A002378 -id:A002378 - cp's OEIS frontend

A078358 Non-oblong numbers: Complement of A002378.

1, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74

Offset: 1

Wolfdieter Lang, Nov 29 2002

The (primitive) period length k(n)=A077427(n) of the (regular) continued fraction of (sqrt(4*a(n)+1)+1)/2 determines whether or not the Diophantine equation (2*x-y)^2 - (1+4*a(n))*y^2 = +4 or -4 is solvable and the approximants of this continued fraction give all solutions. See A077057.
The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003
Infinite series 1/A078358(n) is divergent. Proof: Harmonic series 1/A000027(n) is divergent and can be distributed on two subseries 1/A002378(k+1) and 1/A078358(m). The infinite subseries 1/A002378(k+1) is convergent to 1, so Sum_{n>=1} 1/A078358(n) is divergent. - Artur Jasinski, Sep 28 2008

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Oskar Perron, Die Lehre von den Kettenbrüchen, Teubner, Leipzig, 1913.

a(n)=(A077425(n)-1)/4.

Cf. A049068 (subsequence), A144786.

a078358 n = a078358_list !! (n-1)
a078358_list = filter ((== 0) . a005369) [0..]
-- Reinhard Zumkeller, Jul 04 2014, May 08 2012

Complement[Range[930], Table[n (n + 1), {n, 0, 30}]] (* and *) Table[Ceiling[Sqrt[n]] + n - 1, {n, 900}] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)

a(n)=sqrtint(n-1)+n \\ Charles R Greathouse IV, Jan 17 2013

from operator import sub
from sympy import integer_nthroot
def A078358(n): return n+sub(*integer_nthroot(n,2)) # Chai Wah Wu, Oct 01 2024

4*a(n)+1 is not a square number.
a(n) = ceiling(sqrt(n)) + n -1. - Leroy Quet, Jul 06 2007
A005369(a(n)) = 0. - Reinhard Zumkeller, Jul 05 2014

A048543 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 7's.

Original entry on oeis.org

8, 108, 878, 870, 8813, 27924, 84129, 882483, 2788865, 7601169, 88190572, 421518419, 4204014483, 26034169427, 42305694389, 88022598108, 546605779129, 2788866736108, 69048807207508, 98844816642745, 88187696578, 2516, 5270458024477168, 26772616052167516, 166654612230648203, 521323093463907254

Offset: 1

Patrick De Geest, May 15 1999

			From _Jesse Sealand_, Oct 05 2019: (Start)
a(1) = 8 since A002378(8) = 72;
a(2) = 108 since A002378(36) = 11772;
a(3) = 878 since A002378(182) = 771762;
a(4) = 870 since A002378(1817) = 757770, etc. (End)

Eric Weisstein's World of Mathematics, Pronic Number.

Cf. A048544, A002378.

Cf. A048529, A048531, A048533, A048535, A048537, A048539, A048541, A048545, A048547.

def A048543(n):
    k, m = 1, 2
    while True:
        if str(m).count('7') == n:
            return k
        k += 1
        m += 2*k # Chai Wah Wu, Mar 23 2018

a(13)-a(16) from Lars Blomberg, Jun 10 2011
Name edited by Jesse Sealand, Oct 05 2019
a(17) from Giovanni Resta, Oct 09 2019
a(18)-a(25) from Max Alekseyev, Jul 25 2024

A048531 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 1's.

Original entry on oeis.org

3, 10, 105, 1010, 10203, 33928, 105551, 335128, 1054130, 12313046, 100573943, 333192303, 1054092885, 4261585525, 30184681729, 133083117303, 331829837885, 4136557897710, 11666683809854, 105410706823885, 333183899837885, 3333333637837885, 6466156610468989, 33928029520312720, 138242940872632303

Offset: 1

Patrick De Geest, May 15 1999

			From _Jesse Sealand_, Oct 06 2019: (Start)
a(1) = 3 since A002378(3) = 12;
a(2) = 10 since A002378(10) = 110;
a(3) = 105 since A002378(105) = 11130;
a(4) = 1010 since A002378(1010) = 1021110, etc.
(End).

Eric Weisstein's World of Mathematics, Pronic Number

Cf. A048532, A002378.

Cf. A048529, A048533, A048535, A048537, A048539, A048541, A048543, A048545, A048547.

a(13)-a(16) from Lars Blomberg, Jun 10 2011
Name edited by Jesse Sealand, Oct 06 2019
a(17) from Giovanni Resta, Oct 09 2019
a(18)-a(25) from Max Alekseyev, Jul 24 2024

A048533 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 2's.

Original entry on oeis.org

1, 16, 141, 458, 1458, 10593, 36458, 471933, 471404, 3496458, 14930593, 149072541, 471446966, 149071198, 14907133333, 114989226548, 471404524016, 1510636363333, 14360440530228, 47140441472504, 149071332733333, 1490343155861166, 4712983368549333, 24955404665166666, 149070527690846966

Offset: 1

Patrick De Geest, May 15 1999

			From _Jesse Sealand_, Oct 06 2019: (Start)
a(1) = 1 since A002378(1) = 2;
a(2) = 16 since A002378(16) = 272;
a(3) = 141 since A002378(141) = 20022;
a(4) = 458 since A002378(458) = 210222, etc.
(End).

Eric Weisstein's World of Mathematics, Pronic Number

Cf. A048534, A002378.

Cf. A048529, A048531, A048535, A048537, A048539, A048541, A048543, A048545, A048547.

a(15)-a(16) from Lars Blomberg, Jun 10 2011
Name edited by Jesse Sealand, Oct 06 2019
a(17) from Giovanni Resta, Oct 09 2019
a(18)-a(25) from Max Alekseyev, Jul 25 2024

A048535 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 3's.

Original entry on oeis.org

5, 36, 182, 1817, 4830, 5773, 153079, 577355, 3659694, 18366636, 182574188, 576484036, 1839384036, 17701331569, 36542213005, 485935528988, 1814754484036, 5755287250288, 5773506159694, 182738579159694, 230940107675430, 1820173984357898, 18202838331788627, 57475066797119386, 182305878493709942

Offset: 1

Patrick De Geest, May 15 1999

			From _Jesse Sealand_, Oct 05 2019: (Start)
a(1) = 5 since A002378(5) = 30;
a(2) = 36 since A002378(36) = 1332;
a(3) = 182 since A002378(182) = 33306;
a(4) = 1817 since A002378(1817) = 3303306, etc.
(End)

Eric Weisstein's World of Mathematics, Pronic Number

Cf. A048536, A002378 (oblong numbers).

Cf. A048529, A048531, A048533, A048537, A048539, A048541, A048543, A048545, A048547.

With[{s = 2 Accumulate[Range[0, 10^6]]}, Table[-1 + FirstPosition[s, ?(DigitCount[#, 10, 3] == n &)][[1]], {n, 8}]] (* _Michael De Vlieger, Oct 05 2019 *)

a(13)-a(16) from Lars Blomberg, Jun 10 2011
Name edited by Jesse Sealand, Oct 05 2019
a(17) from Giovanni Resta, Oct 09 2019
a(18)-a(25) from Max Alekseyev, Jul 25 2024

A048537 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 4's.

Original entry on oeis.org

6, 66, 215, 681, 6681, 12018, 201115, 1531159, 6541784, 21058215, 156340764, 675098840, 5868938806, 6666461193, 120186956049, 586862372693, 1115546688777, 4176295540840, 66589372231193, 66634408865198, 210842150539318, 2108185118566309, 21011769217380159, 185591342591574215, 67412496945629038

Offset: 1

Patrick De Geest, May 15 1999

			From _Jesse Sealand_, Oct 06 2019: (Start)
a(1) = 6 since A002378(6) = 42;
a(2) = 66 since A002378(66) = 4422;
a(3) = 215 since A002378(215) = 46440;
a(4) = 681 since A002378(681) = 464442, etc.
(End).

Eric Weisstein's World of Mathematics, Pronic Number

Cf. A048538, A002378.

Cf. A048529, A048531, A048533, A048535, A048539, A048541, A048543, A048545, A048547.

a(13)-a(16) from Lars Blomberg, Jun 10 2011
Name edited by Jesse Sealand, Oct 06 2019
a(17) from Giovanni Resta, Oct 09 2019
a(18)-a(25) from Max Alekseyev, Jul 25 2024

A048539 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 5's.

Original entry on oeis.org

7, 23, 74, 2248, 2292, 23506, 50552, 742667, 2357022, 22706949, 74555050, 703955648, 745020507, 3996943276, 74333434976, 244040827632, 505525029504, 5963015642907, 23548897033949, 156701172827632, 745354652534976, 758666893673274, 7453694087868501, 77198164845775648, 505525028614365003

Offset: 1

Patrick De Geest, May 15 1999

Eric Weisstein's World of Mathematics, Pronic Number

Cf. A048540, A002378.

Cf. A048529, A048531, A048533, A048535, A048537, A048541, A048543, A048545, A048547.

a(14)-a(16) from Lars Blomberg, Jun 10 2011
a(17) added and name edited by Giovanni Resta, Oct 09 2019
a(18)-a(25) from Max Alekseyev, Jul 25 2024

A048541 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 6's.

Original entry on oeis.org

2, 60, 257, 816, 4082, 77887, 258237, 816512, 4546060, 25820081, 103276578, 815209584, 816472087, 2581876578, 24834787435, 258140787487, 1239623599592, 7978512779247, 78515384904912, 121105886176546, 816494743559742, 6608227190863421, 8164653370882732, 81281404248860431, 81649676463937177

Offset: 1

Patrick De Geest, May 15 1999

			From _Jesse Sealand_, Oct 07 2019: (Start)
a(1) = 2 since A002378(2) = 6;
a(2) = 60 since A002378(60) = 3660;
a(3) = 257 since A002378(257) = 66306;
a(4) = 816 since A002378(816) = 666672, etc.
(End).

Eric Weisstein's World of Mathematics, Pronic Number

Cf. A048542, A002378.

Cf. A048529, A048531, A048533, A048535, A048537, A048539, A048543, A048545, A048547.

a(14)-a(16) from Lars Blomberg, Jun 10 2011
Name edited by Jesse Sealand, Oct 07 2019
a(17) from Giovanni Resta, Oct 09 2019
a(18)-a(25) from Max Alekseyev, Jul 25 2024

A048545 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 8's.

Original entry on oeis.org

13, 113, 886, 2861, 9213, 93855, 298053, 1359738, 2967970, 26189480, 220183761, 433461519, 3462498821, 28962801813, 43462498821, 722411301260, 942809540088, 9049247973113, 29354537778480, 195230348268113, 1970200215432144, 2426703296418144, 2233582298883761, 93802389568778480, 242669917148548480

Offset: 1

Patrick De Geest, May 15 1999

			From _Jesse Sealand_, Oct 07 2019: (Start)
a(1) = 13 since A002378(13) = 182;
a(2) = 113 since A002378(113) = 12882;
a(3) = 886 since A002378(886) = 785882;
a(4) = 2861 since A002378(2861) = 8188182, etc.
(End).

Eric Weisstein's World of Mathematics, Pronic Number

Cf. A048546, A002378.

Cf. A048529, A048531, A048533, A048535, A048537, A048539, A048541, A048543, A048547.

a(13)-a(16) from Lars Blomberg, Jun 10 2011
Name edited by Jesse Sealand, Oct 07 2019
a(17) from Giovanni Resta, Oct 09 2019
a(18)-a(25) from Max Alekseyev, Jul 25 2024

A048547 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 9's.

Original entry on oeis.org

9, 31, 890, 2234, 14131, 30331, 173031, 969520, 9958890, 31617890, 360542549, 547631234, 3316547890, 17318486343, 114017542334, 356363452109, 3065925634968, 9995498986343, 44045425415132, 173193446755643, 1414213562370264, 4471015432706534, 31575306806579083, 171463720944111031, 141138230008594765

Offset: 1

Patrick De Geest, May 15 1999

			From _Jesse Sealand_, Oct 07 2019: (Start)
a(1) = 9 since A002378(9) = 90;
a(2) = 31 since A002378(31) = 992;
a(3) = 890 since A002378(890) = 792990;
a(4) = 2234 since A002378(2234) = 4992990, etc.
(End).

Eric Weisstein's World of Mathematics, Pronic Number

Cf. A048548, A002378.

Cf. A048529, A048531, A048533, A048535, A048537, A048539, A048541, A048543, A048545.

a(13)-a(16) from Lars Blomberg, Jun 10 2011
Name edited by Jesse Sealand, Oct 07 2019
a(17) from Giovanni Resta, Oct 09 2019
a(18)-a(25) from Max Alekseyev, Jul 25 2024

cp's OEIS Frontend

A078358 Non-oblong numbers: Complement of A002378.

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A048543 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 7's.

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A048531 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 1's.

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A048533 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 2's.

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A048535 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 3's.

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A048537 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 4's.

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A048539 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 5's.

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A048541 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 6's.

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A048545 a(n) is the smallest value of m such that A002378(m), the m-th oblong number, contains exactly n 8's.

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