A037610 -id:A037610 - cp's OEIS frontend

A261672 Numbers k such that A037610(k) is prime.

4, 7, 52, 100, 136, 388, 30940, 33250

Offset: 1

Felix Fröhlich, Sep 04 2015

The terms are a subset of the terms of A016777, since a term of A037610 can only be prime if it is congruent to 1 modulo 10 and hence congruent to 1 modulo 3. If A037610(k) is congruent to 1 modulo 3, then k is congruent to 1 modulo 3 as well.

No further terms up to 10000.

			A037610(7) = 1231231 is prime, so 7 is a term of the sequence.

Cf. A037610, A062209.

Select[Range@ 500, PrimeQ@ Floor[41/333*10^#] &] (* Michael De Vlieger, Sep 07 2015 *)

a037610(n) = 10^n*41\333
is(n) = ispseudoprime(a037610(n))

a(7)-a(8) from Michael S. Branicky, Jun 28 2023

A057137 Concatenate next digit at right hand end (where the next digit after 9 is again 0).

Original entry on oeis.org

0, 1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 1234567890, 12345678901, 123456789012, 1234567890123, 12345678901234, 123456789012345, 1234567890123456, 12345678901234567, 123456789012345678, 1234567890123456789, 12345678901234567890, 123456789012345678901

Offset: 0

Henry Bottomley, Aug 12 2000

Also called the triangle of the gods (see Pickover link).

See A037610 for a general formula. - Hieronymus Fischer, Jan 03 2013

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 61.

T. D. Noe and Hieronymus Fischer, Table of n, a(n) for n = 0..200 (terms up to 100 from T. D. Noe)
Clifford Pickover, Triangle of the Gods
Index entries for linear recurrences with constant coefficients, signature (10,0,0,0,0,0,0,0,0,1,-10).

Alternative progression for n >= 10 compared with A007908 and A014824.
Cf. A057138 for reverse. Cf. A010879 (decimal digits).
For primes see A120819.

A057137:=n->floor((137174210/1111111111)*10^n); seq(A057137(n), n=0..20); # Wesley Ivan Hurt, Apr 18 2014

a[n_]:=Floor[137174210/1111111111*10^n]; Array[a,19,0] (* Robert G. Wilson v, Apr 18 2014 *)

A057137(n)=sum(i=1,n,i%10*10^(n-i)) \\ M. F. Hasler, Jan 13 2013

A057137(n)=137174210*10^n\1111111111 \\ M. F. Hasler, Jan 13 2013

def A057137(n): s = '0123456789'; return int((n+1)//10*s + s[:(n+1)%10]) # Ya-Ping Lu, Apr 08 2025

a(n) = 10*(a(n-1)-floor[n/10]) + n = floor[A057139(n)/10^(n-1)].

a(n) = floor((137174210/1111111111)*10^n). - Hieronymus Fischer, Jan 03 2013, corrected by M. F. Hasler, Jan 13 2013

A037487 Decimal expansion of a(n) is given by the first n terms of the periodic sequence with initial period 1,2.

Original entry on oeis.org

1, 12, 121, 1212, 12121, 121212, 1212121, 12121212, 121212121, 1212121212, 12121212121, 121212121212, 1212121212121, 12121212121212, 121212121212121, 1212121212121212, 12121212121212121, 121212121212121212, 1212121212121212121, 12121212121212121212

Offset: 1

Clark Kimberling

See A037610 for a general formula. - Hieronymus Fischer, Jan 03 2013

(Smoothly undulating palindromic) primes in this sequence are listed in A092696(n) = (4*10^A062209(n)-7)/33. - M. F. Hasler, Jul 30 2015

Hieronymus Fischer, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (10,1,-10).

Cf. A037610.

Table[FromDigits[PadRight[{},n,{1,2}]],{n,20}] (* or *) LinearRecurrence[ {10,1,-10},{1,12,121},20] (* Harvey P. Dale, Jun 21 2016 *)

A037487(n)=10^n*4\33  \\ - M. F. Hasler, Jan 13 2013

Vec(x*(2*x+1)/((x-1)*(x+1)*(10*x-1)) + O(x^100)) \\ Colin Barker, Apr 30 2014

a(n) = floor((4/33)*10^n). - Hieronymus Fischer, Jan 03 2013

a(n) = 10*a(n-1)+a(n-2)-10*a(n-3). G.f.: x*(2*x+1) / ((x-1)*(x+1)*(10*x-1)). - Colin Barker, Apr 30 2014

A037604 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,2,3.

Original entry on oeis.org

1, 6, 27, 109, 438, 1755, 7021, 28086, 112347, 449389, 1797558, 7190235, 28760941, 115043766, 460175067, 1840700269, 7362801078, 29451204315, 117804817261, 471219269046, 1884877076187, 7539508304749, 30158033218998

Offset: 1

Clark Kimberling

Convolution of A000302 with A010882. - Philippe Deléham, Mar 24 2013

Michael De Vlieger, Table of n, a(n) for n = 1..1661
Index entries for linear recurrences with constant coefficients, signature (4,0,1,-4).
Index entries for 2-automatic sequences.

Cf. A000302, A010882.

Rest@ CoefficientList[Series[x (1 + 2 x + 3 x^2)/((1 - x) (1 - 4 x) (1 + x + x^2)), {x, 0, 23}], x] (* Michael De Vlieger, Mar 19 2021 *)
Table[FromDigits[PadRight[{},n,{1,2,3}],4],{n,30}] (* or *) LinearRecurrence[{4,0,1,-4},{1,6,27,109},30] (* Harvey P. Dale, May 07 2023 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -4,1,0,4]^(n-1)*[1;6;27;109])[1,1] \\ Charles R Greathouse IV, Feb 13 2017

print([3*4**n//7 for n in range(1,24)]) # Karl V. Keller, Jr., Mar 18 2021

G.f.: x*(1+2x+3*x^2)/((1-x)*(1-4x)*(1+x+x^2)). - Philippe Deléham, Mar 24 2013
a(n) = 4*a(n-1) + a(n-3) -4*a(n-4) for n > 5, a(1) = 1, a(2) = 6, a(3) = 27, a(4) = 109, a(5) = 438. - Philippe Deléham, Mar 24 2013
a(n+2) = 6*A033140(n) + A191597(n+2). - Philippe Deléham, Mar 24 2013
A007090(a(n)) = A037610(n). - R. J. Mathar, Apr 27 2015
a(n) = floor(3*4^n/7). - Karl V. Keller, Jr., Mar 18 2021

A037608 Base 8 digits are, in order, the first n terms of the periodic sequence with initial period 1,2,3.

Original entry on oeis.org

1, 10, 83, 665, 5322, 42579, 340633, 2725066, 21800531, 174404249, 1395233994, 11161871955, 89294975641, 714359805130, 5714878441043, 45719027528345, 365752220226762, 2926017761814099, 23408142094512793, 187265136756102346, 1498121094048818771

Offset: 1

Clark Kimberling

			1, 10, 83, 665, 5322, ... in base 8 are 1, 12, 123, 1231, 12312, ...

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,0,1,-8).

Cf. A037610.

Table[FromDigits[PadRight[{},n,{1,2,3}],8],{n,30}] (* Harvey P. Dale, Apr 06 2022 *)

Vec(x*(3*x^2+2*x+1)/((x-1)*(8*x-1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Nov 28 2014

a(n) = 8*a(n-1)+a(n-3)-8*a(n-4). - Colin Barker, Nov 28 2014

G.f.: x*(3*x^2+2*x+1) / ((x-1)*(8*x-1)*(x^2+x+1)). - Colin Barker, Nov 28 2014

A037605 Base 5 digits are, in order, the first n terms of the periodic sequence with initial period 1,2,3.

Original entry on oeis.org

1, 7, 38, 191, 957, 4788, 23941, 119707, 598538, 2992691, 14963457, 74817288, 374086441, 1870432207, 9352161038, 46760805191, 233804025957, 1169020129788, 5845100648941, 29225503244707, 146127516223538, 730637581117691

Offset: 1

Clark Kimberling

Index entries for linear recurrences with constant coefficients, signature (5,0,1,-5).
Index entries for 5-automatic sequences.

Table[FromDigits[PadRight[{},n,{1,2,3}],5],{n,30}] (* or *) LinearRecurrence[{5,0,1,-5},{1,7,38,191},30] (* Harvey P. Dale, Oct 16 2024 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -5,1,0,5]^(n-1)*[1;7;38;191])[1,1] \\ Charles R Greathouse IV, Feb 13 2017

G.f.: x*(1+2*x+3*x^2) / ( (x-1)*(5*x-1)*(1+x+x^2) ). - R. J. Mathar, Apr 27 2015

A007091(a(n)) = A037610(n). - R. J. Mathar, Apr 27 2015

A037606 Base 6 digits are, in order, the first n terms of the periodic sequence with initial period 1,2,3.

Original entry on oeis.org

1, 8, 51, 307, 1844, 11067, 66403, 398420, 2390523, 14343139, 86058836, 516353019, 3098118115, 18588708692, 111532252155, 669193512931, 4015161077588, 24090966465531, 144545798793187, 867274792759124, 5203648756554747

Offset: 1

Clark Kimberling

Index entries for 6-automatic sequences.
Index entries for linear recurrences with constant coefficients, signature (6,0,1,-6).

nn=30;With[{c=PadRight[{},nn,{1,2,3}]},Table[FromDigits[Take[c,n],6],{n,nn}]] (* Harvey P. Dale, Aug 25 2012 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -6,1,0,6]^(n-1)*[1;8;51;307])[1,1] \\ Charles R Greathouse IV, Feb 13 2017

G.f.: x*(1+2*x+3*x^2) / ( (x-1)*(6*x-1)*(1+x+x^2) ). - R. J. Mathar, Apr 27 2015

A007092(a(n)) = A037610(n). - R. J. Mathar, Apr 27 2015

A037609 Base 9 digits are, in order, the first n terms of the periodic sequence with initial period 1,2,3.

Original entry on oeis.org

1, 11, 102, 919, 8273, 74460, 670141, 6031271, 54281442, 488532979, 4396796813, 39571171320, 356140541881, 3205264876931, 28847383892382, 259626455031439, 2336638095282953

Offset: 1

Clark Kimberling

Index entries for 3-automatic sequences.
Index entries for linear recurrences with constant coefficients, signature (9,0,1,-9).

Table[FromDigits[PadRight[{},n,{1,2,3}],9],{n,20}] (* or *) LinearRecurrence[{9,0,1,-9},{1,11,102,919},20] (* Harvey P. Dale, Nov 16 2024 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -9,1,0,9]^(n-1)*[1;11;102;919])[1,1] \\ Charles R Greathouse IV, Feb 13 2017

G.f.: x*(1+2*x+3*x^2) / ( (x-1)*(9*x-1)*(1+x+x^2) ). - R. J. Mathar, Apr 27 2015

A007095(a(n)) = A037610(n). - R. J. Mathar, Apr 27 2015

cp's OEIS Frontend

A261672 Numbers k such that A037610(k) is prime.

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A057137 Concatenate next digit at right hand end (where the next digit after 9 is again 0).

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A037487 Decimal expansion of a(n) is given by the first n terms of the periodic sequence with initial period 1,2.

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A037604 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,2,3.

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A037608 Base 8 digits are, in order, the first n terms of the periodic sequence with initial period 1,2,3.

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A037605 Base 5 digits are, in order, the first n terms of the periodic sequence with initial period 1,2,3.

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A037606 Base 6 digits are, in order, the first n terms of the periodic sequence with initial period 1,2,3.

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