A084006
Squares arising as a concatenation of k and 9's complement of k.
Original entry on oeis.org
36, 81, 1089, 4356, 9801, 110889, 443556, 998001, 11108889, 44435556, 99980001, 1111088889, 4444355556, 9999800001, 111110888889, 444443555556, 999998000001, 11111108888889, 44444435555556, 99999980000001
Offset: 1
Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 23 2003
1089 = 33^2 is a concatenation of 10 and 89, 10+89 = 99.
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f:= proc(k) local F,x,p,t;
p:= 10^k-1;
F:= select(t -> t[2]::odd, ifactors(p)[2]);
x:= mul(t[1],t=F);
seq(j^2*x*p, j=ceil(sqrt((10^(k-1)+1)/x))..floor(sqrt(p/x)))
end proc:
map(f, [$1..20]); # Robert Israel, Sep 09 2020
-
from itertools import count, islice
from math import prod, isqrt
from sympy import factorint
def A084006_gen(): # generator of terms
for l in count(1):
m = 10**l-1
x = prod(p for p, e in factorint(m).items() if e&1)
yield from (j**2*x*m for j in range(isqrt(10**(l-1)//x)+1,isqrt(m//x)+1))
A084006_list = list(islice(A084006_gen(),20)) # Chai Wah Wu, Mar 20 2025
A084004
Squares obtained as a concatenation of k and 10's complement of k.
Original entry on oeis.org
64, 7921, 9604, 164836, 351649, 996004, 19758025, 20647936, 29757025, 30846916, 97990201, 99960004, 1203187969, 1975180249, 3086469136, 4265657344, 8143618564, 9999600004, 115644884356, 131504868496, 132231867769, 140039859961, 173879826121, 339900660100, 391600608400
Offset: 1
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b(n)={my(k=logint(n,10)+1); (n+1)*10^k - n}
{ for(k=1, 10^6, my(x=b(k)); if(issquare(x), print1(x, ", "))) } \\ Andrew Howroyd, Sep 22 2024
Original entry on oeis.org
6, 9, 33, 66, 99, 333, 666, 999, 3333, 6666, 9999, 33333, 66666, 99999, 333333, 666666, 999999, 3333333, 6666666, 9999999, 33333333, 66666666, 99999999, 333333333, 444444444, 555555555, 666666666, 777777777, 888888888, 999999999
Offset: 0
Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 23 2003
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from itertools import count, islice
from math import prod, isqrt
from sympy import factorint
def A084007_gen(): # generator of terms
for l in count(1):
m = 10**l-1
x = prod(p for p, e in factorint(m).items() if e&1)
y = isqrt(x*m)
yield from (j*y for j in range(isqrt(10**(l-1)//x)+1,isqrt(m//x)+1))
A084007_list = list(islice(A084007_gen(),30)) # Chai Wah Wu, Mar 20 2025
Showing 1-3 of 3 results.
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