Displaying 1-10 of 11 results found.
Numbers k such that the concatenation of k with k+2 gives a square.
+10
25
7874, 8119, 69476962, 98010199, 108746354942, 449212110367, 544978035127, 870501316279, 998001001999, 1428394731903223, 1499870932756487, 1806498025502498, 1830668275445687, 1911470478658759, 2255786189655202
COMMENTS
Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 2.
Numbers k such that k concatenated with k-2 gives the product of two numbers which differ by 4.
Numbers k such that k concatenated with k-7 gives the product of two numbers which differ by 6.
EXAMPLE
8119//8121 = 9011^2, where // denotes concatenation.
98010199//98010200 = 99000100 * 99000102.
98010199//98010197 = 99000099 * 99000103.
PROG
(Python)
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
CROSSREFS
Cf. A030465, A102567, A115427, A115428, A115429, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115437.
Numbers k such that the concatenation of k with k+8 gives a square.
+10
25
6001, 6433, 11085116, 44496481, 96040393, 115916930617, 227007035017, 274101929528, 434985419768, 749978863753, 996004003993, 1365379857457948, 1410590590957816, 1762388551055953, 2307340946901148, 2700383162251217
COMMENTS
Also numbers k such that k concatenated with k+7 gives the product of two numbers which differ by 2.
Also numbers k such that k concatenated with k+4 gives the product of two numbers which differ by 4.
Also numbers k such that k concatenated with k-1 gives the product of two numbers which differ by 6.
Also numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 8.
EXAMPLE
6001//6009 = 7747^2, where // denotes concatenation.
96040393//96040400 = 98000200 * 98000202.
96040393//96040397 = 98000199 * 98000203.
96040393//96040392 = 98000198 * 98000204.
CROSSREFS
Cf. A030465, A102567, A115426, A115437, A115428, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115440.
Numbers k such that the concatenation of k with k-2 gives a square.
+10
22
6, 5346, 8083, 10578, 45531, 58626, 2392902, 2609443, 7272838, 51248898, 98009803, 159728062051, 360408196038, 523637103531, 770378933826, 998000998003, 1214959556998, 1434212848998, 3860012299771, 4243705560771
COMMENTS
So there are two equivalent definitions: numbers k such that k concatenated with k-6 gives the product of two numbers which differ by 4; and numbers k such that k concatenated with k-3 gives the product of two numbers which differ by 2.
For each k >= 1, 10^(4*k)-2*10^(3*k)+10^(2*k)-2*10^k+3 is a term.
If k is a term and k-2 has length m, then all prime factors of 10^m+1 must be congruent to 1 or 3 (mod 8). In particular, we can't have m == 2 (mod 4) or m == 3 (mod 6), as in those cases 10^m+1 would be divisible by 101 or 7 respectively. (End)
EXAMPLE
8083_8081 = 8991^2.
98009803_98009800 = 98999900 * 98999902, where _ denotes
concatenation
MAPLE
f:= proc(n) local S;
S:= map(t -> rhs(op(t))^2 mod 10^n+2, [msolve(x^2+2, 10^n+1)]);
op(sort(select(t -> t-2 >= 10^(n-1) and t-2 < 10^n and issqr(t-2 + t*10^n), S)))
end proc:
CROSSREFS
Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115432, A115433, A115434, A115435, A115436, A115442.
Numbers k such that the concatenation of k with k+5 gives a square.
+10
19
1, 4, 20, 31, 14564, 38239, 69919, 120395, 426436, 902596, 7478020, 9090220, 6671332084, 8114264059, 8482227259, 9900250996, 2244338786836, 2490577152964, 2509440638591, 2769448208395, 7012067592220
COMMENTS
Also numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 4.
Also numbers k such that k concatenated with k+4 gives the product of two numbers which differ by 2.
CROSSREFS
Cf. A030465, A102567, A115426, A115437, A115429, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115439.
Numbers k such that the concatenation of k with k+9 gives a square.
+10
18
216, 287, 515, 675, 1175, 4320, 82640, 960795, 1322312, 4049591, 16955015, 34602080, 171010235, 181964891, 183673467, 187160072, 321920055, 326530616, 328818032, 343942560, 470954312, 526023432, 528925616, 534830855
COMMENTS
Also numbers k such that k concatenated with k+8 gives the product of two numbers which differ by 2.
Also numbers k such that k concatenated with k+5 gives the product of two numbers which differ by 4.
CROSSREFS
Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115431, A115432, A115433, A115434, A115435, A115436, A115441.
Numbers k such that the concatenation of k with k-4 gives a square.
+10
14
65, 6653, 9605, 218413, 283720, 996005, 58446925, 99960005, 6086712229, 7385370133, 8478948853, 9999600005, 120178240093, 161171620229, 358247912200, 426843573160, 893417179213, 999996000005, 23376713203604
COMMENTS
The terms of this sequence (k//k-4 = m*m), A116104 (k//k-8 = m*(m+4)) and A116121 (k//k-5 = m*(m+2)) agree as long as the two concatenated numbers k and k-x have the same length. This condition is satisfied for the given terms of all three sequences. - Georg Fischer, Sep 12 2022 Numbers k of the form (y^2+4)/(10^d + 1) where 10^(d-1) <= k - 4 < 10^d and y is a square root of -4 mod (10^d + 1).
Includes 10^(2*d) - 4*10^d + 5 for all d >= 1, as the concatenation of this with 10^(2*d) - 4*10^d + 1 is 10^(4*d) - 4 * 10^(3*d) + 6 * 10^(2*d) - 4 * 10^d + 1 = (10^d - 1)^4.
This is the same sequence as A116104 and A116121. The only possible differences would be if 10^(d-1) + 4 <= k <= 10^(d-1) + 7 or 10^d + 4 <= k <= 10^d + 7, so that k - 4 and k - 8 have different numbers of digits. But in none of those cases can (10^d + 1)*k - 4 be a square:
If k = 10^(d-1) + 4 or 10^d + 4, (10^d + 1)*k - 4 == 6 (mod 9).
If k = 10^(d-1) + 5 or 10^d + 5, (10^d + 1)*k - 4 == 2 (mod 3).
If k = 10^(d-1) + 6 or 10^d + 6, (10^d + 1)*k - 4 == 2 (mod 10).
If k = 10^(d-1) + 7 or 10^d + 7, (10^d + 1)*k - 4 == 3 (mod 10). (End)
MAPLE
f:= proc(d) uses NumberTheory; local m, r;
m:= 10^d + 1;
if QuadraticResidue(-4, m) = -1 then return NULL fi;
r:= ModularSquareRoot(-4, m);
op(sort(select(t -> t >= 10^(d-1)+4 and t < 10^d+4, map(t -> ((r*t mod m)^2+4)/m, convert(RootsOfUnity(2, m), list)))))
end proc:
CROSSREFS
Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115433, A115434, A115435, A115436, A115443.
Numbers k such that the concatenation of k with k-8 gives a square.
+10
12
2137, 2892, 6369, 12217, 21964, 28233, 42312, 4978977, 9571608, 18642249, 32288908, 96039609, 200037461217, 305526508312, 570666416233, 638912248204, 996003996009, 1846991026584, 3251664327537, 4859838227992
EXAMPLE
18642249_18642241 = 43176671^2.
CROSSREFS
Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115432, A115433, A115434, A115436, A115446.
Numbers k such that the concatenation of k with k-5 gives a square.
+10
11
21, 30, 902406, 959721, 6040059046, 6242406405, 9842410005, 9900249006, 15033519988494, 17250863148969, 22499666270469, 27632040031654, 34182546327286, 37487353123861, 52213551379230, 74230108225630
EXAMPLE
902406_902401 = 949951^2.
CROSSREFS
Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115432, A115434, A115435, A115436, A115444.
Numbers k such that the concatenation of k with k-7 gives a square.
+10
11
8, 16, 1337032, 2084503, 2953232, 4023943, 1330033613070195328, 4036108433661798551, 8283744867954114232, 6247320195351414276186411625291, 9452080202814205132771066881607
EXAMPLE
4023943_4023936 = 6343456^2.
CROSSREFS
Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115432, A115433, A115435, A115436, A115445.
Numbers k such that the concatenation of k with k-9 gives a square.
+10
11
50, 5234, 9410, 638370, 994010, 12477933, 41829698, 99940010, 1087279650, 4492494893, 6226356365, 7765453730, 9999400010, 806057802450, 842377434050, 960398039610, 999994000010, 21338126513658, 24752544267698
EXAMPLE
638370_638361 = 798981^2.
CROSSREFS
Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115432, A115433, A115434, A115435, A115447.
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