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.

Showing 1-4 of 4 results.

A228381 Unabridged sub-Kaprekar numbers (A118936, but allowing powers of ten).

Original entry on oeis.org

10, 11, 78, 100, 101, 287, 364, 1000, 1001, 1078, 1096, 1287, 1364, 10000, 10001, 11096, 18183, 100000, 100001, 118183, 336634, 1000000, 1000001, 1336634, 2727274, 10000000, 10000001, 12727274, 19138757, 23529412, 25974026, 97744361, 100000000, 100000001, 120879122
Offset: 1

Views

Author

Hans Havermann, Aug 21 2013

Keywords

Comments

Square roots of A228103.
Excluding powers-of-ten and powers-of-ten-plus-one, what remains may be arranged into pairs (x,y), y>x, where y-x is a power of ten. The x terms correspond to A118938.

Examples

			10^2 = (10-0)^2.
11^2 = (12-1)^2.
78^2 = (6-084)^2.

Links

Crossrefs

Cf. A045913, A118936, A118938, A228103.

Programs

  • Mathematica
    k=3; While[k<10^8, k++; s=k^2; d=IntegerDigits[s]; l=Length[d]; Do[a=FromDigits[Take[d, {1, i}]]; b=FromDigits[Take[d, {i+1, l}]]; If[k==Abs[a-b], Print[k]], {i, l-1}]]
  • PARI
    lista(nn) = my(d, s, t=1, v=List([])); while(t(x>1&&x<=nn), v)); \\ Jinyuan Wang, Jan 02 2025

A384538 Positive integers k >= 10 for which for every pair of nonempty substrings that concatenate to give k one substring divides the other.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 30, 31, 33, 36, 39, 40, 41, 42, 44, 48, 50, 51, 55, 60, 61, 62, 63, 66, 70, 71, 77, 80, 81, 82, 84, 88, 90, 91, 93, 99, 100, 101, 102, 105, 110, 111, 120, 121, 122, 123, 124, 126, 130, 131, 140, 141
Offset: 1

Views

Author

Felix Huber, Jun 09 2025

Keywords

Examples

			324 is a term because 3 divides 24 and 4 divides 32.
105 is a term because 1 divides 05 = 5 and 5 divides 10.
2500 is a term because 2 divides 500. 25 divides 00 = 0 and 250 divides 0.
104 is not a term: Although 1 divides 04 = 4, 4 does not divide 10.

Links

Crossrefs

Supersequence of A384539.
Cf. A102766, A228103.

Programs

  • Maple
    A384538:=proc(n)
    option remember;
    local i,j,k,p,m,q,L;
    if n=1 then
        10
    else
        for k from procname(n-1)+1 do
            L:=ListTools:-Reverse(convert(k,'base',10));
            m:=length(k)-1;
            for j to m do
                p:=parse(cat(seq(L[i],i=1..j)));
                q:=k-p*10^(m+1-length(p));
                if q mod p<>0 and p mod q<>0 then
                    break
            	 elif j=m then
                    return k
                fi
            od
        od
    fi;
end proc;	
seq(A384538(n),n=1..62);
  • PARI
    isok(k) = my(nb=logint(k, 10), d=10); for (i=1, nb, my(sa = k%d, sb=k\d); if ((sa % sb) && (sb % sa), return(0)); d *= 10;); return(1); \\ Michel Marcus, Jun 19 2025
  • Python
    def c(k, m): return (k > 0 and m%k == 0) or (m > 0 and k%m == 0)
def ok(n):
    s = str(n)
    return n > 9 and all(c(int(s[:i]), int(s[i:])) for i in range(1, len(s)))
print([k for k in range(150) if ok(k)]) # Michael S. Branicky, Jun 17 2025

A384539 Zeroless positive integers k for which for every pair of nonempty substrings that concatenate to give k one substring divides the other.

Original entry on oeis.org

11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 24, 26, 28, 31, 33, 36, 39, 41, 42, 44, 48, 51, 55, 61, 62, 63, 66, 71, 77, 81, 82, 84, 88, 91, 93, 99, 111, 121, 122, 123, 124, 126, 131, 141, 142, 147, 151, 153, 155, 161, 162, 164, 168, 171, 181, 182, 183, 186, 189
Offset: 1

Views

Author

Felix Huber, Jun 09 2025

Keywords

Comments

It is conjectured that this sequence is finite and contains 132 terms.
From David A. Corneth, Jun 19 2025: (Start)
A term with at least 3 digits cannot end in 14. Else we can split it in 10*k + 1 and 4 where k >= 1. So we'd need 4 | 10*k + 1. A contradiction.
If a term with at least 5 digits ends in 89 then it is 100*k + 89 where k >= 1000.
We'd need 9 | 1000*k + 8, 89 | 100*k so 89 | k. This largely restrict the possibilities for k. (End)
a(133) > 10^11, if it exists. - Michael S. Branicky, Jun 18 2025
a(133) > 10^25, if it exists. - David A. Corneth, Jun 19 2025

Examples

			168 is a term because 1 divides 68 and 8 divides 16.
4284 is a term because 4 divides 284, 42 divides 84 and 4 divides 428.
222222 is a term because 2 divides 22222, 22 divides 2222, 222 divides 222 and vice versa.

Links

Crossrefs

Subsequence of A052382.
Subsequence of A384538.
Cf. A102766, A228103.

Programs

  • Maple
    A384539:=proc(n)
    option remember;
    local i,j,k,p,m,q,L;
    if n=1 then
        11
    else
        for k from procname(n-1)+1 do
            L:=ListTools:-Reverse(convert(k,'base',10));
            if not member(0,L) then
	        m:=length(k)-1;
	        for j to m do
	            p:=parse(cat(seq(L[i],i=1..j)));
	            q:=k-p*10^(m+1-length(p));
	            if max(p,q) mod min(p,q)<>0 then
	                break
	            elif j=m then
	                return k
	            fi
	        od
	    fi
        od
    fi;
end proc;	
seq(A384539(n),n=1..60);
  • Python
    def c(k, m): return m%k == 0 or k%m == 0
def ok(n):
    s = str(n)
    return n > 9 and "0" not in s and all(c(int(s[:i]), int(s[i:])) for i in range(1, len(s)))
print([k for k in range(200) if ok(k)]) # Michael S. Branicky, Jun 18 2025

A379591 Numbers k whose base-10 digits can be split into two parts, q and r, with k = (|q-r|)^3.

Original entry on oeis.org

1000, 456533, 474552, 1000000, 69426531, 1000000000, 1000000000000, 1000000000000000, 1000000000000000000, 1000000000000000000000, 1000000000000000000000000, 1000000000000000000000000000, 1000000000000000000000000000000, 1000000000000000000000000000000000
Offset: 1

Views

Author

Travis Vasquez, Dec 26 2024

Keywords

Examples

			1000 = (|10-00|)^3.
456533 = (|456-533|)^3.
1000000 = (|100-0000|)^3.

Crossrefs

Cf. A102766, A228103.

Programs

  • Maple
    filter:= proc(x) local m;
for m from 1 to ilog10(x) do
  if x = abs((x mod 10^m) - floor(x/10^m))^3 then return true fi
od;
false
end proc:
select(filter, [seq(i^3,i=1..10^7)]); # Robert Israel, Jan 01 2025
  • PARI
    lista(nn) = my(d, m, q, s, t=1, v=List([])); while(t1&&gcd(f[1], (t+1)/f[1])==1, d=[x[1]^x[2]|x<-f[2]~]; forvec(x=vector(#d, i, [0, 1]), s=lift(chinese(chinese(vector(#d, i, Mod((-1)^x[i], d[i]))), Mod(0, (t+1)/f[1]))); q=(s^3-s)/(t+1); if(q>0&&s+q=s&&q-sx<=nn, v)); \\ Jinyuan Wang, Jan 03 2025

Extensions

a(11)-a(12) from Michael S. Branicky, Jan 01 2025
a(13)-a(14) from Jinyuan Wang, Jan 03 2025
Showing 1-4 of 4 results.

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