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Search: a115922 -id:a115922
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a(n) is the number of k such that k and n*k, taken together, are pandigital.
+10
11
0, 48, 6, 8, 12, 0, 1, 16, 3, 0, 0, 1, 1, 6, 3, 1, 19, 6, 4, 12, 0, 3, 3, 4, 3, 9, 2, 1, 8, 2, 0, 16, 1, 3, 14, 0, 3, 7, 3, 4, 0, 3, 1, 13, 4, 1, 6, 0, 1, 12, 0, 2, 28, 1, 4, 6, 1, 3, 6, 3, 0, 28, 1, 1, 10, 1, 1, 4, 5, 7, 0, 3, 3, 11, 0, 2, 8, 1, 1, 46, 0, 0, 5, 3, 1, 7, 5, 6, 8, 3, 0, 13, 2, 3
OFFSET
1,2
COMMENTS
There are 1549586 nonzero terms in a(n). The largest n for which a(n) > 0 is 987654320. The largest a(n) is a(2) = 48. - Chai Wah Wu, May 24 2015
EXAMPLE
a(7)=1 since there is only one number, k=14076, such that k and 7*k=98532.
a(9)=3 since there are 3 such numbers: 10638, 10647 and 10836.
PROG
(Python)
from itertools import permutations
l = {}
for d in permutations('0123456789', 10):
....if d[0] != '0':
........for i in range(9):
............if d[i+1] != '0':
................q, r = divmod(int(''.join(d[:i+1])), int(''.join(d[i+1:])))
................if not r:
....................if q in l:
........................l[q] += 1
....................else:
........................l[q] = 1
A115927_list = [0]*max(l)
for d in l:
....A115927_list[d-1] = l[d] # Chai Wah Wu, May 24 2015
KEYWORD
nonn,base
AUTHOR
Giovanni Resta, Feb 06 2006
STATUS
approved
Numbers k such that k and 8*k, taken together, are pandigital.
+10
5
10459, 10469, 10537, 10579, 10592, 10674, 10679, 10742, 10794, 10932, 10942, 10953, 10954, 12073, 12307, 12345
OFFSET
1,1
EXAMPLE
10459 and 83672 = 10459*8 together contain all the 10 digits once.
MATHEMATICA
f[a_]:=Join[IntegerDigits[a], IntegerDigits[8*a]]; Select[Range[10000, 100000], ContainsExactly[f[#], {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}]&&Length[f[#]]==10&] (* James C. McMahon, Jul 30 2024 *)
KEYWORD
nonn,base,fini,full
AUTHOR
Giovanni Resta, Feb 06 2006
STATUS
approved
Numbers k such that k and 3*k, taken together, are pandigital.
+10
5
16794, 17694, 20583, 23058, 30582, 32058
OFFSET
1,1
EXAMPLE
16794 and 50382=16794*3 together contain all the 10 digits once.
MATHEMATICA
pdQ[n_]:=Sort[Flatten[Join[{IntegerDigits[n], IntegerDigits[3n]}]]] == Range[0, 9]; Select[Range[10000, 99999], pdQ] (* Harvey P. Dale, Jul 07 2012 *)
PROG
(PARI) {for(n=10234, 49876, #Set(digits(n))==5||next; #Set(digits(n*3))==5 && #Set(concat(digits(n), digits(n*3)))==10 && print1(n", "))} \\ M. F. Hasler, Feb 08 2014
KEYWORD
nonn,base,fini,full
AUTHOR
Giovanni Resta, Feb 06 2006
STATUS
approved
Numbers k such that k and 4*k, taken together, are pandigital.
+10
5
15237, 17235, 17352, 20394, 20439, 21735, 23517, 23715
OFFSET
1,1
EXAMPLE
15237 and 60948=15237*4 together contain all the 10 digits once.
MATHEMATICA
Select[Range[10234, 24987], Sort[Join[IntegerDigits[4 #], IntegerDigits[ #]]] == Range[0, 9]&] (* Harvey P. Dale, Jun 01 2017 *)
PROG
(PARI) {for(n=10234, 24987, #Set(digits(n))==5||next; #Set(digits(n*4))==5 && #Set(concat(digits(n), digits(n*4)))==10 && print1(n", "))} \\ M. F. Hasler, Feb 08 2014
KEYWORD
nonn,base,fini,full
AUTHOR
Giovanni Resta, Feb 06 2006
STATUS
approved
Numbers k such that k and 5*k, taken together, are pandigital.
+10
5
13458, 13584, 13854, 14538, 14586, 14658, 15384, 15846, 15864, 18534, 18546, 18654, 20697, 20769, 20937, 20967, 20973, 26097, 26956, 26958, 26970, 27096, 27609, 27690, 29076, 29356, 29358, 29370, 29536, 29538, 29566, 29586, 29607, 29670, 29706, 29730, 30972, 32097, 32956
OFFSET
1,1
EXAMPLE
13458 and 67290=13458*5 together contain all the 10 digits once.
MATHEMATICA
pdQ[n_]:=Module[{idn=IntegerDigits[n], idn5=IntegerDigits[5n]}, Union[Join[idn, idn5]]==Range[0, 9]]
Select[Range[50000], pdQ] (* Harvey P. Dale, Mar 06 2011 *)
KEYWORD
nonn,base,fini,full
AUTHOR
Giovanni Resta, Feb 06 2006
EXTENSIONS
More terms from Harvey P. Dale, Mar 06 2011
STATUS
approved

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