Dario Piazzalunga - cp's OEIS frontend

A187768 Decimal expansion of square root of 103.

1, 0, 1, 4, 8, 8, 9, 1, 5, 6, 5, 0, 9, 2, 2, 1, 9, 4, 6, 8, 6, 4, 8, 5, 2, 0, 1, 1, 8, 9, 3, 5, 8, 7, 4, 3, 8, 3, 5, 8, 1, 9, 2, 2, 5, 0, 1, 8, 8, 8, 4, 0, 6, 6, 5, 2, 2, 5, 3, 6, 5, 0, 9, 2, 0, 6, 1, 1, 4, 0, 3, 8, 2, 2, 7, 0, 0, 5, 1, 7, 5, 0, 5, 6, 4, 1, 3

Offset: 2

Dario Piazzalunga, Jan 04 2013

Continued fraction expansion of sqrt(103) is A010171.

			10.1488915650922194686485201189358743835819225018884066522...

Cf. A010171, A192106.

RealDigits[Sqrt[103], 10, 90][[1]] (* Bruno Berselli, Jan 04 2013 *)

fpprec:90; ev(bfloat(sqrt(103))); /* Bruno Berselli, Jan 04 2013 */

sqrt(103) \\ Charles R Greathouse IV, Dec 12 2013

A192106 Decimal expansion of square root of 102.

Original entry on oeis.org

1, 0, 0, 9, 9, 5, 0, 4, 9, 3, 8, 3, 6, 2, 0, 7, 7, 9, 5, 3, 3, 6, 3, 3, 8, 5, 9, 1, 7, 0, 6, 9, 6, 0, 0, 7, 1, 0, 6, 0, 3, 8, 9, 8, 9, 6, 4, 4, 7, 9, 6, 1, 2, 9, 4, 1, 8, 5, 3, 0, 2, 4, 7, 6, 2, 3, 2

Offset: 2

Dario Piazzalunga, Dec 30 2012

Continued fraction expansion of sqrt(102) is A040091.

			10.0995049383620779533633859170696007106038989644796129418...

Cf. A040091.

RealDigits[Sqrt[102], 10, 90][[1]] (* Bruno Berselli, Jan 03 2013 *)

fpprec:90; ev(bfloat(sqrt(102))); /* Bruno Berselli, Jan 03 2013 */

sqrt(102) \\ Charles R Greathouse IV, Apr 15 2014

A187744 Numbers whose digital sum is a triangular number.

Original entry on oeis.org

0, 1, 3, 6, 10, 12, 15, 19, 21, 24, 28, 30, 33, 37, 42, 46, 51, 55, 60, 64, 69, 73, 78, 82, 87, 91, 96, 100, 102, 105, 109, 111, 114, 118, 120, 123, 127, 132, 136, 141, 145, 150, 154, 159, 163, 168, 172, 177, 181, 186, 190, 195, 201, 204, 208, 210, 213, 217

Offset: 1

Dario Piazzalunga, Jan 03 2013

Every term with some permutations can become another term of this sequence.
The subsequence of primes begins: 3, 19, 37, 73, 91, 127...
The subsequence of triangular numbers begins: 1, 3, 6, 10, 15, 21, 28, 55...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000217, A007960, A000027.

a187744 n = a187744_list !! (n-1)
a187744_list = filter ((== 1) . a010054 . a007953) [0..]
-- Reinhard Zumkeller, Jan 03 2013

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8 n]]; Select[Range[0, 300], TriangularQ[Total[IntegerDigits[#]]] &] (* T. D. Noe, Jan 03 2013 *)

If decimal expansion of n is x1 x2 ... xk then x1 + x2 + ... xk = T.

A010054(A007953(a(n))) = 1. - Reinhard Zumkeller, Jan 03 2013

A193772 Nonnegative integers whose digital difference is 0.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 202, 211, 220, 303, 312, 321, 330, 404, 413, 422, 431, 440, 505, 514, 523, 532, 541, 550, 606, 615, 624, 633, 642, 651, 660, 707, 716, 725, 734, 743, 752, 761, 770, 808, 817, 826, 835, 844, 853, 862, 871, 880

Offset: 1

Dario Piazzalunga, Jan 02 2013

The subsequence of multiples of 11 begins: 0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 220, 330, 440...

The subsequence of primes begins: 11, 101, 211, 431, 523, 541, 743, 761, 853, ... (see A156307).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A040997, A007953, A156307.

V:= proc(n,s) # n-digits numbers with sum of digits s
     option remember; local i,k;
     `union`(seq(seq(map(t -> i*10^(n-1) + t, procname(k,s-i)),k=1..n-1),i=1..min(s,9)))
end proc:
for s from 0 to 9 do V(1,s) := {s} od:
f:= proc(n) local s,k;
   `union`(seq(seq(map(t -> s*10^(n-1) + t, V(k,s)), k=1..n-1),s=1..9))
end proc:
sort(convert(`union`(seq(f(d),d=1..4)),list)); # Robert Israel, Nov 14 2024

fQ[n_] := Module[{d = IntegerDigits[n]}, d[[1]] == Total[Rest[d]]]; Select[Range[0, 1000], fQ] (* T. D. Noe, Jan 02 2013 *)

If decimal expansion of n is x1 x2 ... xk then x1 - x2 - ... - xk = 0.

Definition edited by Michel Marcus, Oct 26 2014

A185107 a(n) is the first digit of prime(n) minus the sum of the other digits.

Original entry on oeis.org

2, 3, 5, 7, 0, -2, -6, -8, -1, -7, 2, -4, 3, 1, -3, 2, -4, 5, -1, 6, 4, -2, 5, -1, 2, 0, -2, -6, -8, -3, -8, -3, -9, -11, -12, -5, -11, -8, -12, -9, -15, -8, -9, -11, -15, -17, 0, -3, -7, -9, -4, -10, -3, -4, -10, -7, -13

Offset: 1

Dario Piazzalunga, Dec 27 2012

Absolute terms are the same as A042939.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A042939, A007605.

Table[With[{id=IntegerDigits[Prime[n]]},id[[1]]-Total[Rest[id]]],{n,60}] (* Harvey P. Dale, Oct 04 2024 *)

a(n) = {digs = digits(prime(n)); digs[1] - sum(i=2, #digs, digs[i]);} \\ Michel Marcus, Aug 30 2013

a(38) corrected by Michel Marcus, Jun 14 2022

A184328 Primes whose digital product is a positive square.

Original entry on oeis.org

11, 19, 41, 149, 191, 199, 229, 263, 281, 313, 331, 419, 433, 449, 491, 499, 661, 683, 797, 821, 829, 863, 881, 911, 919, 941, 977, 991, 1229, 1289, 1433, 1499, 1559, 1669, 1747, 1889, 1933, 1949, 1999, 2129, 2383, 2693, 2819, 2833, 2963, 3319, 3391, 3413

Offset: 1

Dario Piazzalunga, Dec 24 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A053666, A007954, A007605, A065073, A068395.

[p: p in PrimesUpTo(4000) | not IsZero(t) and IsSquare(t) where t is &*Intseq(p)]; // Bruno Berselli, Dec 25 2012

fQ[n_] := Module[{d = Times @@ IntegerDigits[n]}, d > 0 && IntegerQ[Sqrt[d]]];Select[Prime[Range[1000]], fQ] (* T. D. Noe, Dec 24 2012 *)

Corrected and extended by T. D. Noe, Dec 24 2012

A220588 a(n) = 2^n - n^2 - n.

Original entry on oeis.org

1, 0, -2, -4, -4, 2, 22, 72, 184, 422, 914, 1916, 3940, 8010, 16174, 32528, 65264, 130766, 261802, 523908, 1048156, 2096690, 4193798, 8388056, 16776616, 33553782, 67108162, 134216972, 268434644, 536870042, 1073740894, 2147482656, 4294966240, 8589933470, 17179867994

Offset: 0

Dario Piazzalunga, Dec 16 2012

			a(3) = -4 because 2^3 - 3^2 - 3 = 8 - 9 - 3 = -4.
a(4) = -4 because 2^4 - 4^2 - 4 = 16 - 16 - 4 = -4.
a(5) = 2 because 2^5 - 5^2 - 5 = 32 - 25 - 5 = 2.
a(6) = 22 because 2^6 - 6^2 - 6 = 64 - 36 - 6 = 22.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A015519, A024012, A001580.

Table[2^n - n^2 - n, {n, 0, 32}] (* Alonso del Arte, Dec 16 2012 *)

A220588(n):=2^n-n^2-n$ makelist(A220588(n),n,0,20); /* Martin Ettl, Dec 18 2012 */

Vec((1 - 5*x + 7*x^2 - x^3) / ((1 - x)^3*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Aug 16 2017

a(n) = 2*a(n - 1) + ((n - 3)^2 + 3(n - 3)) = 2*a(n - 1) + A028552(n - 3) for n > 4.
a(n) = (2*a(n-1) + 7*a(n-2))*2 = A015519/2 for n > 4.
From Colin Barker, Aug 16 2017: (Start)
G.f.: (1 - 5*x + 7*x^2 - x^3) / ((1 - x)^3*(1 - 2*x)).
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>3.
(End)

a(3) corrected by Charles A. Dagino, Aug 16 2017

A220425 a(n) = n^2 + 2*n + 2^n.

Original entry on oeis.org

1, 5, 12, 23, 40, 67, 112, 191, 336, 611, 1144, 2191, 4264, 8387, 16608, 33023, 65824, 131395, 262504, 524687, 1049016, 2097635, 4194832, 8389183, 16777840, 33555107, 67109592, 134218511, 268436296, 536871811, 1073742784, 2147484671, 4294968384, 8589935747

Offset: 0

Dario Piazzalunga, Dec 14 2012

Iain Fox, Table of n, a(n) for n = 0..3321
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Table[n^2 + 2*n + 2^n, {n, 0, 50}] (* T. D. Noe, Dec 14 2012 *)

makelist(n^2+2*n+2^n, n, 0, 20); /* Martin Ettl, Dec 15 2012 */

a(n)=n^2+2*n+2^n \\ Charles R Greathouse IV, Oct 07 2015

first(n) = Vec((1 - 4*x^2 + x^3)/((1 - x)^3 * (1 - 2*x)) + O(x^n)) \\ Iain Fox, Aug 08 2018

G.f.: (x^3-4*x^2+1) / ((x-1)^3*(2*x-1)). - Colin Barker, May 09 2013

E.g.f.: exp(x) * (exp(x) + x*(3 + x)). - Iain Fox, Aug 08 2018

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User: Dario Piazzalunga

A187768 Decimal expansion of square root of 103.

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A192106 Decimal expansion of square root of 102.

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A187744 Numbers whose digital sum is a triangular number.

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A193772 Nonnegative integers whose digital difference is 0.

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A185107 a(n) is the first digit of prime(n) minus the sum of the other digits.

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A184328 Primes whose digital product is a positive square.

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A220588 a(n) = 2^n - n^2 - n.

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