k = Product_{i=1..n-1} a(i).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A368326 a(0) = 0; a(n) = (1+a(n-1))^4 for n > 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A368327 a(0) = 0; a(n) = (1+a(n-1))^5 for n > 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A360826 a(1) = 1, a(n) = (k+1)(2k+1), where k = Product_{i=1..n-1} a(i).