A199685 -id:A199685 - cp's OEIS frontend

A033819 Trimorphic numbers: n^3 ends with n. Also m-morphic numbers for all m > 5 such that m-1 is not divisible by 10 and m == 3 (mod 4).

0, 1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999, 1249, 3751, 4375, 4999, 5001, 5625, 6249, 8751, 9375, 9376, 9999, 18751, 31249, 40625, 49999, 50001, 59375, 68751, 81249, 90624, 90625

Offset: 1

David W. Wilson

n is in this sequence iff it occurs in one of A002283, A007185, A016090, A198971, A199685, A216092, A216093, A224473, A224474, A224475, A224476, A224477, and A224478. - Eric M. Schmidt, Apr 08 2013

Let q(n) = floor(a(n)^3 / 10^A055642(a(n))), where A055642(n) is the number of digits in the decimal expansion of n. As well, let na and nb denote the indices of the preceding and next terms that begin with a 9. Then (1/q(n)) * (a(n)^4 - a(n)^3 - a(n)^2 + a(n)) - 2*a(n)^2 + a(n) + q(n) + 1 = a(na+nb-n)^2 - a(na+nb-n) - q(na+nb-n). - Christopher Hohl, Apr 08 2019

			376^3 = 53157376 which ends with 376.

S. Premchaud, A class of numbers, Math. Student, 48 (1980), 293-300.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
Shyam Sunder Gupta, Elegance of Squares, Cubes, and Higher Powers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 2, 29-81.
Eric Weisstein's World of Mathematics, Trimorphic Number
Index entries for sequences related to automorphic numbers

Cf. A074194, A215558 (cubes of the terms).

[n: n in [0..10^5] | Intseq(n^3)[1..#Intseq(n)] eq Intseq(n)]; // Bruno Berselli, Apr 04 2013

Do[x=Floor[N[Log[10, n], 25]]+1; If[Mod[n^3, 10^x] == n, Print[n]], {n, 1, 10000}]
Select[Range[100000],PowerMod[#,3,10^IntegerLength[#]]==#&](* Harvey P. Dale, Nov 04 2011 *)
Select[Range[0, 10^5], 10^IntegerExponent[#^3-#, 10]>#&] (* Jean-François Alcover, Apr 04 2013 *)

A224474 (2*16^(5^n) - 1) mod 10^n: a sequence of trimorphic numbers ending in 1.

Original entry on oeis.org

1, 51, 751, 8751, 18751, 218751, 4218751, 74218751, 574218751, 3574218751, 63574218751, 163574218751, 163574218751, 80163574218751, 480163574218751, 7480163574218751, 87480163574218751, 487480163574218751, 5487480163574218751, 15487480163574218751

Offset: 1

Eric M. Schmidt, Apr 07 2013

a(n) is the unique positive integer less than 10^n such that a(n) + 1 is divisible by 2^n and a(n) - 1 is divisible by 5^n.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Trimorphic Number
Index entries for sequences related to automorphic numbers

Cf. A033819. Corresponding 10-adic number is A063006. The other trimorphic numbers ending in 1 are included in A199685 and A224476.

def A224474(n) : return crt(-1, 1, 2^n, 5^n)

a(n) = (2 * A016090(n) - 1) mod 10^n.

A224476 (2*16^(5^n) + (10^n)/2 - 1) mod 10^n: a sequence of trimorphic numbers ending (for n > 1) in 1.

Original entry on oeis.org

6, 1, 251, 3751, 68751, 718751, 9218751, 24218751, 74218751, 8574218751, 13574218751, 663574218751, 5163574218751, 30163574218751, 980163574218751, 2480163574218751, 37480163574218751, 987480163574218751, 487480163574218751, 65487480163574218751

Offset: 1

Eric M. Schmidt, Apr 07 2013

a(n) is the unique positive integer less than 10^n such that a(n) + 2^(n-1) + 1 is divisible by 2^n and a(n) - 1 is divisible by 5^n.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Trimorphic Number
Index entries for sequences related to automorphic numbers

Cf. A033819. Converges to the 10-adic number A063006. The other trimorphic numbers ending in 1 are included in A199685 and A224474.

def A224476(n) : return crt(2^(n-1)-1, 1, 2^n, 5^n)

a(n) = (A224474(n) + 10^n/2) mod 10^n.

A301912 Numbers k such that the decimal representation of k ends that of the sum of the first k cubes.

Original entry on oeis.org

0, 1, 5, 25, 76, 376, 500, 625, 876, 1876, 2500, 5001, 5625, 9376, 15625, 25001, 40625, 50001, 62500, 65625, 71876, 75001, 90625, 109376, 171876, 265625, 375001, 390625, 500001, 765625, 875001, 890625, 1171876, 2265625, 2890625, 4062500, 4375001, 5000001

Offset: 1

Robert Dawson, Mar 28 2018

For j >= 3, 1 + 5*10^j = A199685(j) is in the sequence, so the sequence is infinite. - Vaclav Kotesovec, Mar 29 2018
From Robert Dawson, Apr 12 2018: (Start)
This sequence is the union of the following ten subsequences.
Terms in  have fewer than d digits: they are always terms of the sequence, and always appear elsewhere, as an earlier term of the same subsequence or a related subsequence. (However, the d-th terms of the subsequences are always distinct for any d > 4.) Dashes replace certain solutions to the congruences for small values of d for which certain other divisibility criteria are not met. The integers n_0(d) and n_1(d) are the even and odd zeros of n^2+3n+4 (mod 2^d) (note that by Hensel's Lemma these always exist and each is unique).
(i)   p(d) satisfying 2^d| p(d) - n_0(d), 5^d |p(d):
  (0,<0>,500,2500,62500,62500,4062500,14062500,...)
(ii)  q(d) satisfying 2^{d-1}|q(d)-1, 5^d|q(d) for d != 3:
  (0,25,-,<625>,40625,390625,2890625,12890625,...)
(iii) q(d) + 5x10^{d-1} for d != 2:
  (5,-, 625,5625,90625, 890625,7890625, 62890625,...)
(iv)  q'(d) satisfying 2^{d-1}|q'(d) - n_1(d), 5^d|q'(d), for d != 1,3:
  (-,25,-,<625>,15625,265625,2265625,47265625,...)
(v)   q'(d) + 5x10^{d-1} for d != 2:
  (5,-,625,5625,65625,765625,7265625,97265625,...)
(vi)  r(d) satisfying 2^d|r(d), 5^d|r(d)-1 for d >= 2
  (-,76,376,9376,<9376>,109376,7109376,87109376,...) = A016090(d)
(vii) r'(d) satisfying 2^d|r'(d) - n_0(d), 5^d|r'(d)-1 for d >= 2:
  (-,76,876,1876,71876,171876,1171876,<1171876>,...)
(viii)s(d) := 5x10^{d-1}+1 for d >= 4:
  (-,-,-,5001,50001,500001,5000001,50000001,...) = A199685(d-1)
(ix)  t(d) satisfying 2^{d-1}|t(d)-n_0(d), 5^d|t(d)-1:
  (1,<1>,<1>,<1>,25001,375001,4375001,34375001,...)
(x)   t(d) + 5x10^{d-1} for d >= 4:
  (-,-,-,5001,75001,875001,9375001,84375001,...)
For d > 4, the sequence A301912 has at most 10 and at least 5 terms with d digits. The maximum is first attained for d=7. The minimum is first attained for d=168. (End)

			The sum of the first five cubes is 225, which ends in 5, so 5 is in the sequence.

Vaclav Kotesovec, Table of n, a(n) for n = 1..61
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
Wikipedia, Hensel's lemma

Cf. A000537, A199685.

seq = {}; Do[If[StringTake[ToString[k^2*(k+1)^2/4], -StringLength[ToString[k]]] == ToString[k], seq = Join[seq, {k}]], {k, 0, 1000000}]; seq (* Vaclav Kotesovec, Mar 29 2018 *)

A301912_list, k, n = [], 1, 1
while len(A301912_list) < 100:
    if n % 10**(len(str(k))) == k:
        A301912_list.append(k)
    k += 1
    n += k**3 # Chai Wah Wu, Mar 30 2018

Corrected and extended by Vaclav Kotesovec, Mar 29 2018

A181607 Numbers n with k digits such that n^2 == 1 (mod 10^k).

Original entry on oeis.org

1, 9, 49, 51, 99, 249, 251, 499, 501, 749, 751, 999, 1249, 3751, 4999, 5001, 6249, 8751, 9999, 18751, 31249, 49999, 50001, 68751, 81249, 99999, 218751, 281249, 499999, 500001, 718751, 781249, 999999, 4218751, 4999999, 5000001, 5781249, 9218751

Offset: 1

Robert G. Wilson v, Nov 01 2010

Least term of n digits: 1, 49, 249, 1249, 18751, 218751, 4218751, ..., .
If n of k digits is present then 10^k-n is present.
The union of A002283, A198971, A199685, A224473, A224474, A224475, and A224476 (except that this sequence omits 0, 4, and 6). - Eric M. Schmidt, Jan 26 2016

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A007185, A016090, A033819.

Table[ Select[ Range[10^(k - 1), 10^k - 1], Mod[ #^2, 10^k] == 1 &], {k, 7}] // Flatten

A384094 Numbers whose square has digit sum 9 and no trailing zero.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 39, 45, 48, 51, 102, 105, 111, 201, 249, 318, 321, 348, 351, 501, 549, 1002, 1005, 1011, 1101, 1149, 1761, 2001, 4899, 5001, 10002, 10005, 10011, 10101, 10149, 11001, 14499, 20001, 50001, 100002, 100005, 100011, 100101, 101001, 110001, 200001, 375501, 500001, 1000002

Offset: 1

M. F. Hasler, Jun 15 2025

All numbers of the form 10^a + 10^b + 1 (i.e., A052216+1 = 3*A237424) and of the form 10^a + 5*10^b with min(a, b) = 0 (i.e., A133472 U A199685), are in this sequence. Terms not of this form are (9, 18, 39, 45, 48, 249, 318, 321, 348, 351, 549, 1149, 1761, 4899, 10149, 14499, 375501, ...), see subsequence A384095. (Is this sequence finite? What is the next term?)

Is it true that no number > 1049 = A215614(6) has a square with digit sum less than 9, other than the trivial 1 and 4?

Cf. A004159 (sum of digits of n^2), A215614 (sumdigits(n^2) = 7), A133472 (10^n + 5), A199685 (5*10^n + 1), A052216 (10^a + 10^b), A237424 ((10^a + 10^b + 1)/3).

See also: A058414 (digits(n^2) in {0,1,4}).

select( {is_A384094(n)=n%10 && sumdigits(n^2)==9}, [1..10^5])

A384095 Numbers other than {10^a + 10^b + 1} and {10^a + 5*10^b, min(a, b) = 0} whose square has digit sum 9 and no trailing zero.

Original entry on oeis.org

9, 18, 39, 45, 48, 249, 318, 321, 348, 351, 549, 1149, 1761, 4899, 10149, 14499, 375501

Offset: 1

M. F. Hasler, Jun 15 2025

The definition excludes the two "regular" subsequences of A384094, namely A052216+1 = 3*A237424 and A133472 U A199685, which provide most of its terms.
Is it true that no number > 1049 = A215614(6) has a square with digit sum less than 9, other than the trivial 1 and 4?
The next term, if it exists, is a(18) > 10^8.
a(18) > 10^14 if it exists. - Robert Israel, Jun 15 2025
a(18) > 10^40 if it exists. - Chai Wah Wu, Jun 19 2025

Cf. A004159 (sum of digits of n^2), A384094 (sumdigits(n^2) = 9), A133472 (10^n+5), A199685 (5*10^n + 1), A052216 (10^a+10^b), A237424 ((10^a+10^b+1)/3).

See also: A215614 (sumdigits(n^2) = 7), A058414 (digits(n²) ⊂ {0,1,4}).

extend:= proc(a,d) local i,s;
    s:= convert(convert(a,base,10),`+`);
    op(select(t -> numtheory:-quadres(t,10^d)=1, [seq(i*10^(d-1)+a, i=0 .. 9 - s)]))
end proc:
istriv:= proc(n) local L;
   L:= subs(0=NULL,convert(n,base,10));
   member(L, [[4],[5],[6],[1,1],[1,1,1],[1,2],[2,1],[1,5],[5,1]])
end proc:
R:= NULL:
A:= [1,4,5,6,9]:
for d from 2 to 20 do
  A:= map(extend,A,d);
  V:= select(t -> t > 10^(d-1) and issqr(t) and convert(convert(t,base,10),`+`)=9, A);
  if V <> [] then V:= sort(remove(istriv,map(sqrt,V))); R:= R,op(V); fi
od:
R;# Robert Israel, Jun 15 2025

select( {is_A384095(n)=n%10 && sumdigits(n^2)==9 && !bittest(36938, fromdigits(Set(digits(n))))}, [1..10^5])

cp's OEIS Frontend

A033819 Trimorphic numbers: n^3 ends with n. Also m-morphic numbers for all m > 5 such that m-1 is not divisible by 10 and m == 3 (mod 4).

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A224474 (2*16^(5^n) - 1) mod 10^n: a sequence of trimorphic numbers ending in 1.

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A224476 (2*16^(5^n) + (10^n)/2 - 1) mod 10^n: a sequence of trimorphic numbers ending (for n > 1) in 1.

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A301912 Numbers k such that the decimal representation of k ends that of the sum of the first k cubes.

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A181607 Numbers n with k digits such that n^2 == 1 (mod 10^k).

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A384094 Numbers whose square has digit sum 9 and no trailing zero.

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A384095 Numbers other than {10^a + 10^b + 1} and {10^a + 5*10^b, min(a, b) = 0} whose square has digit sum 9 and no trailing zero.

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