A115426
Numbers k such that the concatenation of k with k+2 gives a square.
Original entry on oeis.org
7874, 8119, 69476962, 98010199, 108746354942, 449212110367, 544978035127, 870501316279, 998001001999, 1428394731903223, 1499870932756487, 1806498025502498, 1830668275445687, 1911470478658759, 2255786189655202
Offset: 1
8119//8121 = 9011^2, where // denotes concatenation.
98010199//98010200 = 99000100 * 99000102.
98010199//98010197 = 99000099 * 99000103.
Cf.
A030465,
A102567,
A115427,
A115428,
A115429,
A115430,
A115431,
A115432,
A115433,
A115434,
A115435,
A115436,
A115437.
-
from itertools import count, islice
from sympy import sqrt_mod
def A115426_gen(): # generator of terms
for j in count(0):
b = 10**j
a = b*10+1
for k in sorted(sqrt_mod(2,a,all_roots=True)):
if a*(b-2) <= k**2-2 < a*(a-3):
yield (k**2-2)//a
A115426_list = list(islice(A115426_gen(),40)) # Chai Wah Wu, Feb 20 2024
A115429
Numbers k such that the concatenation of k with k+8 gives a square.
Original entry on oeis.org
6001, 6433, 11085116, 44496481, 96040393, 115916930617, 227007035017, 274101929528, 434985419768, 749978863753, 996004003993, 1365379857457948, 1410590590957816, 1762388551055953, 2307340946901148, 2700383162251217
Offset: 1
6001//6009 = 7747^2, where // denotes concatenation.
96040393//96040400 = 98000200 * 98000202.
96040393//96040397 = 98000199 * 98000203.
96040393//96040392 = 98000198 * 98000204.
Cf.
A030465,
A102567,
A115426,
A115437,
A115428,
A115430,
A115431,
A115432,
A115433,
A115434,
A115435,
A115436,
A115440.
A115430
Numbers k such that the concatenation of k with k+9 gives a square.
Original entry on oeis.org
216, 287, 515, 675, 1175, 4320, 82640, 960795, 1322312, 4049591, 16955015, 34602080, 171010235, 181964891, 183673467, 187160072, 321920055, 326530616, 328818032, 343942560, 470954312, 526023432, 528925616, 534830855
Offset: 1
Cf.
A030465,
A102567,
A115426,
A115437,
A115428,
A115429,
A115431,
A115432,
A115433,
A115434,
A115435,
A115436,
A115441.
A116107
Numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 7.
Original entry on oeis.org
52, 63716866, 48793687600063875363014809897052, 60020753655608135708762056127156, 60446518621981165303188950156776, 71135436903815748345367595855336, 72876856643103028189103298533248
Offset: 1
63716866//63716864 = 79822843 * 79822848, where // denotes concatenation.
A116163
Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 1.
Original entry on oeis.org
1, 5, 61, 65479, 84289, 106609, 225649, 275599, 453589, 1869505, 2272555, 2738291, 3221951, 1667833021, 2475062749, 2525062249, 3500010739, 9032526511, 9225507211, 1753016898055, 1860598847399, 3233666953849, 3379207972471, 5632076031055, 5823639407489
Offset: 1
1 is a member since 12 = 3*4; also 10 = 2*5.
5 is a member since 56 = 7*8; also 54 = 6*9.
-
Union @@ ((y /. List@ ToRules@ Reduce[x (x+1) == 10^# y +y+1 && x>0 && 10^(#-1) <= y+1 < 10^#, {x,y}, Integers]) & /@ Range[13] /. y->{}) (* Giovanni Resta, Jul 08 2018 *)
A116171
Numbers k such that k concatenated with k+2 gives the product of two numbers which differ by 3.
Original entry on oeis.org
8, 52, 63716866, 48793687600063875363014809897052, 60020753655608135708762056127156, 60446518621981165303188950156776, 71135436903815748345367595855336, 72876856643103028189103298533248
Offset: 1
63716866//63716868 = 79822844 * 79822847, where // denotes concatenation.
63716866//63716870 = 79822845 * 79822846.
A116179
Numbers k such that k concatenated with k+3 gives the product of two numbers which differ by 5.
Original entry on oeis.org
1, 3, 81, 1353, 3997, 7723, 23761, 26271, 76771, 1415683, 3890571, 8495497, 1066870443, 1239366513, 4198438981, 4534273891, 6502317141, 6918679731, 2199164200036329043, 2820114781174460091, 5500888421709400741
Offset: 1
A116185
Numbers k such that k concatenated with k+4 gives the product of two numbers which differ by 3.
Original entry on oeis.org
150, 186, 324, 376, 666, 2046, 3000, 82650, 100384, 466716, 1322316, 4049584, 67820074, 110003884, 135734074, 156502836, 196043286, 213017754, 238849000, 261405396, 289940826, 310507774, 365294050, 398891964, 446667216
Offset: 1
-
cc:=proc(x,y) local s: s:=proc(m) nops(convert(m,base,10)) end: x*10^s(y)+y: end: a:=proc(n) if type(sqrt(9+4*cc(n,n+4)),integer) then n else fi end: seq(a(n),n=1..500000); # very slow; cc yields the concatenation of x and y; - Emeric Deutsch, May 05 2007
A116309
Numbers k such that k*(k+3) gives the concatenation of two numbers m and m+3.
Original entry on oeis.org
40, 58, 32262232, 67737766, 79321056, 3341093417798787499093, 3861488851737861033961, 4747922651210186579787, 5252077348789813420211, 6138511148262138966037, 6658906582201212500905, 7232275368591793618231
Offset: 1
79321056 * 79321059 = 62918301//62918304, where // denotes concatenation.
-
As:= {}:
for m from 2 to 62 do
acands:= map(t -> rhs(op(t)), [msolve(a*(a+3)=3, 10^m+1)]);
bcands:= map(t -> t*(t+3) mod 10^m, acands);
good:= select(t -> bcands[t]>=10^(m-1), [$1..nops(acands)]);
As:= As union convert(acands[good],set);
od:
sort(convert(As,list)); # Robert Israel, Aug 20 2019
A116113
Numbers k such that k concatenated with k-7 gives the product of two numbers which differ by 8.
Original entry on oeis.org
95, 216, 287, 515, 675, 995, 1175, 4320, 9995, 82640, 99995, 960795, 999995, 1322312, 4049591, 9999995, 16955015, 34602080, 99999995, 171010235, 181964891, 183673467, 187160072, 321920055, 326530616, 328818032
Offset: 1
8533818720//8533818713 = 9237867023 * 9237867031, where // denotes concatenation.
Showing 1-10 of 16 results.
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