A114258
Numbers k such that k^2 contains exactly 2 copies of each digit of k.
Original entry on oeis.org
72576, 406512, 415278, 494462, 603297, 725760, 3279015, 4065120, 4152780, 4651328, 4915278, 4927203, 4944620, 4972826, 4974032, 4985523, 4989323, 5002245, 5016125, 6032970, 6214358, 6415002, 6524235, 7257600, 9883667
Offset: 1
72576 is in the sequence since its square 5267275776 contains four 7's, two 2's, two 5's and two 6's.
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from math import isqrt
from itertools import count, islice
def A114258_gen(): # generator of terms
for l in count(1):
a = isqrt(10**((l<<1)-1))
if (a9:=a%9):
a -= a9
for b in range(a,10**l,9):
for c in (0,2):
k = b+c
if sorted(str(k)*2)==sorted(str(k**2)):
yield k
A114258_list = list(islice(A114258_gen(),20)) # Chai Wah Wu, Feb 27 2024
A370675
Number of unordered pairs of n-digit numbers k1, k2 such that their product has the same multiset of digits as in both k1 and k2 together.
Original entry on oeis.org
0, 7, 156, 3399, 112025, 4505706, 213002162
Offset: 1
For n=2 the a(2)=7 solutions are:
15 * 93 = 1395
21 * 60 = 1260
21 * 87 = 1827
27 * 81 = 2187
30 * 51 = 1530
35 * 41 = 1435
80 * 86 = 6880
Cf.
A114258,
A370676 (number of such pairs with possibly unequal number of digits).
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a370675(n) = {my (np=0, n1=10^(n-1), n2=10*n1-1); for (k1=n1, n2, my(s1=digits(k1)); for (k2=k1, n2, my (s2=digits(k2)); my(sp=digits(k1*k2)); if (#s1+#s2==#sp && vecsort(concat(s1,s2)) == vecsort(sp), np++))); np} \\ Hugo Pfoertner, Feb 26 2024
A370678
a(n) is the number of pairs x <= y of n-digit numbers such that the number of distinct digits in their product is less than in their concatenation.
Original entry on oeis.org
10, 1395, 147718, 15187437, 1530456465, 152653821364
Offset: 1
a(1) = 10: 8 products 1*2, ..., 1*9, 2*3, 2*4 with 1 digit in x*y and 2 digits in x|y.
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\\ returns [number of products, [a(n), A370679(n), A370680(n)]]
a370678_80(n) = {my (m=0, c=vector(3), n1=10^(n-1), n2=10*n1-1); for (k1=n1, n2, my (s1=digits(k1)); for (k2=k1, n2, my (s2=digits(k2), cs=#Set(digits(k1*k2)), d=cs-#Set(concat(s1,s2))); c[sign(d)+2]++; m++)); [m,c]}
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def A370678(n):
a = 10**(n-1)
b, c = 10*a, 0
for x in range(a,b):
s = set(str(x))
for y in range(x,b):
if len(s|set(str(y))) > len(set(str(x*y))):
c += 1
return c # Chai Wah Wu, Feb 28 2024
A370679
a(n) is the number of pairs x <= y of n-digit numbers such that the number of distinct digits in their product is the same as in their concatenation.
Original entry on oeis.org
29, 1674, 136854, 12082393, 1136370471, 111392993465
Offset: 1
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See A370678.
-
def A370679(n):
a = 10**(n-1)
b, c = 10*a, 0
for x in range(a,b):
s = set(str(x))
for y in range(x,b):
if len(s|set(str(y))) == len(set(str(x*y))):
c += 1
return c # Chai Wah Wu, Feb 28 2024
A370680
a(n) is the number of pairs x <= y of n-digit numbers such that the number of distinct digits in their product is greater than in their concatenation.
Original entry on oeis.org
6, 1026, 120878, 13234670, 1383218064, 140953635171
Offset: 1
a(1) = 6: 6 squares 4*4, ..., 9*9 with 2 distinct digits in x*y = 16, 25, ..., and 1 digit in their concatenation.
-
See A370678.
-
def A370680(n):
a = 10**(n-1)
b, c = 10*a, 0
for x in range(a,b):
s = set(str(x))
for y in range(x,b):
if len(s|set(str(y))) < len(set(str(x*y))):
c += 1
return c # Chai Wah Wu, Feb 28 2024
Showing 1-5 of 5 results.
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