cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A091581 Number of partitions of n into distinct decimal palindromes.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 17, 19, 21, 23, 26, 27, 30, 32, 34, 36, 37, 39, 40, 42, 42, 44, 44, 45, 45, 47, 47, 47, 49, 48, 50, 50, 52, 52, 55, 55, 58, 60, 60, 64, 65, 68, 69, 73, 73, 77, 78, 82, 84, 84, 88, 88, 92, 92, 96, 96, 100, 100, 105, 107, 107, 113
Offset: 0

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Author

Reinhard Zumkeller, Jan 22 2004

Keywords

Comments

Not the same as A088670: a(n) > A088670(n) for n > 101.

Examples

			n=13: there are A000009(13)=18 partitions of 13 into distinct integers, 4 of them contain non-palindromes: 13=12+1, 13=10+3, 13=10+2+1 and 13 itself, therefore a(13)=18-4=14;
for n=14 there are a(14)=17 partitions into palindromes: 11+3 = 11+2+1 = 9+5 = 9+4+1 = 9+3+2 = 8+6 = 8+5+1 = 8+4+2 = 8+3+2+1 = 7+6+1 = 7+5+2 = 7+4+3 = 7+4+2+1 = 6+5+3 = 6+5+2+1 = 6+4+3+1 = 5+4+3+2.

Links

Crossrefs

Cf. A091580, A046489.

Extensions

a(0)=1 prepended by Alois P. Heinz, Sep 17 2018

A046485 Sum of first n palindromic primes A002385.

Original entry on oeis.org

2, 5, 10, 17, 28, 129, 260, 411, 592, 783, 1096, 1449, 1822, 2205, 2932, 3689, 4476, 5273, 6192, 7121, 17422, 27923, 38524, 49835, 61246, 73667, 86388, 99209, 112540, 126371, 140302, 154643, 169384, 184835, 200386, 216447, 232808, 249369, 266030, 283501
Offset: 1

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Author

Patrick De Geest, Sep 15 1998

Keywords

Comments

The subsequence of prime partial sum of palindromic primes begins: 2, 5, 17, 5273, 7121, 154643, 283501. What is the smallest nontrivial (i.e., multidigit) palindromic prime partial sum of palindromic primes? [Jonathan Vos Post, Feb 07 2010]

Links

Crossrefs

Cf. A002385, A007504, A046489.
Cf. A000040, A007500, A006567, A016041, A029732, A117697. [Jonathan Vos Post, Feb 07 2010]

Programs

  • Mathematica
    t = {}; b = 10; Do[p = Prime[n]; i = IntegerDigits[p, b]; If[i == Reverse[i], AppendTo[t, p];(*Print[p.FromDigits[i]]*)], {n, 4000}]; Accumulate[t] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
Accumulate[Select[Prime[Range[10000]],IntegerDigits[#]==Reverse[ IntegerDigits[#]]&]] (* Harvey P. Dale, Aug 10 2013 *)

Formula

a(n) = Sum_{i=1..n} A002385(i) = Sum_{i=1..n} {p prime and R(p) = p, i.e., primes whose decimal expansion is a palindrome}. [Jonathan Vos Post, Feb 07 2010]

Extensions

Offset set to 1 by R. J. Mathar, Feb 21 2010

A046488 Palindromes that are the sum of the first n palindromes for some n.

Original entry on oeis.org

1, 3, 6, 111, 353, 7557, 2376732
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1998

Keywords

Comments

No additional terms up through the partial sums of all palindromes less than or equal to 10^7-1. - Harvey P. Dale, Jun 23 2018

Examples

			a(4) = 111 as 111 = 1+2+3+4+5+6+7+8+9+11+22+33.

Links

Crossrefs

Cf. A002113, A046486, A046487.

Programs

  • Mathematica
    palQ[n_] := Reverse[x = IntegerDigits[n]] == x; Select[Accumulate[Select[Range[25000], palQ]], palQ] (* Jayanta Basu, Jun 26 2013 *)
Select[Accumulate[Select[Range[10^6],PalindromeQ]],PalindromeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 23 2018 *)

Formula

a(n) = A046489(A046486(n)-1). - R. J. Mathar, Sep 09 2015

Extensions

Edited by Jonathan Sondow, Nov 28 2017

A277209 Partial sums of repdigit numbers (A010785).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 56, 78, 111, 155, 210, 276, 353, 441, 540, 651, 873, 1206, 1650, 2205, 2871, 3648, 4536, 5535, 6646, 8868, 12201, 16645, 22200, 28866, 36643, 45531, 55530, 66641, 88863, 122196, 166640, 222195, 288861, 366638, 455526, 555525, 666636, 888858, 1222191, 1666635, 2222190
Offset: 0

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Author

Ilya Gutkovskiy, Oct 05 2016

Keywords

Comments

More generally, the ordinary generating function for the partial sums of numbers that are repdigits in base k (for k > 1) is (Sum_{m = 1..(k-1)} m*x^m)/((1 - x)*(1 - x^(k-1))*(1 - k*x^(k-1))).

Examples

			a(0)=0;
a(1)=0+1=1;
a(2)=0+1+2=3;
a(3)=0+1+2+3=6;
...
a(10)=0+1+2+3+4+5+6+7+8+9+11=56;
a(11)=0+1+2+3+4+5+6+7+8+9+11+22=78;
a(12)=0+1+2+3+4+5+6+7+8+9+11+22+33=111, etc.

Links

Crossrefs

Cf. A000217, A010785, A027828, A046489.

Programs

  • Mathematica
    CoefficientList[Series[x (1 + 2 x + 3 x^2 + 4 x^3 + 5 x^4 + 6 x^5 + 7 x^6 + 8 x^7 + 9 x^8)/((1 - x) (1 - 10 x^9) (1 - x^9)), {x, 0, 50}], x]

Formula

G.f.: x*(1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 8*x^7 + 9*x^8)/((1 - x)*(1 - x^9)*(1 - 10*x^9)).
a(n) = A000217(n) for n < 10.
a(n) = A046489(n-1) for n < 19.

A309325 Numbers that are the sum of two successive palindromes.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 20, 33, 55, 77, 99, 121, 143, 165, 187, 200, 212, 232, 252, 272, 292, 312, 332, 352, 372, 393, 414, 434, 454, 474, 494, 514, 534, 554, 574, 595, 616, 636, 656, 676, 696, 716, 736, 756, 776, 797, 818, 838, 858, 878, 898, 918, 938, 958, 978
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 23 2019

Keywords

Links

Crossrefs

Cf. A002113, A046489, A046497, A260255, A261906.

Programs

  • Mathematica
    Total /@ Partition[Select[Range[0, 500], PalindromeQ], 2, 1]
  • Python
    from itertools import chain, count, islice
def A309325_gen(): # generator of terms
    c = 0
    for a in chain.from_iterable(chain((int((s:=str(d))+s[-2::-1]) for d in range(10**l,10**(l+1))), (int((s:=str(d))+s[::-1]) for d in range(10**l,10**(l+1)))) for l in count(0)):
        yield c+(c:=a)
A309325_list = list(islice(A309325_gen(),20)) # Chai Wah Wu, Jun 23 2022

Formula

a(n) = A002113(n) + A002113(n+1).
Showing 1-5 of 5 results.

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