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 1-3 of 3 results.

A306265 Terms in A323711 such that deleting any existing 9 or 0 digit in decimal notation does not result in a term of A323711.

Original entry on oeis.org

142857, 285714, 15623784, 15843762, 17438256, 17562438, 18243756, 21584376, 23784156, 24375618, 24381756, 25617438, 137965842, 139657842, 157836042, 157836204, 157839642, 157840362, 157842036, 157860342, 157862034, 157963842, 158379642, 159637842, 160357842
Offset: 1

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Author

Chai Wah Wu, Feb 01 2019

Keywords

Comments

Terms in A323711 can be generated by inserting 9's and 0's under certain conditions (see comment in A323711). These are the terms that are not formed this way and can be considered primitive terms of A323711. The first term containing the digit 0 is a(15) = 157836042 and the first term containing the digit 9 is a(13) = 137965842.
Terms in A323711 without the digits 0 and 9 are in the sequence.

Examples

			A323711(3) = 1402857. Deleting the digit 0 results in 142857 which is A323711(1). Similarly, deleting 0's and 9' from A323711(3)-A323711(20) results in another term of A323711 and a(3) = A323711(21).
		

Crossrefs

Cf. A323711.

A344436 Numbers k such that k, 2*k, 3*k, 4*k, 5*k and 6*k are anagrams and no digit of k is zero.

Original entry on oeis.org

142857, 1429857, 14299857, 142999857, 1429999857, 14299999857, 142857142857, 142999999857, 1428571429857, 1429857142857, 1429999999857, 14285714299857, 14298571429857, 14299857142857, 14299999999857, 137428291864557, 137464282918557, 142829186455737
Offset: 1

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Author

Bhupendra Kumar Singh, May 19 2021

Keywords

Comments

All terms are divisible by 9.
a(1) = 143*999 = 1287*111;
a(2) = 143*9999 = 1287*1111;
a(7) = 143*999000999 = 1287*111000111; etc.
a(n) = k is odd. Proof: If k is even then 5*k ends in 0, which is forbidden by definition. - David A. Corneth, May 22 2021

Examples

			142857, 1429857, and 14299857 are in the sequence:
.
      k        2*k       3*k       4*k       5*k       6*k
  --------  --------  --------  --------  --------  --------
    142857    285714    428571    571428    714285    857142
   1429857   2859714   4289571   5719428   7149285   8579142
  14299857  28599714  42899571  57199428  71499285  85799142
		

Crossrefs

Programs

  • PARI
    isok(k) = {my(d = vecsort(digits(k))); vecmin(d) && (d==vecsort(digits(2*k))) && (d==vecsort(digits(3*k))) && (d==vecsort(digits(4*k))) && (d==vecsort(digits(5*k))) && (d==vecsort(digits(6*k)));} \\ Michel Marcus, Jun 01 2021

Extensions

Data corrected by David A. Corneth, May 22 2021

A373177 Integers k such that 2k + 1 and 4k + 3 are anagrams of k.

Original entry on oeis.org

15632, 126530, 130265, 150632, 152630, 156329, 162530, 163025, 1265030, 1265300, 1265309, 1300265, 1302650, 1302659, 1500632, 1502630, 1506329, 1526300, 1526309, 1563299, 1566332, 1625030, 1625300, 1625309, 1630025, 1630250, 1630259, 1656332, 12650030
Offset: 1

Views

Author

Gonzalo Martínez, May 26 2024

Keywords

Comments

The terms of this sequence begin with decimal digits 1 or 2, otherwise 4*k + 3 has more digits than k and cannot be an anagram. The first term whose first digit is 2 is a(3931) = 2055114278.
This sequence has infinitely many terms, since 1500*10^m + 632 is a term for all positive integers m.
All terms == 8 (mod 9). - Hugo Pfoertner, May 27 2024

Examples

			15632 is a term, since 2*15632 + 1 = 31265 and 4*15632 + 3 = 62531 are both permutations of the digits of 15632.
		

Crossrefs

Programs

  • Maple
    filter:= proc(n) local L;
      L:= sort(convert(n,base,10));
      sort(convert(2*n+1,base,10))=L
      and sort(convert(4*n+3,base,10))=L
    end proc:
    R:= NULL: count:= 0:
    for d from 1 while count < 100 do
     for x from 10^(d-1) + 7 by 9 to (10^d-3)/4 while count < 100 do
       if filter(x) then R:= R,x; count:= count+1 fi
    od od:
    R; # Robert Israel, May 27 2024
  • Mathematica
    sid[n_] := Sort[IntegerDigits[n]]; Select[Range[13000000], sid[#] == sid[2*# + 1] == sid[4*# + 3] &] (* Amiram Eldar, May 27 2024 *)
  • PARI
    isok(k) = my(d=vecsort(digits(k))); (d == vecsort(digits(2*k+1))) && (d == vecsort(digits(4*k+3))); \\ Michel Marcus, May 28 2024
  • Python
    from itertools import count, islice
    def agen(): # generator of terms
        for e in count(1):
            for k in range(10**(e-1), 10**e//4):
                if sorted(str(k)) == sorted(str(2*k+1)) == sorted(str(4*k+3)):
                    yield k
    print(list(islice(agen(), 30))) # Michael S. Branicky, May 26 2024
    
Showing 1-3 of 3 results.