A071251 -id:A071251 - cp's OEIS frontend

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

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.

A035132 Nonsquarefree palindromes.

Original entry on oeis.org

4, 8, 9, 44, 88, 99, 121, 171, 212, 232, 242, 252, 272, 292, 333, 343, 363, 404, 414, 424, 444, 464, 484, 525, 575, 585, 616, 636, 656, 666, 676, 686, 696, 747, 808, 828, 848, 868, 888, 909, 999, 1331, 1881, 2112, 2332, 2552, 2662, 2772, 2992, 3663, 3773

Offset: 1

Patrick De Geest, Nov 15 1998

			4114 = 2 * 11^2 * 17.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Squarefree.

Intersection of A002113 and A013929.
Supersequence of A077572.
Cf. A005117, A035133, A071251, A075804 (first differences).

sfQ[n_]:=Max[Transpose[FactorInteger[n]][[2]]]>1; palQ[n_]:=FromDigits[Reverse[IntegerDigits[n]]]==n; Select[Range[2,3776],sfQ[#] && palQ[#] &] (* Jayanta Basu, May 12 2013 *)
Select[Range[4000], PalindromeQ[#] && !SquareFreeQ[#] &] (* Amiram Eldar, Feb 25 2021 *)

from itertools import product
from sympy.ntheory.factor_ import core
def palsthru(maxdigits):
  midrange = [[""], [str(i) for i in range(10)]]
  for digits in range(1, maxdigits+1):
    for p in product("0123456789", repeat=digits//2):
      left = "".join(p)
      if len(left) and left[0] == '0': continue
      for middle in midrange[digits%2]: yield int(left+middle+left[::-1])
def okpal(p): return p > 3 and core(p, 2) != p
print(list(filter(okpal, palsthru(4)))) # Michael S. Branicky, Apr 08 2021

A075803 Differences between adjacent palindromic squarefree numbers.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 11, 11, 22, 11, 11, 24, 10, 20, 10, 10, 10, 20, 10, 11, 20, 40, 20, 21, 10, 10, 30, 20, 10, 10, 41, 20, 20, 20, 11, 10, 20, 10, 10, 10, 30, 11, 20, 20, 61, 10, 10, 10, 20, 10, 10, 10, 10, 21, 20, 20, 20, 20, 21, 10, 10, 10, 10, 10, 10, 10, 12, 110, 110, 220

Offset: 1

Jani Melik, Oct 13 2002

			a(1)=2-1=1, a(3)=5-3=2, a(6)=11-7=4.

Cf. A037010.

First differences of A071251.

test := proc(n) local d; d := convert(n,base,10); return ListTools[Reverse](d)=d and numtheory[mobius](n)<>0; end; s := []; for n from 1 to 1500 do if test(n) then s := [op(s),n]; end; od; a := [op(2..-1,s)-op(1..-2,s)];

Edited by Dean Hickerson, Oct 21 2002

cp's OEIS Frontend

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

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A035132 Nonsquarefree palindromes.

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A075803 Differences between adjacent palindromic squarefree numbers.

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