cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 51-56 of 56 results.

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

Views

Author

M. F. Hasler, Jun 15 2025

Keywords

Comments

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

Crossrefs

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}).

Programs

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

A279771 Numbers n such that the sum of digits of 11n equals 11.

Original entry on oeis.org

19, 28, 37, 46, 55, 64, 73, 82, 190, 280, 370, 460, 550, 640, 730, 820, 919, 928, 937, 946, 955, 964, 973, 982, 991, 1819, 1828, 1837, 1846, 1855, 1864, 1873, 1882, 1891, 1900, 2728, 2737, 2746, 2755, 2764, 2773, 2782, 2791, 2800, 3637, 3646, 3655, 3664
Offset: 1

Views

Author

M. F. Hasler, Dec 23 2016

Keywords

Comments

Inspired by A088404 = A069537/2 through A088410 = A069543/8.

Crossrefs

Cf. A007953 (digital sum), Digital sum of m*n equals m: A088404 = A069537/2, A088405 = A052217/3, A088406 = A063997/4, A088407 = A069540/5, A088408 = A062768/6, A088409 = A063416/7, A088410 = A069543/8.
Cf. A005349 (Niven or Harshad numbers), A245062 (arranged in rows by digit sums).
Numbers with given digital sum: A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225 (14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A279772 (sumdigits(2n) = 4), A279773 (sumdigits(3n) = 6), A279774 (sumdigits(4n) = 8), A279775 (sumdigits(5n) = 10), A279776 (sumdigits(6n) = 12), A279770 (sumdigits(7n) = 14), A279768 (sumdigits(8n) = 16), A279769 (sumdigits(9n) = 18), A279777 (sumdigits(9n) = 27).

Programs

  • Mathematica
    Select[Range@ 3664, Total@IntegerDigits[11 #] == 11 &] (* Michael De Vlieger, Dec 23 2016 *)
  • PARI
    is(n)=sumdigits(11*n)==11

A306855 Primes of the form 10^i + 10^j - 1.

Original entry on oeis.org

19, 109, 199, 1009, 1999, 10009, 10099, 100999, 199999, 1000099, 1000999, 19999999, 1000000009, 1000009999, 1000099999, 1009999999, 10000000999, 10000099999, 10999999999, 100999999999, 1000000009999, 1000000999999, 1099999999999, 10000000000099, 10009999999999, 100000000000099, 100000009999999
Offset: 1

Views

Author

Robert Israel, Mar 13 2019

Keywords

Comments

Primes p such that p+1 is in A052216.
Primes of the following form in base 10: 1, followed by 0 or more 0's, then 1 or more 9's.

Links

Crossrefs

Cf. A052216. Subsequence of A199329.

Programs

  • Maple
    select(isprime, [seq(seq(10^n+10^m-1, m=1..n),n=1..15)]);
  • Mathematica
    Select[Flatten[Table[FromDigits[Join[PadRight[{1},n,0],PadRight[{},m,9]]],{n,20},{m,20}]],PrimeQ]//Sort (* Harvey P. Dale, May 15 2021 *)

A344214 Numbers k such that repeated iterations of f(m) = (digsum(f(m-1)))^2 + 1 starting from f(1) = k will eventually yield 5 before any other single-digit number.

Original entry on oeis.org

5, 11, 15, 18, 19, 20, 24, 27, 28, 33, 36, 37, 39, 42, 45, 46, 48, 51, 54, 55, 57, 60, 63, 64, 66, 69, 72, 73, 75, 78, 81, 82, 84, 87, 90, 91, 93, 96, 99, 101, 105, 108, 109, 110, 114, 117, 118, 123, 126, 127, 129, 132, 135, 136, 138, 141, 144, 145, 147, 150, 153, 154
Offset: 1

Views

Author

Joseph Brown, May 11 2021

Keywords

Comments

f(x) = digsum(x)^2 + 1 < x for x >= 400, and all iterations terminate in a single digit or lead to the cycle 65 -> 122 -> 26. - Michael S. Branicky, May 14 2021

Examples

			11 is in the list because (1+1)^2 + 1 = 5.
12 is not in the list because repeatedly iterating the function starting with f(1) = 12 will yield 2 before 5.
13 is not in the list because it will never yield 5. Specifically, 13 -> 17 -> 65 -> 122 -> 26 -> 65 -> ... .

Crossrefs

Subsequence of A344208.
Cf. A007953, A052216, A052224, A344208, A118881.

Programs

  • Mathematica
    Select[Range@100,Last@NestWhileList[Total[IntegerDigits@#]^2+1&,#,#>10&&#!=26&]==5&] (* Giorgos Kalogeropoulos, May 12 2021 *)
  • Python
    def f(n):
    s = 0
    while n > 0:
        s, n = s+n%10, n//10
    return s*s+1
n, pota = 0, 0
while n < 62:
    a, repf, i, ii = pota, 0, 0, 4
    while a > 9 and a != repf:
        a, i = f(a), i+1
        if i == ii:
            repf, ii = a, 2*ii
    if a == 5:
        n = n+1
        print(pota, end = ", ")
    pota = pota+1 # A.H.M. Smeets, May 13 2021
  • Python
    def f(x): return sum(map(int, str(x)))**2 + 1
def ok(n):
  iter = n  # set to f(n) if number of iterations must be >= 1
  while iter > 9:
    if iter in {65, 122, 26}: return False
    iter = f(iter)
  return iter == 5
print(list(filter(ok, range(1, 155)))) # Michael S. Branicky, May 19 2021

A356520 Numbers k such that A000005(A007953(k)) = A007953(k).

Original entry on oeis.org

1, 2, 10, 11, 20, 100, 101, 110, 200, 1000, 1001, 1010, 1100, 2000, 10000, 10001, 10010, 10100, 11000, 20000, 100000, 100001, 100010, 100100, 101000, 110000, 200000, 1000000, 1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000, 10000000, 10000001
Offset: 1

Views

Author

Ctibor O. Zizka, Aug 10 2022

Keywords

Comments

Union of A011557 and A052216. I.e., numbers with digital sum 1 or 2. - David A. Corneth, Aug 10 2022

Examples

			k = 101; A000005(A007953(101)) = A007953(101) = 2, thus k = 101 is in the sequence.

Crossrefs

Cf. A000005, A007953, A011557, A052216, A101318, A306509, A356061.

Programs

  • Mathematica
    Select[Range[1,10000001], Plus @@ IntegerDigits[#] < 3 &] (* Amiram Eldar, Aug 10 2022 *)
  • PARI
    isok(k) = my(s=sumdigits(k)); numdiv(s) == s; \\ Michel Marcus, Aug 10 2022
  • PARI
    is(n) = my(s = sumdigits(n)); s == 1 || s == 2 \\ David A. Corneth, Aug 10 2022
  • Python
    from itertools import count, islice
def agen():
    for i in count(0):
        yield from [10**i] + [10**i + 10**j for j in range(i+1)]
print(list(islice(agen(), 37))) # Michael S. Branicky, Aug 10 2022

A055378 Table read by antidiagonals: T(n,k) = n^trinv(k)+n^(k-((trinv(k)*(trinv(k)-1))/2)) where trinv (k) = floor((1+sqrt(1+8*k))/2) and with 0^0 = 1.

Original entry on oeis.org

2, 1, 2, 0, 2, 2, 1, 2, 3, 2, 0, 2, 4, 4, 2, 0, 2, 5, 6, 5, 2, 1, 2, 6, 10, 8, 6, 2, 0, 2, 8, 12, 17, 10, 7, 2, 0, 2, 9, 18, 20, 26, 12, 8, 2, 0, 2, 10, 28, 32, 30, 37, 14, 9, 2, 1, 2, 12, 30, 65, 50, 42, 50, 16, 10, 2, 0, 2, 16, 36, 68, 126, 72, 56, 65, 18, 11, 2, 0, 2, 17, 54, 80, 130
Offset: 0

Views

Author

Henry Bottomley, Jun 22 2000

Keywords

Examples

			a(50) = T(5,4) = 5^2+5^1 = 30

Crossrefs

Rows include A010054 (apart from initial term), A007395 and A048645 (offset). Subsequent rows are sums of two powers of a given number and also rewritings of A052216 from a particular base to base 10. Columns include A007395, A000027, A005843, A002522, A002378, A001105, A001093, A034262, A011379, A033431, A002523.

Formula

T(n, k) = n^A025581(k)+n^A002262(k)
Previous Showing 51-56 of 56 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.