A008509 -id:A008509 - cp's OEIS frontend

A083571 Numbers k such that A008509(k) is prime.

2, 3, 5, 10, 12, 72

Offset: 1

N. J. A. Sloane, Jun 14 2003

Patrick De Geest, Subsets of palindromic triangulars

palQ[n_]:=Reverse[x=IntegerDigits[n]]==x; t={}; c=0; Do[If[palQ[n*(n+1)/2],c=c+1; If[PrimeQ[n],AppendTo[t,c]]],{n,10^3}]; t (* Jayanta Basu, May 14 2013 *)

ispal(n)=n=digits(n); for(i=1, #n\2, if(n[i]!=n[#n+1-i], return(0))); 1
for(k=1,1e9,if(ispal(k*(k+1)/2),n++;if(isprime(k),print1(n", ")))) \\ Charles R Greathouse IV, May 15 2013

A003098 Palindromic triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 55, 66, 171, 595, 666, 3003, 5995, 8778, 15051, 66066, 617716, 828828, 1269621, 1680861, 3544453, 5073705, 5676765, 6295926, 35133153, 61477416, 178727871, 1264114621, 1634004361, 5289009825, 6172882716, 13953435931

Offset: 1

N. J. A. Sloane

The only known terms with an even number 2*m of digits that are the concatenation of two palindromes with m digits are 55, 66 and 828828 (see David Wells entry 828828). - Bernard Schott, Apr 29 2022

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Charles W. Trigg, Palindromic Triangular Numbers, J. Rec. Math., 6 (1973), 146-147.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, p. 73 and p. 178, entry 828828 (Rev. ed. 1997)

T. D. Noe, Table of n, a(n) for n = 1..148 (from Patrick De Geest)
Patrick De Geest, Palindromic Triangulars
Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 83-125.

Cf. A008509.

Intersection of A000217 and A002113.

palQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn]]; Select[ Accumulate[ Range[200000]],palQ]  (* Harvey P. Dale, Mar 23 2011 *)
Select[Accumulate[Range[0,170000]],PalindromeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 15 2019 *)

list(lim)=my(v=List(),d); for(n=0,(sqrt(8*lim+1)-1)/2, d=digits(n*(n+1)/2); if(d==Vecrev(d), listput(v,n*(n+1)/2))); Vec(v) \\ Charles R Greathouse IV, Jun 23 2017

A003098_list = [m for m in (n*(n+1)//2 for n in range(10**5)) if str(m) == str(m)[::-1]] # Chai Wah Wu, Sep 03 2021

A054263 Number of palindromic triangular numbers with n digits.

Original entry on oeis.org

3, 2, 3, 3, 2, 2, 6, 2, 1, 4, 7, 0, 4, 4, 12, 5, 6, 2, 3, 2, 6, 3, 6, 2, 2, 4, 3, 2, 5, 0, 3, 2, 1, 4, 3, 1, 10, 1, 4, 0

Offset: 1

Patrick De Geest, Apr 15 2000

			a(10) = 4: 1264114621, 1634004361, 5289009825, 6172882716.

P. De Geest, Palindromic Triangulars

Cf. A000217, A003098, A008509, A008510.

A028336 Positive numbers k such that k*(k+1) is a palindrome.

Original entry on oeis.org

1, 2, 16, 77, 538, 1621, 2457, 5291, 5313, 52008, 142401, 143498, 1610151, 1713543, 4670028, 5218983, 15137566, 15282411, 15814148, 47370058, 50702751, 142594226, 166691108, 1694576061, 2554554552, 25541432472, 47878213558, 77714915542, 155482156418

Offset: 1

Patrick De Geest

For additional terms, see the De Geest link.

P. De Geest, Palindromic pronic numbers of the form n(n+1)
Eric Weisstein's World of Mathematics, Palindromic Number.
Eric Weisstein's World of Mathematics, Pronic Number.

Cf. A008509, A028337.

Select[Range[500000], PalindromeQ[#(#+1)] &] (* or *) Select[Range[50000], IntegerDigits[#(#+1)] == Reverse[ IntegerDigits[#(#+1)]] &] (* G. C. Greubel, Nov 24 2016 *)

isok(k) = my(d = digits(k*(k+1))); Vecrev(d) == d; \\ Michel Marcus, Nov 09 2017

More terms from Jon E. Schoenfield, Nov 09 2017

A050722 Palindromic triangular numbers arising from A050721 and A083571.

Original entry on oeis.org

3, 6, 66, 5995, 15051, 15199896744769899151

Offset: 1

Patrick De Geest, Sep 15 1999

P. De Geest, Subsets of palindromic triangulars

Cf. A008509, A008510, A050721.

A068641 Smallest n-digit palindromic triangular number, or 0 if no such number exists.

Original entry on oeis.org

1, 55, 171, 3003, 15051, 617716, 1269621, 35133153, 178727871, 1264114621, 13953435931, 0, 1313207023131, 19895044059891, 114401848104411, 1250444114440521, 11121736463712111, 357961407704169753

Offset: 1

Amarnath Murthy, Feb 28 2002

			a(4) = 3003 = 77*78/2 is the smallest 4-digit palindromic triangular number.
a(12) = 0 as no 12-digit palindromic triangular number exists.

P. De Geest, Exhaustive list of Palindromic Triangulars

Cf. A068642, A068643, A068644, A000217, A003098, A008509, A008510, A054263.

Edited by Patrick De Geest, Mar 24 2002

A068642 Index of the smallest n-digit palindromic triangular number, or 0 if no such number exists.

Original entry on oeis.org

1, 10, 18, 77, 173, 1111, 1593, 8382, 18906, 50281, 167053, 0, 1620621, 6307938, 15126258, 50008881, 149142458, 846122222, 2480116437, 5513600773, 14667896198, 49786655918, 246644446642, 529670494286, 2466444446642

Offset: 1

Amarnath Murthy, Feb 28 2002

			a(4) = 77 as 77*78/2 = 3003 is the smallest 4-digit palindromic triangular number.
a(12) = 0 as no 12-digit palindromic triangular number exists.

P. De Geest, Exhaustive list of Palindromic Triangulars

Cf. A068641, A068643, A068644, A000217, A003098, A008509, A008510, A054263.

Edited by Patrick De Geest, Mar 24 2002

A068643 Largest n-digit palindromic triangular number, or 0 if no such number exists.

Original entry on oeis.org

6, 66, 666, 8778, 66066, 828828, 6295926, 61477416, 178727871, 6172882716, 87350505378, 0, 68742000024786, 82078700787028, 684866959668486, 8208268228628028, 67898244444289876, 514816979979618415

Offset: 1

Amarnath Murthy, Feb 28 2002

			a(4) = 8778 = 132*133/2 is the largest 4-digit palindromic triangular number.
a(12) = 0 as no 12-digit palindromic triangular number exists.

P. De Geest, Exhaustive list of Palindromic Triangulars

Cf. A068641, A068642, A068644, A000217, A003098, A008509, A008510, A054263.

Edited by Patrick De Geest, Mar 24 2002

A068644 Index of the largest n-digit palindromic triangular number, or 0 if no such number exists.

Original entry on oeis.org

3, 11, 36, 132, 363, 1287, 3548, 11088, 18906, 111111, 417972, 0, 3707883, 12812392, 37009916, 128127032, 368505751, 1014708805, 3567632391, 11151642876, 36657342048, 104561417190, 417898160427, 1325269593372

Offset: 1

Amarnath Murthy, Feb 28 2002

			a(4) = 132 as 132*133/2 = 8778 is the largest 4-digit palindromic triangular number.
a(12) = 0 as no 12-digit palindromic triangular number exists.

P. De Geest, Exhaustive list of Palindromic Triangulars

Cf. A068641, A068642, A068643, A000217, A003098, A008509, A008510, A054263.

Edited by Patrick De Geest, Mar 24 2002

A350988 Numbers k such that the k-th triangular number is a binary palindrome.

Original entry on oeis.org

0, 1, 2, 5, 6, 9, 17, 21, 25, 33, 42, 65, 90, 129, 170, 257, 341, 357, 450, 513, 693, 893, 1025, 1365, 1397, 1445, 1617, 1670, 1750, 2049, 2730, 4097, 5418, 5985, 8193, 10397, 10922, 16385, 17313, 21717, 21845, 31749, 32769, 40637, 43605, 51537, 63482, 65537, 76217

Offset: 1

Amiram Eldar, Jan 28 2022

This sequence is infinite since 2^k+1 is a term for all k>1 (Trigg, 1974).

			2 is a term since A000217(2) = 2*(2+1)/2 = 3 = 11_2 is a triangular number and also a binary palindromic number.
5 is a term since A000217(5) = 5*(5+1)/2 = 15 = 1111_2 is a triangular number and also a binary palindromic number.

Amiram Eldar, Table of n, a(n) for n = 1..128
Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.
Maciej Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT], 2008.

The binary version of A008509.
A000051 \ {3} is a subsequence.
Cf. A000217, A003098, A006995, A350987.

Select[Range[0, 10^5], PalindromeQ[IntegerDigits[#*(# + 1)/2, 2]] &]

isok(k) = my(b=binary(k*(k+1)/2)); b == Vecrev(b); \\ Michel Marcus, Jan 28 2022

def ok(n): b = bin(n*(n+1)//2)[2:]; return b == b[::-1]
print([k for k in range(80000) if ok(k)]) # Michael S. Branicky, Jan 28 2022

A000217(a(n)) = A350987(n).

cp's OEIS Frontend

A083571 Numbers k such that A008509(k) is prime.

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A003098 Palindromic triangular numbers.

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A054263 Number of palindromic triangular numbers with n digits.

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A028336 Positive numbers k such that k*(k+1) is a palindrome.

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A050722 Palindromic triangular numbers arising from A050721 and A083571.

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A068641 Smallest n-digit palindromic triangular number, or 0 if no such number exists.

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A068642 Index of the smallest n-digit palindromic triangular number, or 0 if no such number exists.

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A068643 Largest n-digit palindromic triangular number, or 0 if no such number exists.

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A068644 Index of the largest n-digit palindromic triangular number, or 0 if no such number exists.

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A350988 Numbers k such that the k-th triangular number is a binary palindrome.

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