A177814 -id:A177814 - cp's OEIS frontend

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

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

A177816 Numbers k such that k^3 divides 16^(k^2) + 1.

Original entry on oeis.org

1, 17, 707489, 5030929, 6029713, 209372172193, 250938565921, 1413292053713, 1784415176081, 24025953593297, 48948914347889, 1423524187400657, 5817190224008753, 49446116858851553, 74262006382962977

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

17 divides a(n) for n > 1.

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).

Terms a(6) onward from Max Alekseyev, May 16 2010

A177817 Numbers k such that k^3 divides 17^(k^2) + 1.

Original entry on oeis.org

1, 3, 9, 21, 39, 63, 117, 273, 819, 1467, 2067, 3081, 4269, 6201, 7299, 9243, 10269, 12807, 14469, 16959, 19071, 20421, 21567, 23877, 29883, 43407, 48711, 50877, 51093, 55497, 64701, 89649, 94887, 118713, 133497, 142947, 146133, 149331

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

3 divides a(n) for n > 1.

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).

A177818 Numbers k such that k^3 divides 18^(k^2) + 1.

Original entry on oeis.org

1, 19, 397841, 1152331, 3566699, 24128658809, 74683110361, 216316727651, 1339092172657, 7967201553697

Offset: 1

Alexander Adamchuk, May 14 2010

19 divides a(n) for n > 1.

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).

a(6)-a(10) from Max Alekseyev, May 16 2010

A177819 Numbers k such that k^3 divides 19^(k^2) + 1.

Original entry on oeis.org

1, 5, 55, 1265, 11255, 59455, 123805, 395755, 635255, 874115, 1028555, 1456015, 2847515, 3201715, 3841805, 4353305, 6655055, 6987805, 13443155, 16825765, 23656765, 36370015, 41083405, 66919765, 68432705, 100126015, 123012395

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

5 divides a(n) for n > 1.

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).

Select[Range[123020000],PowerMod[19,#^2,#^3]==#^3-1&] (* Harvey P. Dale, May 20 2021 *)

More terms from Max Alekseyev, May 16 2010

A177820 Numbers k such that k^3 divides 20^(k^2) + 1.

Original entry on oeis.org

1, 3, 7, 21, 381, 903, 921, 2667, 5789, 6447, 17367, 18543, 73703, 114681, 116967, 208443, 221109, 277221, 746781, 797349, 818769, 855141, 871347, 1459101, 2205609, 2354961, 5090367, 5331669, 5692701, 6099429, 7611387, 8710041

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

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).

A136372 Prime factors of terms of A127263.

Original entry on oeis.org

3, 19, 571, 9137, 41113, 174763, 274081, 1236787, 7485811, 23474953, 34630009, 47822393, 69171211, 92346689, 160465489

Offset: 1

Alexander Adamchuk, Dec 27 2007

From Alexander Adamchuk, May 14 2010: (Start)
3 divides A127263(n) for n > 1.
19 divides A127263(n) for n > 2. (End)

			A127263(1) = 1,
A127263(2) = 3,
A127263(3) = 57 = 3*19,
A127263(4) = 32547 = 3*19*571,
A127263(5) = 9961491 = 3*19*174763,
A127263(6) = 297381939 = 3*19*571*9137.

Cf. A127263, A128677, A128678, A128679, A128680, A128681, A128682, A128683, A128684, A128685, A136373.

Cf. A177813, A177814, A177816, A177817, A177818, A177819, A177820. - Alexander Adamchuk, May 14 2010

Edited and extended by Max Alekseyev, May 14 2010

A185732 Accumulation array of the polygonal number array (A086270), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 10, 9, 3, 20, 24, 15, 4, 35, 50, 42, 22, 5, 56, 90, 90, 64, 30, 6, 84, 147, 165, 140, 90, 39, 7, 120, 224, 273, 260, 200, 120, 49, 8, 165, 324, 420, 434, 375, 270, 154, 60, 9, 220, 450, 612, 672, 630, 510, 350, 192, 72, 10, 286, 605, 855, 984, 980, 861, 665, 440, 234, 85, 11, 364, 792, 1155, 1380, 1440, 1344, 1127, 840, 540, 280, 99, 12, 455, 1014, 1518, 1870, 2025, 1980, 1764, 1428, 1035, 650, 330, 114, 13, 560, 1274, 1950, 2464, 2750, 2790, 2604, 2240

Offset: 1

Clark Kimberling, Feb 01 2011

This is the (first) accumulation array of A086270; the second is A185733. See A144112 for the definition of accumulation array.

			Northwest corner:
1....4....10...20...35
2....9....24...50...90
3....15...42...90...165
4....22...64...140..260
5....30...90...200..375

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Rows 1 to 5: A000292, A006002, A059270, A177814, 5*A002411.
Columns 1 to 4: A000027, A055999, A067728, 10*A000096.
Cf. A144112, A086270, A185732.

f[n_,k_]:=k+n*k(k-1)/2;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* Array A086270 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten  (* A086270 *)
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* acc. arr. of {f(n,k)} *)
Factor[s[n,k]]  (* formula for A185732 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* acc. arr. A185732 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* A185732 *)

T(n,k) = k*(k+1)*n*(n+1)*(k*n-n+k+5)/12.

A292339 Numbers k such that k^2 divides 14^k + 1.

Original entry on oeis.org

1, 3, 5, 15, 183, 355, 505, 915, 1065, 1515, 12165, 35855, 64965, 92415, 107565, 134139, 201995, 605985, 670695, 742065, 863715, 1228665, 1674633, 1675365, 2269515, 6561465, 8373165, 9074865, 13772865, 16521423, 20401495, 24197155, 25709455, 29234715, 36965085, 47619345, 52686615, 61204485, 67740195, 70694445, 72591465

Offset: 1

Max Alekseyev, Sep 16 2017

nonn

Max Alekseyev, Table of n, a(n) for n = 1..92

Subsequence of A015965.

Cf. A128394, A177814.

cp's OEIS Frontend

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

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

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A177816 Numbers k such that k^3 divides 16^(k^2) + 1.

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A177817 Numbers k such that k^3 divides 17^(k^2) + 1.

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A177818 Numbers k such that k^3 divides 18^(k^2) + 1.

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A177819 Numbers k such that k^3 divides 19^(k^2) + 1.

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A177820 Numbers k such that k^3 divides 20^(k^2) + 1.

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A136372 Prime factors of terms of A127263.

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A185732 Accumulation array of the polygonal number array (A086270), by antidiagonals.

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A292339 Numbers k such that k^2 divides 14^k + 1.

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