A034792 -id:A034792 - cp's OEIS frontend

A076818 Lexicographically earliest sequence of pairwise coprime tetrahedral numbers.

1, 4, 35, 969, 302621, 437989, 657359, 939929, 3737581, 6435689, 9290431, 21084251, 26536591, 39338069, 44101441, 61690919, 112805879, 289442201, 439918931, 1008077071, 1103914379, 1220664491, 1369657969, 1504148881, 1779510701, 1868223839, 2252547431

Offset: 1

Shyam Sunder Gupta, Nov 19 2002

nonn

Previous name was: Tetrahedral numbers ((k^3-k)/6) which are coprime to each smaller number in this sequence.

Sierpinski proved that any finite set of pairwise coprime tetrahedral numbers can be extended by adding an additional tetrahedral number which is coprime with all the elements of the set. Therefore this sequence is infinite. - Amiram Eldar, Mar 01 2019

			35 is a term because it is the least tetrahedral number after 4 which is coprime to 1 and 4.

W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970, Problem 43.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000292, A034792.

t[n_] := n (n + 1) (n +2)/6; s = {1}; While[Length[s] < 50, k = s[[-1]] + 1; While[Max[GCD[t[k], t /@ s]] > 1, k++]; AppendTo[s, k]]; t /@ s (* Amiram Eldar, Mar 01 2019 *)

v=vector(1000); n=0; for(i=1, 540537, t=i*(i+1)*(i+2)/6; for(j=2, n, if(gcd(t,v[j])>1, next(2))); n++; v[n]=t); v \\ Donovan Johnson, Oct 10 2013

Edited by Don Reble, Nov 03 2005

New name from Amiram Eldar, Mar 02 2019

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

Original entry on oeis.org

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

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A076818 Lexicographically earliest sequence of pairwise coprime tetrahedral numbers.

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

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cp's OEIS Frontend

A076818 Lexicographically earliest sequence of pairwise coprime tetrahedral numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

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