A261422 -id:A261422 - cp's OEIS frontend

A262526 Any number greater than a(n) can conjecturally be represented in more ways by sums of three base-10 palindromes than a(n).

1, 2, 3, 4, 98, 120, 142, 164, 172, 192, 212, 223, 2082, 2102, 2203, 2213, 130282, 130992, 131392, 131492, 131592, 131742, 131752, 131792, 131902, 132002, 132102, 132192, 132202, 132482, 132502, 132602, 132662, 132672, 132752, 132782, 132802

Offset: 1

Hugo Pfoertner, Sep 25 2015

The corresponding representation counts are provided in A262527. Positions of latest occurrence of increasing minima of representation counts in A261132. The sequence provides numerical evidence for the validity of the conjecture that every number is the sum of three palindromes.

			a(5)=98 because A261132(k)>5 for all k>98.
a(7)=142 because A261132(k)>A262527(7)=8 for all k>142.

Hugo Pfoertner, Table of n, a(n) for n = 1..1072

Cf. A262527, A261132, A002113, A262524, A262525.

See A261422, A262544, A262545 for another approach.

A262527 Conjectured minimum number of ways to represent a number >= A262526(i) by sums of three base-10 palindromes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 11, 13, 14, 15, 16, 17, 18, 21, 27, 29, 35, 41, 45, 48, 51, 54, 55, 56, 61, 63, 64, 65, 74, 75, 79, 80, 82, 83, 85, 86, 87, 91, 111, 112, 113, 114, 115, 116, 118, 120, 121, 124, 127, 133, 134, 138, 140, 141, 142, 145, 147, 150, 153, 165, 169, 171, 174, 175, 177, 179, 180, 183, 184, 185

Offset: 1

Hugo Pfoertner, Sep 25 2015

The sequence is obtained by sorting the counts A261132 into increasing order together with their positions of occurrence. If a new record in the sorted A261132 is found, the index of its latest occurrence in A261132 becomes the next term in A262526 and the corresponding value of A261132 becomes a(n).
7 is not in the sequence, because the latest occurrence of 7 is at A261132(64), whereas the latest occurrence of 6 had already set the record to A262526(6) = 120.
a(7) = 8 corresponds to the latest occurrence of 8 at A261132(142), thus A262526(7) = 142.

			a(5) = 5 because A261132(k) > 5 for all k > A262526(5) = 98.

Hugo Pfoertner, Table of n, a(n) for n = 1..1072
Hugo Pfoertner, Least number of representations by sums of three palindromes.

Cf. A002113, A261132, A262524, A262525, A262526.

See A261422, A262544, A262545 for another approach.

A109326 Smallest positive number that requires n steps to be represented as a sum of palindromes using the greedy algorithm.

Original entry on oeis.org

1, 10, 21, 1022, 101023, 1000101024

Offset: 1

David Wasserman, Aug 11 2005

Index of first occurrence of n in A088601.
Presumably this sequence is unbounded. - N. J. A. Sloane, Aug 28 2015
The greedy algorithm means iteration of A261424 until a palindrome is reached. For n = 3, 4, ... we have a(n+1) = 10^L(n) + a(n) + 1 with L(n) = 2^(n-2) + 1 = length(a(n))*2 - 3 for n > 3. We have a(7) <= 10^17 + 1000101025, a(8) <= 10^33 + 10^17 + 1000101026, a(9) <= 10^65 + 10^33 + 10^17 + 1000101027, a(10) <= 10^129 + 10^65 + 10^33 + 10^17 + 1000101028, etc, with conjectured equality. - M. F. Hasler, Sep 08 2015, edited Sep 09 2018

M. F. Hasler, Sum of palindromes, OEIS wiki, Sept. 2015.

Cf. A088601, A002113, A261422, A261423.

# uses functions in A088601
def afind(limit):
    record = 0
    for i in range(1, limit+1):
        steps = A088601(i)
        if steps > record: print(i, end=", "); record = steps
afind(10**6) # Michael S. Branicky, Jul 12 2021

a(n) = Sum_{0 <= k <= n-3} 10^(2^k+1) + n - 82, for n > 2 (conjectured). - M. F. Hasler, Sep 08 2015

Edited by N. J. A. Sloane, Aug 28 2015

A261680 Number of ordered quadruples (u,v,w,x) of binary palindromes (see A006995) with u+v+w+x=n.

Original entry on oeis.org

1, 4, 6, 8, 13, 16, 22, 28, 34, 44, 50, 60, 59, 72, 70, 80, 92, 88, 114, 96, 125, 104, 152, 120, 172, 144, 188, 152, 215, 144, 242, 160, 272, 172, 302, 180, 329, 216, 352, 240, 388, 228, 430, 228, 442, 212, 476, 192, 506, 228, 496, 248, 540, 252, 582, 276, 592

Offset: 0

N. J. A. Sloane, Sep 04 2015

Conjecture: a(n)>0: every number is the sum of four binary palindromes. (Compare A261422, A261675.)

N. J. A. Sloane, Table of n, a(n) for n = 0..9999
Aayush Rajasekaran, Jeffrey Shallit, and Tim Smith, Sums of Palindromes: an Approach via Nested-Word Automata, preprint arXiv:1706.10206 [cs.FL], June 30 2017.

Cf. A006995, A261422, A261675, A261679.

G.f. = (Sum_{p in A006995} x^p)^4.

A282584 Number of compositions (ordered partitions) of n into decimal palindromes (A002113).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1022, 2042, 4081, 8156, 16300, 32576, 65104, 130112, 260032, 519681, 1038595, 2075660, 4148259, 8290402, 16568581, 33112734, 66176648, 132255728, 264316464, 528243231, 1055707644, 2109858797, 4216606912, 8426997041, 16841569684, 33658308890, 67266993433

Offset: 0

Ilya Gutkovskiy, Feb 19 2017

			a(4) = 8 because we have [4], [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2] and [1, 1, 1, 1].

Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for sequences related to palindromes
Index entries for sequences related to compositions

Cf. A002113, A091580, A091581, A260254, A261422.

nmax = 37; CoefficientList[Series[1/(1 - Sum[Boole[PalindromeQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=2} x^A002113(k)).

A282585 Number of ways to write n as an ordered sum of 3 squarefree palindromes (A071251).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 7, 9, 12, 19, 21, 21, 18, 24, 27, 28, 18, 18, 19, 24, 15, 10, 6, 12, 12, 12, 9, 9, 12, 15, 18, 12, 9, 7, 15, 15, 15, 9, 12, 15, 18, 18, 12, 9, 9, 18, 15, 12, 0, 9, 9, 9, 0, 0, 0, 6, 6, 9, 12, 9, 12, 15, 18, 18, 12, 9, 13, 18, 18, 18, 9, 15, 18, 21, 18, 12, 9, 15, 21, 21, 21, 9, 18, 21, 24, 18

Offset: 0

Ilya Gutkovskiy, Feb 19 2017

Every number can be written as the sum of 3 palindromes (see A261132 and A261422).
Conjecture: a(n) > 0 for any sufficiently large n.
Additional conjecture: every number > 3 can be written as the sum of 4 squarefree palindromes.

			a(22) = 6 because we have [11, 6, 5], [11, 5, 6] [6, 11, 5], [6, 5, 11], [5, 11, 6] and [5, 6, 11].

Ilya Gutkovskiy, Extended graphical example
Ilya Gutkovskiy, Extended graphical example for additional conjecture
Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Squarefree
Index entries for sequences related to palindromes

Cf. A002113, A005117, A035137, A071251, A091580, A091581, A260254, A261131, A261132, A261422, A280210, A282584.

nmax = 85; CoefficientList[Series[Sum[Boole[SquareFreeQ[k] && PalindromeQ[k]] x^k, {k, 1, nmax}]^3, {x, 0, nmax}], x]

G.f.: (Sum_{k>=1} x^A071251(k))^3.

A261916 Smallest p such that n can be written as n = p+q+r where p>=q>=r>=0 are palindromes.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 11, 11, 11, 11, 22, 11, 22, 22, 22, 22, 22, 22, 22, 22, 22, 33, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 44, 22, 33, 33, 33, 33, 33, 33, 33, 33, 33, 55, 22, 33, 33, 33, 33, 33

Offset: 0

N. J. A. Sloane, Sep 11 2015

Every number is the sum of three palindromes.

			Initial values of n,p,q,r are:
0 0 0 0
1 1 0 0
2 1 1 0
3 1 1 1
4 2 1 1
5 2 2 1
6 2 2 2
7 3 3 1
...
25 9 9 7
26 9 9 8
27 9 9 9
28 11 11 6
29 11 11 7
30 11 11 8
...
33 11 11 11
34 22 11 1
...

David Consiglio, Jr., Table of n, a(n) for n = 0..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, arXiv: 1602.06208 [math.NT], 2017, Math. Comp. 87 (2018), 3023-3055.
David Consiglio, Jr., Python program
James Grime and Brady Haran, Every Number is the Sum of Three Palindromes, Numberphile video (2018)

Cf. A002113, A261422, A261132.

If "smallest" is changed to "largest" we get a sequence which agrees with the palindromic floor function A261423 for at least 300 terms.

Edited by Alois P. Heinz, Dec 29 2018

A282845 Number of ways to write n as an ordered sum of 6 prime power palindromes (A084092).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 6, 21, 56, 126, 246, 432, 702, 1077, 1576, 2232, 3072, 4112, 5352, 6801, 8422, 10197, 12102, 14117, 16146, 18177, 20112, 21882, 23382, 24661, 25566, 26136, 26316, 26181, 25560, 24677, 23436, 21981, 20226, 18486, 16536, 14642, 12702, 10962, 9166, 7662, 6222, 5042, 3912, 3096, 2306, 1746, 1236, 921, 600

Offset: 0

Ilya Gutkovskiy, Feb 22 2017

Is there k which satisfies a(n) > 0 for all n > k?

			a(7) = 6 because we have:
[2, 1, 1, 1, 1, 1]
[1, 2, 1, 1, 1, 1]
[1, 1, 2, 1, 1, 1]
[1, 1, 1, 2, 1, 1]
[1, 1, 1, 1, 2, 1]
[1, 1, 1, 1, 1, 2]

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for sequences related to palindromes

Cf. A000961, A002113, A002385, A084092, A260254, A261131, A261132, A261422, A282585.

nmax = 55; CoefficientList[Series[(x + Sum[Boole[PrimePowerQ[k] && PalindromeQ[k]] x^k, {k, 1, nmax}])^6, {x, 0, nmax}], x]

G.f.: (Sum_{k>=1} x^A084092(k))^6.

cp's OEIS Frontend

A262526 Any number greater than a(n) can conjecturally be represented in more ways by sums of three base-10 palindromes than a(n).

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A262527 Conjectured minimum number of ways to represent a number >= A262526(i) by sums of three base-10 palindromes.

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A109326 Smallest positive number that requires n steps to be represented as a sum of palindromes using the greedy algorithm.

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A261680 Number of ordered quadruples (u,v,w,x) of binary palindromes (see A006995) with u+v+w+x=n.

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A282584 Number of compositions (ordered partitions) of n into decimal palindromes (A002113).

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A282585 Number of ways to write n as an ordered sum of 3 squarefree palindromes (A071251).

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A261916 Smallest p such that n can be written as n = p+q+r where p>=q>=r>=0 are palindromes.

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A282845 Number of ways to write n as an ordered sum of 6 prime power palindromes (A084092).

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