A056524 -id:A056524 - cp's OEIS frontend

A241946 Numbers n equal to the sum of all the four-digit numbers formed without repetition from the digits of n.

1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 3003, 3113, 3223, 3333, 3443, 3553, 3663, 3773, 3883, 3993, 4004, 4114, 4224, 4334, 4444, 4554, 4664, 4774, 4884, 4994, 5005, 5115, 5225

Offset: 1

Michel Lagneau, May 03 2014

Let d(1)d(2)... d(q) denote the decimal expansion of a number n. Any decimal expansion of four-digits d(i)d(j)d(k)d(l) formed from the digits of n is such that ij>k>l.
This sequence is interesting because it contains more than just the only trivial palindromic values 1001, 1111, 1221,... The sequence is given by the union of subsets {palindromes with four digits from A056524} union {37323, 48015, 72468, 152658} and contains 94 elements. The last four elements are non-palindromic numbers.
But the generalization of this problem seems difficult, for example the case with the sum of all the three-digit numbers formed without repetition from the digits of n gives only 90 palindromic numbers 101, 111, 121,..., 989,999.

			37323 is in the sequence because 37323 =  2373 + 3233 + 3237 + 3273 + 3323 + 3373 + 3723 + 3732 + 3733 + 7323.

Michel Lagneau, Table of n, a(n) for n = 1..93

Cf. A241899.

with(numtheory):
for n from 1000 to 10000 do:
     lst:={}:k:=0:x:=convert(n,base,10):n1:=nops(x):
        for i from 1 to n1 do:
          for j from i+1 to n1 do:
            for m from j+1 to n1 do:
              for q from m+1 to n1 do:
            lst:=lst union {x[i]+10*x[j]+100*x[m]+1000*x[q]}:
            od:
          od:
        od:
        od:
           for a from n1 by -1 to 1 do:
             for b from a-1 by -1 to 1 do:
               for c from b-1 by -1 to 1 do:
                 for d from c-1 by -1 to 1 do:
               lst:=lst union
               {x[a]+10*x[b]+100*x[c]+1000*x[d]}:
               od:
             od:
            od:
            od:
           n2:=nops(lst):s:=sum('lst[i]', 'i'=1..n2):
           if s=n
             then
             printf(`%d, `,n):
             else
           fi:
  od:

A352535 Numbers m such that A257588(m) = 0.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 220, 330, 354, 440, 453, 550, 660, 770, 880, 990, 1001, 1100, 1111, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 1221, 1331, 1441, 1487, 1551, 1575, 1661, 1771, 1784, 1881, 1991, 2002, 2112, 2200, 2211, 2222, 2233, 2244, 2255, 2266, 2277

Offset: 1

Bernard Schott, Mar 20 2022

If m is a term, 10*m is also a term; so, terms with no trailing zeros are all primitive terms.

Palindromes with even number of digits (A056524) are all terms.

			354 is a term since 3^2 - 5^2 + 4^2 = 0 (with Pythagorean triple (3,4,5)).
1487 is a term since 1^2 - 4^2 + 8^2 - 7^2 = 0.

Cf. A003132, A257588.

Subsequences: A056524, A333440, A338754.

f[n_] := Abs @ Total[(d = IntegerDigits[n]^2) * (-1)^Range[Length[d]]]; Select[Range[0, 2300], f[#] == 0 &] (* Amiram Eldar, Mar 20 2022 *)

from itertools import count, islice
def A352535_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda m: not sum(int(d)**2*(-1 if i % 2 else 1) for i, d in enumerate(str(m))), count(max(startvalue,0)))
A352535_list = list(islice(A352535_gen(),30)) # Chai Wah Wu, Mar 24 2022

A257588(a(n)) = 0.

A071272 Palindromes in A066492.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 101, 111, 121, 131, 141, 151, 161, 171, 181, 202, 212, 222, 232, 242, 252, 262, 272, 303, 313, 323, 333, 343, 353, 363, 404, 414, 424, 434, 444, 454, 505, 515, 525, 535, 545, 606, 616, 626, 636, 707, 717, 727, 808, 818, 909, 10001

Offset: 1

Amarnath Murthy, Jun 07 2002

Cf. A056524, A066492.

More terms from Sean A. Irvine, Jul 07 2024

A240468 Sum of the distinct prime divisors of the palindromes having an even number of digits.

Original entry on oeis.org

11, 13, 14, 13, 16, 16, 18, 13, 14, 31, 112, 51, 11, 142, 61, 162, 41, 33, 192, 33, 16, 114, 66, 53, 42, 13, 23, 144, 30, 34, 294, 304, 115, 324, 47, 51, 18, 364, 14, 33, 30, 16, 210, 114, 39, 66, 51, 53, 240, 36, 50, 35, 113, 19, 117, 119, 26, 123, 125, 36, 152, 296, 16, 306, 162, 117, 20

Offset: 1

Michel Lagneau, Apr 06 2014

a(n) = Sopf(A056524(n)) = A008472(A056524(n)).
There exists a subsequence of squares such that 16, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, ...
There exists a subsequence of primes such that 11, 13, 19, 23, 31, 41, 47, 53, 59, 61, 67, 71, 73, 83, 89, 97, 109, 113, 131, 137, 139, 149,... but the subsequence of primes 17, 29, 37, 43, 101, 317, 433, 439, 487, 569,... is not included in the sequence.

			a(11) = 112 because Sopf(A056524(11)) = Sopf(1111) = A008472(1111) = 112.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A008472, A056524, A240453, A240454.

with(numtheory):for n from 1 to 100 do:x:=convert(n,base,10):n1:=nops(x): s:=sum('x[i]*10^(n1-i)', 'i'=1..n1):y:=n*10^n1+s:z:=factorset(y):n2:=nops(z):s1:=sum('z[j]', 'j'=1..n2):printf(`%d, `,s1):od:

Join[{11},d[n_]:=IntegerDigits[n];Rest[Total[Transpose[FactorInteger[Plus[FromDigits[Join[x=d[#],Reverse[x]]]]]][[1]]]&/@Range[100]]]

A372149 Palindrome numbers consisting only of odd digits.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 33, 55, 77, 99, 111, 131, 151, 171, 191, 313, 333, 353, 373, 393, 515, 535, 555, 575, 595, 717, 737, 757, 777, 797, 919, 939, 959, 979, 999, 1111, 1331, 1551, 1771, 1991, 3113, 3333, 3553, 3773, 3993, 5115, 5335, 5555, 5775, 5995, 7117, 7337, 7557, 7777, 7997

Offset: 1

James S. DeArmon, Apr 20 2024

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from James S. DeArmon)
James S. DeArmon, Common LISP code to create A372149

Intersection of A002113 and A014261.

Cf. A056524, A056525.

Select[Range[8000], PalindromeQ[#] && Times@@Boole[OddQ[IntegerDigits[#]]] == 1 &] (* Stefano Spezia, Apr 30 2024 *)

from itertools import count, islice, product
def agen(): # generator of terms
    for d in count(1):
        for p in product("13579", repeat=d//2):
            left = "".join(p)
            for mid in [[""], "13579"][d&1]:
                yield int(left + mid + left[::-1])
print(list(islice(agen(), 60))) # Michael S. Branicky, Jun 01 2025

Missing 1551 inserted by Stefano Spezia, Apr 30 2024

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A241946 Numbers n equal to the sum of all the four-digit numbers formed without repetition from the digits of n.

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A352535 Numbers m such that A257588(m) = 0.

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A071272 Palindromes in A066492.

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A240468 Sum of the distinct prime divisors of the palindromes having an even number of digits.

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Maple

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A372149 Palindrome numbers consisting only of odd digits.

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Python

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