A128681 -id:A128681 - cp's OEIS frontend

A128678 Numbers k such that k^3 divides 4^(k^2) + 1.

1, 5, 205, 505, 20705, 40505, 168305, 408545, 1342505, 1660705, 2084645, 4091005, 10201505, 16322105, 16750345, 16998805, 37217545, 49985405, 55042705, 117165905, 135593005, 167731205, 186678805, 210549145, 218301905, 418261705

Offset: 1

Alexander Adamchuk, Mar 31 2007

5 divides a(n) for n > 1.

Cf. A127263, A128677, A128679, A128680, A128681, A128682, A128683, A128684, A128685, A292330.

Select[Range[5*10^6], Mod[ 4^(#^2)+1, #^3]==0 &] (* G. C. Greubel, Jan 17 2018 *)

More terms from Ryan Propper, Jan 13 2008

A128683 Numbers k such that k^3 divides 10^(k^2) + 1.

Original entry on oeis.org

1, 11, 253, 11891, 35167, 45023, 47927, 96569, 276859, 640343, 1035529, 1102321, 1652849, 2221087, 6367757, 16339367, 19588019, 26499979, 30096121, 48669863, 51809087, 74901409, 88023001, 89007677, 104391089, 105080767, 143938531

Offset: 1

Alexander Adamchuk, Mar 31 2007

11 divides a(n) for n > 1.

Chai Wah Wu, Table of n, a(n) for n = 1..92

Cf. A127263, A128677, A128678, A128679, A128680, A128681, A128682, A128684, A128685, A292335.

Select[Range[5*10^6], Mod[ 10^(#^2)+1, #^3]==0 &] (* G. C. Greubel, Jan 18 2018 *)

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 26 2010

Terms a(16) onward from Max Alekseyev and Alexander Adamchuk, May 12 2010

A128677 Least k>p such that (kp)^3 divides (p-1)^(kp)^2+1 for prime p = A000040(n).

Original entry on oeis.org

19, 41, 29, 23, 79, 41617, 20939, 47, 40427, 4093, 4441, 2543, 1033, 659, 2612032921, 394502321, 14958421, 17957, 569, 14747, 12641, 167, 174263, 100493, 285629

Offset: 2

Alexander Adamchuk, Mar 30 2007, Mar 31 2007, Apr 09 2007

For every prime p>2, p^3 divides (p-1)^(p^2)+1 and furthermore p divides all numbers n>1 such that n^3 divides (p-1)^(n^2)+1.
a(27)>10^15. - Max Alekseyev, Nov 30 2017
Some further terms: a(28)-a(36) = {857, 3271, 7243979, 509, 263, 43019, 38921, 2683, 312055091}. a(38)-a(43) = {7499, 88588425539, 9689, 359, 1087, 383}. a(45)-a(61) = {931417, 40597, 2111, 2677, 14983, 261061, 1302937, 479, 17935703, 503, 4227137, 39398453, 2153, 1627, 1109, 28663, 1699}. a(63)-a(69) = {1229, 1867, 78877, 500861, 1987, 62683, 2777}. a(71)-a(75) = {275884327, 719, 44041, 3122698559, 15161}. a(77)-a(80) = {907927, 202471, 5788837, 16361}.
a(n) <= A177996(n).
A000040(n) divides (a(n) - 1)/2. The quotients (a(n)-1)/2/A000040(n) are listed in A136374.

			a(2) = A127263(3)/3 = 57/3 = 19.

Cf. A127263, A128678, A128679, A128680, A128681, A128682, A128683, A128684, A128685, A136374.

a[n_] := Module[{p, k}, p = Prime[n]; k = p + 1;
   While[! Divisible[(p - 1)^(k p)^2 + 1, (k p)^3], k++]; k];
Table[a[n], {n, 2, 15}] (* Robert Price, Mar 23 2020 *)

a(n) = smallest prime divisor of (p-1)^(p^2)+1 other than p, where p=A000040(n).

a(16)-a(26), a(39), a(74) from Max Alekseyev, May 16 2010

A128679 Numbers k such that k^3 divides 5^(k^2) + 1.

Original entry on oeis.org

1, 3, 21, 609, 903, 2667, 9429, 15501, 18543, 26187, 77343, 108507, 114681, 159663, 212541, 273441, 318507, 405447, 537747, 797349, 1197483, 1357503, 1398747, 1731387, 2354961, 3146703, 3325749, 4451181, 4630227, 4665801, 5192943

Offset: 1

Alexander Adamchuk, Mar 31 2007

3 divides a(n) for n > 1. 7 divides a(n) for n > 2.

Cf. A127263 (numbers k such that k^3 divide 2^(k^2)+1).
Cf. A128677 (least k > p such that (k*p)^3 divides (p-1)^(k*p)^2+1, where p = prime(n) > 2).
Cf. A128678, A128680, A128681, A128682, A128683, A128684, A128685.

Select[Range[5*10^6], Mod[ 5^(#^2)+1, #^3]==0 &] (* G. C. Greubel, Jan 18 2018 *)

More terms from Ryan Propper, Dec 31 2007

A128680 Numbers k such that k^3 divides 6^(k^2) + 1.

Original entry on oeis.org

1, 7, 203, 1379, 11977, 39991, 577129, 2359469, 4780447, 6521291, 6640739, 9904979, 21511301, 28434413, 34050611, 113694413, 189117439, 222281549, 250163599, 282046373, 391803601, 941748059, 1269166759, 1308225583

Offset: 1

Alexander Adamchuk, Mar 31 2007

7 divides a(n) for n > 1.

Cf. A127263 (numbers k such that k^3 divide 2^(k^2)+1).
Cf. A128677 (least k > p such that (k*p)^3 divides (p-1)^(k*p)^2+1, where p = prime(n) > 2).
Cf. A128678, A128679, A128681, A128682, A128683, A128684, A128685.

Select[Range[5*10^6], Mod[ 6^(#^2)+1, #^3]==0 &] (* G. C. Greubel, Jan 18 2018 *)

More terms from Ryan Propper, Jan 13 2008

A128682 Numbers k such that k^3 divides 9^(k^2) + 1.

Original entry on oeis.org

1, 5, 505, 5905, 596405, 1971005, 199071505, 515151005, 581457505, 618181105, 2327756905, 11855474405, 19555113485, 31361954905, 101901244565, 235103447405, 305354088005, 395331063745, 600905819065, 608393336905, 686701313405, 730071885005

Offset: 1

Alexander Adamchuk, Mar 31 2007

nonn

5 divides a(n) for n > 1.

Cf. A127263, A128677, A128678, A128679, A128680, A128681, A128683, A128684, A128685.

Select[Range[5*10^6], Mod[ 9^(#^2)+1, #^3]==0 &] (* G. C. Greubel, Jan 18 2018 *)

a(5)-a(6) from J. Mulder (jasper.mulder(AT)planet.nl), Jan 26 2010

Terms a(7) onward from Max Alekseyev, May 14 2010

A128684 Numbers k such that k^3 divides 11^(k^2) + 1.

Original entry on oeis.org

1, 3, 111, 24753, 1770231, 23401797, 39402669, 65498547, 298100379, 440377293, 1788478509, 5218600731, 8786795187, 14606175981, 24342867543, 45340639419, 98204136339, 5032808252679, 5293823142513, 6241891310439, 7013907181779

Offset: 1

Alexander Adamchuk, Mar 31 2007

nonn

3 divides a(n) for n > 1.

Cf. A127263, A128677, A128678, A128679, A128680, A128681, A128682, A128683, A128685.

Select[Range[5*10^6], Mod[ 11^(#^2)+1, #^3]==0 &] (* G. C. Greubel, Jan 18 2018 *)

a(5) from J. Mulder (jasper.mulder(AT)planet.nl), Jan 26 2010

Terms from a(6) onward from Max Alekseyev, May 14 2010

A128685 Numbers k such that k^3 divides 12^(k^2) + 1.

Original entry on oeis.org

1, 13, 1027, 52741, 468481, 2890927, 4166539, 37009999, 228383233, 999884119, 1345997497, 1900627417, 2126334691, 4499474941, 5037873529, 11728490839, 104180336299, 105897267463, 150149565943, 926550776281, 4056529870783

Offset: 1

Alexander Adamchuk, Mar 31 2007

13 divides a(n) for n > 1.

Cf. A127263, A128677, A128678, A128679, A128680, A128681, A128682, A128683, A128684.

Select[Range[5*10^6], Mod[ 12^(#^2)+1, #^3]==0 &] (* G. C. Greubel, Jan 18 2018 *)
Select[Range[5*10^6],PowerMod[12,#^2,#^3]==#^3-1&] (* The program generates the first 7 terms of the sequence. *) (* Harvey P. Dale, Aug 10 2025 *)

16 more terms from Ryan Propper, Dec 03 2007

Terms a(17) onward from Max Alekseyev, May 14 2010

A127263 Numbers k such that k^3 divides 2^(k^2)+1.

Original entry on oeis.org

1, 3, 57, 32547, 9961491, 297381939, 1338104811, 3942759027, 5688011361, 8920514307, 9146532873, 40253706489, 243640690617, 764039295291, 1127102902923, 1556475424971, 2251315404417, 3005607686883, 5222670270483

Offset: 1

Max Alekseyev, Mar 27 2007, Mar 29 2007, Apr 18 2007

nonn

If k belongs to this sequence, then so does (2^(k^2)+1)/k^2.
From Alexander Adamchuk, May 14 2010: (Start)
3 divides a(n) for n>1.
19 divides a(n) for n>2. (End)

ArtOfProblemSolving forum, n^3 | (2^{n^2}+1)

Cf. A006521, A093546, A093665, A136372, A128677.

Cf. A128678, A128679, A128680, A128681, A128682, A128683, A128684, A128685, A177813, A177814, A177816, A177817, A177818, A177819, A177820.

Select[Range[100000], Divisible[2^(#^2) + 1, #^3] &] (* Robert Price, Mar 23 2020 *)

a(7) from Ryan Propper, Jan 01 2008

a(8)-a(19) from Max Alekseyev, May 14 2010

A177813 Numbers k such that k^3 divides 13^(k^2) + 1.

Original entry on oeis.org

1, 7, 203, 11977, 154553, 353423, 3049963, 4482037, 5192537, 7170569, 9904979, 20851957, 33461911, 35852033, 69262991, 88448927, 160274303, 264440183, 306359683, 381231473, 423063571, 978466699, 1974252749, 2115269947, 4647954787, 5218486693, 6824905927, 7803226417, 9040206917, 10041409007, 11055712717, 12483960629, 17244568439, 47773414171, 57280493557, 57729535241, 58257187051, 62418389453, 67340133077

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

7 divides a(n) for n > 1.

Prime factors of a(n) in the order of their appearance are {7, 29, 59, 22079, 1741, 435709, 25579, 35323, 827, 164837, 176611, 5783, 435709, 22896329, 54461639, 4820033, ...}. - Alexander Adamchuk, May 16 2010

Cf. A127263 (k^3 divides 2^(k^2) + 1).
Cf. A128678, A128679, A128680, A128681, A128682, A128683, A128684, A128685.
Cf. A177813, A177814, A177816, A177817, A177818, A177819, A177820.
Cf. A128677 (least k > p such that (k*p)^3 divides (p-1)^(k*p)^2+1, where p = prime(n) > 2).

Corrected and extended by Max Alekseyev, May 16 2010
a(8) = 4482037 (suggested by Max Alekseyev) missing in original sequence inserted by Alexander Adamchuk, May 16 2010
More terms from Max Alekseyev, Feb 14 2012

cp's OEIS Frontend

A128678 Numbers k such that k^3 divides 4^(k^2) + 1.

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A128683 Numbers k such that k^3 divides 10^(k^2) + 1.

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A128677 Least k>p such that (kp)^3 divides (p-1)^(kp)^2+1 for prime p = A000040(n).

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A128679 Numbers k such that k^3 divides 5^(k^2) + 1.

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A128680 Numbers k such that k^3 divides 6^(k^2) + 1.

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A128682 Numbers k such that k^3 divides 9^(k^2) + 1.

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A128684 Numbers k such that k^3 divides 11^(k^2) + 1.

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A128685 Numbers k such that k^3 divides 12^(k^2) + 1.

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A127263 Numbers k such that k^3 divides 2^(k^2)+1.

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