Leonardo Sznajder - cp's OEIS frontend

A359048 a(n) is the minimum denominator d such that the decimal expansion of n/d is eventually periodic with periodicity not equal to zero.

3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 9, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 9, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 11, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 9, 3, 3, 7, 3, 3, 7

Offset: 1

Leonardo Sznajder, Dec 14 2022

a(n) is the smallest prime power p^e that does not divide n, where p is a prime that doesn't divide 10, and e >= 1. - Jon E. Schoenfield, Dec 24 2022

			For n=21, a(21) = 9 because 21/9 = 2.333... (periodic) and 9 is the first number with that property for numerator 21. That's because 21/2 = 10.5, 21/3 = 7, 21/4 = 5.25, 21/5 = 4.2, 21/6 = 3.5, 21/7 = 3 and 21/8 = 2.625.

Antti Karttunen, Table of n, a(n) for n = 1..20000

f:= proc(n) local d;
for d from 3 by 2 do
  if (n mod d <> 0) and (d mod 5 <> 0) and nops(numtheory:-factorset(d))=1 then return d fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Jan 19 2023

a(n) = for(d=1, oo, my(p); if (isprimepower(d, &p) && (10 % p) && (n % d), return(d))); \\ Michel Marcus, Dec 28 2022

More terms from Michel Marcus, Dec 28 2022

A340459 a(n) is the sum of the numbers adjacent to n in a triangle in which the nonnegative integers are placed from top to bottom and from left to right.

Original entry on oeis.org

3, 9, 10, 18, 26, 23, 31, 44, 50, 40, 48, 68, 74, 80, 61, 69, 98, 104, 110, 116, 86, 94, 134, 140, 146, 152, 158, 115, 123, 176, 182, 188, 194, 200, 206, 148, 156, 224, 230, 236, 242, 248, 254, 260, 185, 193, 278, 284, 290, 296, 302, 308, 314, 320, 226, 234, 338

Offset: 0

Leonardo Sznajder, Jan 10 2021

nonn

The triangle of nonnegative integers begins:
      0
     1 2
    3 4 5
   6 7 8 9
     ...

			For n=4:
- the numbers adjacent to 4 are 1, 2, 3, 5, 7 and 8,
- so a(4) = 1 + 2 + 3 + 5 + 7 + 8 = 26.

Cf. A214177.

T[i_,j_]:=Binomial[i+1,2]+j; a[i_,j_]:=If[j-1>=0,T[i,j-1],0]+If[i-1>=0&&j-1>=0, T[i-1,j-1],0]+If[i-1>=0&&j<=i-1,T[i-1,j],0]+If[j+1<=i,T[i,j+1],0]+T[i+1,j]+T[i+1,j+1]; Flatten[Table[a[i,j],{i,0,12},{j,0,i}]] (* Stefano Spezia, Jan 28 2021 *)

More terms from Stefano Spezia, Jan 28 2021

A296420 Period of last digit of multiples of n.

Original entry on oeis.org

1, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 10, 5, 10, 5, 2, 5, 10

Offset: 0

Leonardo Sznajder, Dec 11 2017

The list is periodic, with period 10.

			a(6)=5 because multiples of 6 are 6, 12, 18, 24, 30, 36, 42 and the last digits of those numbers are 6,2,8,4,0,6,2,... with a period of 5.

Cf. A054531.

Array[10/GCD[#, 10] &, 77] (* Michael De Vlieger, Dec 23 2017 *)

a(n) = 10/gcd(n, 10) \\ Iain Fox, Dec 11 2017

More terms from Michael De Vlieger, Dec 23 2017.

Term a(0) = 1 prepended by Halfdan Skjerning, Jun 18 2019

A244151 0-additive sequence: start with a(1) = 2; thereafter, a(n) = smallest number not already in sequence which is not the sum of any previous two terms.

Original entry on oeis.org

2, 3, 4, 8, 9, 14, 15, 20, 21, 26, 27, 32, 33, 38, 39, 44, 45, 50, 51, 56, 57, 62, 63, 68, 69, 74, 75, 80, 81, 86, 87, 92, 93, 98, 99, 104, 105, 110, 111, 116, 117, 122, 123, 128, 129, 134, 135, 140, 141, 146, 147, 152, 153, 158, 159, 164, 165, 170, 171, 176, 177, 182, 183, 188, 189, 194, 195, 200, 201, 206, 207, 212, 213

Offset: 1

Leonardo Sznajder, Jun 21 2014

As A033627, but first term is 2.

4 and the numbers in A047243. - Joerg Arndt, Jun 22 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A033627, A047243.

f[s_List] := Block[{k = s[[-1]] + 1, ss = Union[Plus @@@ Subsets[s, {2}]]}, While[ MemberQ[ss, k], k++]; Append[s, k]]; Nest[f, {2}, 70] (* Robert G. Wilson v, Jun 23 2014 *)

Vec(x*(x^5+3*x^3-x^2+x+2)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Jun 26 2014

a(2n) = 6(n-1)+2 & a(2n+1) = 6(n-1)+3 for n>1. - Robert G. Wilson v, Jun 23 2014
From Colin Barker, Jun 26 2014: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 6.
G.f.: x*(x^5 + 3*x^3 - x^2 + x + 2)/((x - 1)^2*(x + 1)). (End)
E.g.f.: (x^3 + 3*x^2 + 30*x + 24)/6 + (3*x - 4)*cosh(x) + 3*(x - 2)*sinh(x). - Stefano Spezia, Apr 15 2023

Added terms >= 20, Joerg Arndt, Jun 22 2014

A161367 Primes such that prime(k)-prime(k-1)+1 is not prime.

Original entry on oeis.org

97, 127, 307, 331, 367, 397, 409, 457, 487, 499, 691, 709, 727, 751, 769, 787, 853, 877, 907, 919, 937, 967, 991, 1117, 1171, 1201, 1361, 1381, 1423, 1447, 1531, 1567, 1579, 1597, 1657, 1693, 1741, 1831, 1861, 1987, 2011, 2053, 2161, 2203, 2221, 2251, 2281, 2371, 2437, 2467

Offset: 1

Leonardo Sznajder, Jun 08 2009

nonn

			a(1)=97 because it is the first prime, prime(k), such that prime(k)-prime(k-1)+1 is not prime: 97-89+1=15, and 15 is not prime.

Cf. A000040 (the prime numbers), A001223 (difference between primes).

isok(p) = isprime(p) && !isprime(p-precprime(p-1)+1); \\ Michel Marcus, May 12 2024

More terms from Michel Marcus, May 12 2024

A160536 a(n) = Fibonacci(n) + n^2.

Original entry on oeis.org

0, 2, 5, 11, 19, 30, 44, 62, 85, 115, 155, 210, 288, 402, 573, 835, 1243, 1886, 2908, 4542, 7165, 11387, 18195, 29186, 46944, 75650, 122069, 197147, 318595, 515070, 832940, 1347230, 2179333, 3525667, 5704043, 9228690, 14931648, 24159186, 39089613, 63247507

Offset: 0

Leonardo Sznajder, May 18 2009

			a(6) = Fibonacci(6) + 6^2 = 8 + 36 = 44.

Bruno Berselli, Table of n, a(n) for n = 0..1000 (terms 0..285 from Vincenzo Librandi).
Math Marathon (MM95) (In Russian) [Vladimir Joseph Stephan Orlovsky, May 12 2010]
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Equals A000045 + A000290.

Cf. A001611, A002062, A212272.

[ Fibonacci(n)+n^2: n in [0..40] ]; // Klaus Brockhaus, May 22 2009

Table[Fibonacci[n]+n^2,{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)

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

a(n) = a(n-4) - a(n-3) - 2*a(n-2) + 3*a(n-1) - 2 for n > 3; a(0)=0, a(1)=2, a(2)=5, a(3)=11. - Klaus Brockhaus, May 22 2009

G.f.: x*(2-3*x+x^2-2*x^3) / ((1-x)^3*(1-x-x^2)). - Klaus Brockhaus, May 22 2009

Edited and extended by Klaus Brockhaus, May 22 2009

A139330 A-numbers of sequences in the OEIS not beginning with a 1.

Original entry on oeis.org

4, 14, 20, 22, 28, 30, 32, 33, 35, 36, 37, 38, 40, 43, 45, 49, 51, 52, 54, 57, 58, 65, 66, 71, 73, 75, 76, 78, 81, 87, 93, 94, 96, 100, 101, 102, 103, 107, 114, 120, 129, 130, 131, 133, 134, 139, 147, 150, 153, 155, 159, 163, 167, 173, 176, 181, 183, 184, 185

Offset: 1

Leonardo Sznajder, Jun 05 2008

			a(3)=20 because the third sequence not begining with a "1" is A000020.

Nathaniel Johnston, Table of n, a(n) for n = 1..1000 (as of Sep 27 2011)

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User: Leonardo Sznajder

A359048 a(n) is the minimum denominator d such that the decimal expansion of n/d is eventually periodic with periodicity not equal to zero.

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A340459 a(n) is the sum of the numbers adjacent to n in a triangle in which the nonnegative integers are placed from top to bottom and from left to right.

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A296420 Period of last digit of multiples of n.

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A244151 0-additive sequence: start with a(1) = 2; thereafter, a(n) = smallest number not already in sequence which is not the sum of any previous two terms.

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A161367 Primes such that prime(k)-prime(k-1)+1 is not prime.

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A160536 a(n) = Fibonacci(n) + n^2.

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A139330 A-numbers of sequences in the OEIS not beginning with a 1.

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