A056964 -id:A056964 - cp's OEIS frontend

A249280 Repeatedly apply 'Reverse and add' to n. a(n) gives the number of steps needed to reach a sum containing each digit from 0 to 9 at least once.

38, 37, 35, 36, 54, 34, 45, 35, 48, 53, 52, 33, 51, 44, 32, 34, 50, 47, 43, 52, 33, 51, 44, 32, 34, 50, 47, 43, 42, 33, 51, 44, 32, 34, 50, 47, 43, 42, 31, 51, 44, 32, 34, 50, 47, 43, 42, 31, 41, 44, 32, 34, 50, 47, 43, 42, 31, 41, 33, 32, 34, 50, 47, 43, 42

Offset: 1

Felix Fröhlich, Oct 26 2014

Conjecture 1: a(n) exists for all n.
Conjecture 2: There exists an upper bound c such that a(n) < c for all n.
The conjectures seem highly likely, especially since a(n) = 0 for almost all n. A lower bound for c is a(1418993) = 73. (Checked to 10^9.) - Charles R Greathouse IV, Oct 28 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Felix Fröhlich, C++ program for this sequence

Cf. A056964.

Table[Length[NestWhileList[#+IntegerReverse[#]&,n,Min[DigitCount[#]] == 0&]]-1,{n,70}] (* Harvey P. Dale, Aug 20 2022 *)

fromdigits(v,b=10)=subst(Pol(v),'x,b) \\ needed for gp < 2.63 or so
A056964(n)=fromdigits(Vecrev(digits(n)))+n
ispan(n)=#Set(digits(n))==10
a(n)=my(k); while(!ispan(n), n=A056964(n); k++); k \\ Charles R Greathouse IV, Oct 28 2014

A277258 Numbers such that n+R(n) | n*R(n), where R(n) is the digits reverse of n.

Original entry on oeis.org

2, 4, 6, 8, 22, 44, 66, 88, 110, 132, 198, 202, 212, 220, 222, 231, 232, 242, 252, 262, 264, 272, 282, 292, 297, 330, 396, 404, 414, 424, 434, 440, 444, 454, 462, 464, 474, 484, 494, 495, 550, 594, 606, 616, 626, 636, 646, 656, 660, 666, 676, 686, 693, 696, 770

Offset: 1

Paolo P. Lava, Oct 07 2016

This sequence contains all positive terms of A029951. So the sequence is infinite. - Altug Alkan, Oct 07 2016

			R(132) = 231, (132 * 231) / (132 + 231) = 30492 / 363 = 84.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A004086, A029951, A056964, A061205, A131058.

R:=proc(w) local x, y, z; x:=w; y:=0; for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end: P:= proc(q) local n; for n from 1 to q do if type(n*R(n)/(n+R(n)),integer) then print(n); fi; od; end: P(10^4);

Select[Range@ 770, Function[r, Mod[# r, # + r] == 0]@ FromDigits@ Reverse@ IntegerDigits@ # &] (* Michael De Vlieger, Oct 14 2016 *)

R(n) = eval(concat(Vecrev(Str(n))));
isok(n) = n*R(n) % (n+R(n)) == 0; \\ Michel Marcus, Oct 15 2016

A298972 Number of positive integers k < n such that n occurs in the Reverse-and-Add trajectory of k.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 2, 2, 0, 1, 0, 4, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Felix Fröhlich, Jan 30 2018

Number of integers k < n such that n occurs in row k of A243238.
For n > 0, a(n) = 0 iff n is a term of A067031.
For n > 0, a(n) > 0 iff n is a term of A067030.

			For n = 22: There exist 4 positive integers k < 22 such that 22 occurs in the Reverse-and-Add trajectory of k, namely 5, 10, 11 and 20, so a(22) = 4.

Index entries for sequences related to Reverse and Add!

Cf. A056964, A067030, A067031, A243238.

Block[{nn = 85, s}, s = Array[Union@ NestWhileList[# + IntegerReverse@ # &, #, # < nn &, 1, nn] &, nn]; Array[Count[Take[s, # - 1], #, 2] &, nn + 1, 0]] (* Michael De Vlieger, Feb 01 2018 *)

a(n) = my(i=0); for(k=1, n-1, my(x=k); while(x < n, x=x+eval(concat(Vecrev(Str(x))))); if(x==n, i++)); i

A333523 Number of iterations of Reverse And Add needed to reach a number divisible by n (or 0 if such a number is never reached).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 61, 1, 3, 8, 34, 22, 8, 17, 2, 8, 119, 14, 1, 17, 7, 110, 7, 12, 33, 34, 158, 28, 12, 1, 60, 11, 12, 50, 79, 7, 129, 64, 13, 42, 1, 4, 89, 131, 8, 14, 81, 30, 19, 125, 12, 1, 88, 13, 33, 67, 232, 26, 27, 24, 123, 59, 1, 24, 59, 36, 206, 148, 12, 217, 90, 97

Offset: 1

Daniel Starodubtsev, Mar 26 2020

If n is a palindrome > 0, a(n) = 1. See A002113.

a(n) > 0 for n < 10000.

			a(12) = 3, because 12 takes 3 iterations (12 -> 33 -> 66 -> 132) to become 132, which is divisible by 12.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A002113, A016016, A056964.

radd(n) = fromdigits(Vecrev(digits(n)))+n; \\ A056964
a(n) = {my(i=1, k=n, x); while((x=radd(n)) % k, i++; n=x); i;} \\ Michel Marcus, Apr 11 2020

A338739 Number of true-palindromic compositions of n.

Original entry on oeis.org

1, 2, 2, 4, 4, 8, 8, 16, 16, 31, 32, 62, 63, 124, 126, 248, 252, 496, 504, 991, 1007, 1982, 2013, 3960, 4023, 7914, 8040, 15816, 16068, 31609, 32112, 63171, 64180, 126251, 128266, 252318, 256347, 504268, 512324, 1007801, 1023909, 2014131, 2046338, 4025329, 4089724

Offset: 1

Michel Marcus, Nov 06 2020

A true-palindromic composition or true-palindrome to be a composition whose digit-comma-sequence is the same whether read from left to right or right to left. [Shapcott p. 35]

			(12, 6, 21) is a true-palindromic composition of 39.
(126, 621) is a true-palindromic composition of 747.

Caroline Shapcott, An introduction to true-palindromic compositions, Australasian Journal of Combinatorics, Volume 60(1) (2014), Pages 35-49.

Cf. A002113, A004086, A056964.

Cf. A016116 (symmetric compositions), A338740.

rev(n) = Vecrev(n=digits(n)); \\ A004086
ispal(n) = Vecrev(n=digits(n))==n; \\ A002113
radd(n) = fromdigits(Vecrev(digits(n))) + n; \\ A056964
lista(nn) = my(x='x+O('x^(nn))); Vec(sum(k=0, nn, if (ispal(k), x^k))/(1 - sum(k=1, nn, if (k%10, x^radd(k)))) - 1);

Shapcott gives a g.f on p. 3, and 1 should be subtracted to get sequence for n>=1.

A338740 Number of hairpin compositions of n.

Original entry on oeis.org

0, 0, 0, 0, 2, 4, 14, 28, 70, 140, 310, 621, 1302, 2607, 5335, 10675, 21593

Offset: 1

Michel Marcus, Nov 06 2020

A hairpin composition with t parts is a composition that is true-palindromic except for a part-sequence in the center of length at least 1 but no more than t-2. [Shapcott p. 46]

Caroline Shapcott, An introduction to true-palindromic compositions, Australasian Journal of Combinatorics, Volume 60(1) (2014), Pages 35-49.

Cf. A002113, A004086, A056964.

Cf. A016116 (symmetric compositions), A338739.

A344871 a(n) is the least number that can be represented in exactly n ways as the sum of a prime and its digit reversal.

Original entry on oeis.org

1, 4, 44, 88, 1090, 3212, 4334, 2992, 5995, 4994, 7997, 9779, 5104, 11110, 11891, 10109, 11000, 10780, 108880, 110500, 252142, 278872, 296692, 293282, 308902, 287782, 411103, 289982, 466664, 281072, 457754, 398893, 298892, 462154, 517814, 494384, 299992, 707806, 471064, 476674, 487784, 467764

Offset: 0

J. M. Bergot and Robert Israel, May 31 2021

If the reversal of p is another prime, p+reversal(p) and reversal(p)+p are both counted.

a(n) is the first number that occurs exactly n times in A061227.

			a(4) = 1090 because 1090 = 149+941 = 347+743 = 743+347 = 941+149, and this is the least number with exactly four such representations.

Robert Israel, Table of n, a(n) for n = 0..92

Cf. A004086, A061227, A056964.

revdigs:= proc(n) local L,t;
L:= convert(n,base,10);
add(L[-t]*10^(t-1),t=1..nops(L));
end proc:
V:= Vector(10^6):
p:= 1:
do
  p:= nextprime(p);
  if p > 9*10^5 then break fi;
  r:= p+revdigs(p);
  if r <= 10^6 then V[r]:= V[r]+1 fi
od:
A:= Array(0..64):
for i from 1 to 10^6 do
  if V[i] <= 64 and A[V[i]] = 0 then A[V[i]]:= i fi
od:
convert(A,list);

A345410 a(n) is the least number that is the sum of an emirp and its reversal in exactly n ways.

Original entry on oeis.org

44, 1090, 10450, 5104, 88888, 10780, 289982, 299992, 482174, 478874, 868868, 499994, 1073270, 1087790, 1071070, 1069970, 10904990, 10794980, 1091090, 10892990, 1100000, 29955992, 1101100, 26688662, 31022002, 27599572, 46400354, 44688644, 29821792, 45289244, 30122092, 26988962

Offset: 1

J. M. Bergot and Robert Israel, Jun 18 2021

Interchanging an emirp and its reversal is not counted as a different way.
a(n) is the least number k such that there are exactly n unordered pairs of distinct primes (p,p') such that p' is the digit reversal of p and p+p' = k.
Are terms not divisible by 3? Amiram Eldar finds proof they are; A056964(n) = n + reverse(n) is divisible by 3 if and only if n is divisible by 3. But emirps are primes (other than 3) so they are not divisible by 3. - David A. Corneth, Jun 19 2021

			a(3) = 10450 because 10450 = 1229+9221 = 1409+9041 = 3407+7043.

David A. Corneth, Table of n, a(n) for n = 1..423
David A. Corneth, A few examples

Cf. A006567, A345408, A345409.

revdigs:= proc(n) local L,i; L:= convert(n,base,10); add(L[-i]*10^(i-1),i=1..nops(L)) end proc:
isemirp1:= proc(n) local r;
if not isprime(n) then return false fi;
r:= revdigs(n);
r > n and isprime(r)
end proc:
E:= select(isemirp1, [seq(seq(seq(i*10^d+j,j=1..10^d-1,2),i=[1,3,7,9]),d=1..5)]):
V:= sort(map(t -> t+revdigs(t),E)):
N:= nops(V):
W:= Vector(16):
i:= 1:
while i < N do
for j from 1 to N-i while V[i+j]=V[i] do od:
if j <= 16 and W[j] = 0 then W[j]:= V[i] fi;
  i:= i+j;
od:
convert(W,list);

from itertools import product
from collections import Counter
from sympy import isprime, nextprime
def epgen(start=1, end=float('inf')): # generates unique emirp/prime pairs
    digits = 2
    while True:
      for first in "1379":
        for last in "1379":
          if last < first: continue
          for mid in product("0123456789", repeat=digits-2):
            strp = first + "".join(mid) + last
            revstrp = strp[::-1]
            if strp >= revstrp: continue
            p = int(strp)
            if p > end: return
            revp = int(strp[::-1])
            if isprime(p) and isprime(revp): yield (p, revp)
      digits += 1
def aupto(lim):
    alst = []
    c = Counter(sum(ep) for ep in epgen(1, lim) if sum(ep) <= lim)
    r = set(c.values())
    for i in range(1, max(r)+1):
        if i in r: alst.append(min(s for s in c if c[s] == i))
        else: break
    return alst
print(aupto(11*10**5)) # Michael S. Branicky, Jun 19 2021

More terms from David A. Corneth, Jun 18 2021

A350575 Squarefree numbers k such that k + (k reversed) is also squarefree.

Original entry on oeis.org

1, 3, 5, 7, 10, 11, 14, 15, 19, 21, 23, 30, 33, 34, 37, 41, 42, 43, 46, 51, 55, 58, 59, 61, 67, 69, 70, 73, 77, 78, 82, 85, 86, 87, 89, 91, 94, 95, 101, 102, 105, 106, 109, 111, 115, 118, 119, 130, 131, 134, 138, 139, 141, 142, 146, 149, 151, 155, 158, 159, 161, 166, 170, 174, 178, 181, 182, 185, 190, 191, 194, 195, 199

Offset: 1

Jean-François Alcover, Jan 07 2022

This is to squarefree numbers what A061783 is to primes.

			14 is a term since it's squarefree and so is 14 + 41 = 55.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A004086, A005117, A056964, A061783.

R:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||n):
q:= n-> andmap(numtheory[issqrfree], [n, n+R(n)]):
select(q, [$1..200])[];  # Alois P. Heinz, Jan 07 2022

okQ[n_] := SquareFreeQ[n] && SquareFreeQ[n + IntegerReverse[n]];
Select[Range[200], okQ]

isok(m) = issquarefree(m) && issquarefree(m+fromdigits(Vecrev(digits(m)))); \\ Michel Marcus, Jan 07 2022

from sympy.ntheory.factor_ import core
def squarefree(n): return core(n, 2) == n
def ok(n): return squarefree(n) and squarefree(n + int(str(n)[::-1]))
print([k for k in range(1, 200) if ok(k)]) # Michael S. Branicky, Jan 07 2022

A359011 Numbers k such that k^2 + the reversal of k^2 is a square.

Original entry on oeis.org

0, 231, 9426681, 8803095102, 56017891104, 4811618419542

Offset: 1

Michel Marcus, Dec 11 2022

Cf. A056964, A061230.

a(1)-a(5) from Douglas McNeil and Zak Seidov, Nov 09 2010

a(6) from Giovanni Resta, Sep 26 2011

cp's OEIS Frontend

A249280 Repeatedly apply 'Reverse and add' to n. a(n) gives the number of steps needed to reach a sum containing each digit from 0 to 9 at least once.

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A277258 Numbers such that n+R(n) | n*R(n), where R(n) is the digits reverse of n.

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A298972 Number of positive integers k < n such that n occurs in the Reverse-and-Add trajectory of k.

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A333523 Number of iterations of Reverse And Add needed to reach a number divisible by n (or 0 if such a number is never reached).

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A338739 Number of true-palindromic compositions of n.

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A338740 Number of hairpin compositions of n.

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A344871 a(n) is the least number that can be represented in exactly n ways as the sum of a prime and its digit reversal.

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Maple

A345410 a(n) is the least number that is the sum of an emirp and its reversal in exactly n ways.

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