A171744 -id:A171744 - cp's OEIS frontend

A333965 Numbers k such that k * A171744(k) increases to a record.

1, 2, 5, 6, 8, 12, 14, 16, 18, 20, 25, 27, 32, 43, 45, 46, 52, 58, 70, 71, 77, 81, 91, 105, 109, 149, 158, 176, 240, 247, 297, 303, 401, 421, 431, 531, 536, 542, 543, 608, 617, 622, 640, 643, 667, 677, 685, 713, 720, 748, 751, 1028, 1085, 1203, 1282, 1320, 1466, 1600

Offset: 1

David A. Corneth, Aug 14 2020

			6 is in the sequence as A171744(6) = 22 and 6*22 = 132. This is a record; no n less than 6 produces a product of at least 132.

Robert Israel, Table of n, a(n) for n = 1..1064

Cf. A171744.

f:= proc(p) local t,k;
    t:= 1;
    for k from 1 do
      t:= t*p;
      if nops(convert(convert(t,base,10),set))=10 then return k fi;
    od
end proc:
R:= NULL: m:= 0: p:= 1: count:= 0:
for k from 1 while count < 100 do
  p:= nextprime(p);
  v:= k*f(p);
  if v > m then
    R:= R,k;
    m:= v;
    count:= count+1
  fi
od:
R; # Robert Israel, Aug 05 2024

With[{s = Array[Block[{p = Prime@ #, k = 1}, While[Min@ DigitCount[p^k] == 0, k++]; k #] &, 1600]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Aug 21 2020 *)

f(n) = {my(k=1, p=prime(n)); while(#Set(digits(p^k))<10, k++); k; } \\ A171744
lista(nn) = {my(m=0, x); for (n=1, nn, x = n*f(n); if (x >m, m = x; print1(n, ", ")););} \\ Michel Marcus, Jan 26 2021

A178303 Smallest multiple of 13 such that decimals digits 1, ..., k (k = 1, ..., 9) and 0 appear in any order.

Original entry on oeis.org

13, 182, 312, 2314, 14235, 125346, 1234675, 12348765, 123456879, 1023457968

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), May 24 2010, May 26 2010

Divisibility of a number N by 13: add the digits of N in alternate blocks of three from right to left, then subtract the two sums:

9th term: 123 456 879: (123 + 879) - 456 = 546, 546: 54 + (6 * 4) = 78 = 6 * 13.

			         1 * 13 =         13
        14 * 13 =        182
        24 * 13 =        312
       178 * 13 =       2314
      1095 * 13 =      14235
      9642 * 13 =     125346
     94975 * 13 =    1234675
    949905 * 13 =   12348765
   9496683 * 13 =  123456879
  78727536 * 13 = 1023457968

Faith Javane, Zahlenmystik: Das Handbuch der Numerologie, Goldmann - Arkana, Frankfurt, 1995
Helmut Kracke, Mathe - musische Knobelisken. Tüfteleien für Tüftler und Laien, Dümmler, Bonn, 1982
Hugo Steinhaus, Kaleidoskop der Mathematik, Deutscher Verlag der Wissenschaften, Berlin, 1959

Cf. A171768, A171744.

A360656 Least k such that the decimal representation of 2^k contains all possible n-digit strings.

Original entry on oeis.org

68, 975, 16963, 239697, 2994863

Offset: 1

Hans Havermann, Feb 15 2023

			2^68 = 295147905179352825856 is the least power of 2 containing all ten decimal digits, so a(1) = 68 = A171744(1).
2^975 is the least power of 2 containing all 100 two-digit strings, so a(2) = 975.

Cf. A171744, A360623.

def a(n, starte=0):
    e, p2, t = starte, 2**starte, 10**n
    while True:
        s2, ss = str(p2), set()
        for i in range(len(s2)-n+1):
            ss.add(s2[i:i+n])
            if len(ss) == t:
                return e
        e += 1
        p2 *= 2
print([a(n) for n in range(1, 4)]) # Michael S. Branicky, Feb 22 2023

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A333965 Numbers k such that k * A171744(k) increases to a record.

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A178303 Smallest multiple of 13 such that decimals digits 1, ..., k (k = 1, ..., 9) and 0 appear in any order.

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A360656 Least k such that the decimal representation of 2^k contains all possible n-digit strings.

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Python