keyword:hear - cp's OEIS frontend

A229818 Even bisection gives sequence a itself, n->a(2(3n+k)-1) gives k-th differences of a for k=1..3 with a(n)=n for n<2.

0, 1, 1, -1, 1, -1, -1, 0, 1, -2, -1, 6, -1, -2, 0, 4, 1, -8, -2, 2, -1, -4, 6, 6, -1, -2, -2, 2, 0, -1, 4, 0, 1, 1, -8, -1, -2, 1, 2, 0, -1, -4, -4, 1, 6, -4, 6, 8, -1, -3, -2, 4, -2, 2, 2, 1, 0, 6, -1, -20, 4, 7, 0, -14, 1, 20, 1, -7, -8, 6, -1, -3, -2, -1

Offset: 0

Alois P. Heinz, Sep 30 2013

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A005590, A229817, A229819, A229820, A229821, A229822, A229823, A229824, A229825.

a:= proc(n) option remember; local m, q, r;
      m:= (irem(n, 6, 'q')+1)/2;
      `if`(n<2, n, `if`(irem(n, 2, 'r')=0, a(r),
      add(a(q+m-j)*(-1)^j*binomial(m, j), j=0..m)))
    end:
seq(a(n), n=0..100);

a[n_] := a[n] = Module[{m, q, r, q2, r2}, {q, r} = QuotientRemainder[n, 6]; m = (r+1)/2; If[n<2, n, {q2, r2} = QuotientRemainder[n, 2]; If[r2 == 0, a[q2], Sum[a[q+m-j]*(-1)^j*Binomial[m, j], {j, 0, m}]]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 08 2017, translated from Maple *)

{M=Map(); a(n)= n&&n>>=valuation(n, 2); my(r); mapisdefined(M, n, &r) && return(r); r=if(n<2, n, my(m=n%6, k=n\6); if(1==m, a(k+1)-a(k), 3==m, a(k+2)-2*a(k+1)+a(k), a(k+3)-3*a(k+2)+3*a(k+1)-a(k))); mapput(~M, n, r); r;} \\ Ruud H.G. van Tol, Nov 19 2024

a(2*n)   = a(n),
a(6*n+1) = a(n+1) - a(n),
a(6*n+3) = a(n+2) - 2*a(n+1) + a(n),
a(6*n+5) = a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n).

A260643 Start a spiral of numbers on a square grid, with the initial square as a(1) = 1. a(n) is the smallest positive integer not equal to or previously adjacent (horizontally/vertically) to its neighbors. See the Comments section for a more exact definition.

Original entry on oeis.org

1, 2, 3, 4, 2, 5, 3, 6, 7, 1, 8, 7, 4, 8, 5, 6, 4, 9, 7, 10, 1, 9, 8, 11, 3, 12, 11, 10, 12, 13, 1, 12, 14, 9, 10, 14, 1, 15, 6, 13, 2, 16, 3, 17, 11, 13, 5, 14, 2, 11, 6, 14, 13, 9, 15, 18, 2, 19, 5, 15, 16, 4, 17, 20, 2, 21, 3, 18, 16, 17, 5, 20, 4, 19, 6

Offset: 1

Peter Kagey, Nov 11 2015

A more detailed definition from Antti Karttunen, Dec 09 2015: (Start)
After a(1) = 1, for the next term always choose the smallest number k >= 1 such that neither k and a(n-1) nor k and a(A265400(n)) [in case A265400(n) > 0] are equal, and neither of these pairs occur anywhere adjacent to each other (horizontally or vertically) in so far constructed spiral. Here A265400(n) gives the index of the nearest horizontally or vertically adjacent inner neighbor of the n-th term in spiral, or 0 if n is one of the corner cases A033638.
The condition "... do not occur anywhere adjacent to each other (horizontally or vertically) in so far constructed spiral" can be more formally stated as: there is no such 1 < j < n, for which either the unordered pair {a(j),a(j-1)} or [in case A265400(j) > 0] also the unordered pair {a(j),a(A265400(j))} would be equal to either of the unordered pair {k,a(n-1)} or the unordered pair {k,a(A265400(n))} [in case A265400(n) > 0], where k is the term chosen for a(n). (See also my reference Scheme-implementation.)
(End)

			a(8) = 6 because pairs {1,2}, {1,4} and {1,5} already occur, the immediately adjacent terms are 1 and 3, thus neither number can be used, so the smallest usable number is 6.
a(12) = 7 because 1 and 2 are already adjacent to 8; 2, 4, 5, and 6 are already adjacent to 3.
The following illustration is the timeline of spiral's construction step-by-step:
        |      |   3  |  43  | 243  | 243  |     |  243  |  243  |  2437
    1   |  12  |  12  |  12  |  12  | 512  |     |  512  |  5128 |  5128
        |      |      |      |      |      | ... |  3671 |  3671 |  3671
        |      |      |      |      |      |     |       |       |
  a(1)=1|a(2)=2|a(3)=3|a(4)=4|a(5)=2|a(6)=5|     |a(10)=1|a(11)=8|a(12)=7
Indices of this spiral are shown below using the base-36 system, employing as its placeholder values the digits 0-9 and letter A-Z. The 1 at the center is where the spiral starts:
            ZYXWV
           HGFEDU
           I543CT
           J612BS
           K789AR
           LMNOPQ

Peter Kagey, Table of n, a(n) for n = 1..10000
Antti Karttunen, R6RS-Scheme program for computing this sequence (with a naive algorithm)
Peter Kagey, Ruby program for computing this sequence.

Cf. A033638, A123663, A265400, A265579.
Cf. A272573 (analogous sequence on a hexagonal tiling).
Cf. A265414 (positions of records, where n occurs for the first time), A265415 (positions of ones).

A325794 Number of divisors of n minus the sum of prime indices of n.

Original entry on oeis.org

1, 1, 0, 1, -1, 1, -2, 1, -1, 0, -3, 2, -4, -1, -1, 1, -5, 1, -6, 1, -2, -2, -7, 3, -3, -3, -2, 0, -8, 2, -9, 1, -3, -4, -3, 3, -10, -5, -4, 2, -11, 1, -12, -1, -1, -6, -13, 4, -5, -1, -5, -2, -14, 1, -4, 1, -6, -7, -15, 5, -16, -8, -2, 1, -5, 0, -17, -3, -7

Offset: 1

Gus Wiseman, May 23 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with sum A056239(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of positive terms are A325795.
Positions of nonnegative terms are A325796.
Positions of negative terms are A325797.
Positions of nonpositive terms are A325798.
Positions of 1's are A325792.
Positions of 0's are A325793.
Positions of -1's are A325694.
Cf. A000005, A002033, A056239, A112798, A299702, A325780, A325781.

Table[DivisorSigma[0,n]-Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]],{n,100}]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A325794(n) = (numdiv(n)-A056239(n)); \\ Antti Karttunen, May 26 2019

a(n) = A000005(n) - A056239(n).

A140263 Permutation of nonnegative integers obtained by interleaving A117967 and A117968.

Original entry on oeis.org

0, 1, 2, 5, 7, 3, 6, 4, 8, 17, 22, 15, 21, 16, 23, 11, 19, 9, 18, 10, 20, 14, 25, 12, 24, 13, 26, 53, 67, 51, 66, 52, 68, 47, 64, 45, 63, 46, 65, 50, 70, 48, 69, 49, 71, 35, 58, 33, 57, 34, 59, 29, 55, 27, 54, 28, 56, 32, 61, 30, 60, 31, 62, 44, 76, 42, 75, 43, 77, 38, 73, 36

Offset: 0

Antti Karttunen, May 19 2008, originally described in a posting at the SeqFan mailing list on Sep 15 2005

Inverse: A140264. Bisections: A117967 & A117968. a(n) = A140265(n+1)-1.

Cf. A004488, A117966.

from sympy import ceiling
from sympy.ntheory.factor_ import digits
def a004488(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3)
def a117968(n):
    if n==1: return 2
    if n%3==0: return 3*a117968(n/3)
    elif n%3==1: return 3*a117968((n - 1)/3) + 2
    else: return 3*a117968((n + 1)/3) + 1
def a117967(n): return 0 if n==0 else a117968(-n) if n<0 else a004488(a117968(n))
def a001057(n): return -(-1)**n*ceiling(n/2)
def a(n): return a117967(a001057(n)) # Indranil Ghosh, Jun 07 2017

a(n) = A117967(A001057(n)). (Assuming that the domain of A117967 is the whole Z line.)

A240830 a(n)=1 for n <= s+k; thereafter a(n) = Sum(a(n-i-s-a(n-i-1)),i=0..k-1) where s=0, k=7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 7, 7, 7, 7, 7, 7, 7, 13, 13, 13, 13, 13, 13, 19, 13, 19, 19, 19, 19, 25, 19, 25, 19, 25, 25, 31, 25, 31, 25, 31, 25, 31, 31, 37, 31, 37, 31, 37, 37, 37, 37, 43, 37, 43, 43, 43, 43, 43, 43, 49, 49, 49, 49, 49, 49, 49, 55, 55, 55, 55, 55, 55, 61, 55, 61, 61, 61, 61, 67, 61, 67, 61, 67, 67, 73

Offset: 1

N. J. A. Sloane, Apr 16 2014

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Joseph Callaghan, John J. Chew III, and Stephen M. Tanny, On the behavior of a family of meta-Fibonacci sequences, SIAM Journal on Discrete Mathematics 18.4 (2005): 794-824. See Table 2.1.
Index entries for Hofstadter-type sequences

Same recurrence as A240828, A120503 and A046702.
See also A240831, A240832.
Callaghan et al. (2005)'s sequences T_{0,k}(n) for k=1 through 7 are A000012, A046699, A046702, A240835, A241154, A241155, A240830.

#T_s,k(n) from Callaghan et al. Eq. (1.7).
s:=0; k:=7;
a:=proc(n) option remember; global s,k;
if n <= s+k then 1
else
    add(a(n-i-s-a(n-i-1)),i=0..k-1);
fi; end;
t1:=[seq(a(n),n=1..100)];

A240830[n_]:=A240830[n]=If[n<=7,1,Sum[A240830[n-i-A240830[n-i-1]],{i,0,6}]];
Array[A240830,100] (* Paolo Xausa, Dec 06 2023 *)

A051933 Triangle T(n,m) = Nim-sum (or XOR) of n and m, read by rows, 0<=m<=n.

Original entry on oeis.org

0, 1, 0, 2, 3, 0, 3, 2, 1, 0, 4, 5, 6, 7, 0, 5, 4, 7, 6, 1, 0, 6, 7, 4, 5, 2, 3, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 9, 10, 11, 12, 13, 14, 15, 0, 9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0

Offset: 0

N. J. A. Sloane, Dec 20 1999

			{0},
{1,0},
{2,3,0},
{3,2,1,0}, ...

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
J. H. Conway, On Numbers and Games, Academic Press, p. 52.

Reinhard Zumkeller, Rows n = 0..127 of triangle, flattened first 50 rows by R. J. Mathar
Index entries for sequences related to Nim-sums

Cf. A224915 (row sums), A003987 (array), A051910 (Nim-product).

Other triangles: A080099 (AND), A080098 (OR), A265705 (IMPL), A102037 (CNIMPL), A002262 (k).

import Data.Bits (xor)
a051933 n k = n `xor` k :: Int
a051933_row n = map (a051933 n) [0..n]
a051933_tabl = map a051933_row [0..]
-- Reinhard Zumkeller, Aug 02 2014, Aug 13 2013

using IntegerSequences
A051933Row(n) = [Bits("XOR", n, k) for k in 0:n]
for n in 0:10 println(A051933Row(n)) end  # Peter Luschny, Sep 25 2021

nimsum := proc(a,b) local t1,t2,t3,t4,l; t1 := convert(a+2^20,base,2); t2 := convert(b+2^20,base,2); t3 := evalm(t1+t2); map(x->x mod 2, t3); t4 := convert(evalm(%),list); l := convert(t4,base,2,10); sum(l[k]*10^(k-1), k=1..nops(l)); end; # memo: adjust 2^20 to be much bigger than a and b
AT := array(0..N,0..N); for a from 0 to N do for b from a to N do AT[a,b] := nimsum(a,b); AT[b,a] := AT[a,b]; od: od:
# Alternative:
A051933 := (n, k) -> Bits:-Xor(n, k):
seq(seq(A051933(n, k), k=0..n), n=0..12); # Peter Luschny, Sep 23 2019

Flatten[Table[BitXor[m, n], {m, 0, 12}, {n, 0, m}]] (* Jean-François Alcover, Apr 29 2011 *)

More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

A105870 Fibonacci sequence (mod 7).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, 1, 1, 2, 3

Offset: 0

Shyam Sunder Gupta, May 05 2005

Sequence is periodic with Pisano period 16 = A001175(7).

			a(5) = 5 because Fibonacci(5) = 5.
a(6) = 1 because Fibonacci(6) = 8 and 8 mod 7 = 1.
a(7) = 6 because Fibonacci(7) = 13 and 13 mod 7 = 6.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Brady Haran, Fibonacci Tartan and Bagpipes, Numberphile video (2013). The music by Alan Stewart at 1:53 to 3:20 has pitch based on this sequence.
Wayne Peng, ABC Implies There are Infinitely Many non-Fibonacci-Wieferich Primes - An Application of ABC Conjecture over Number Fields, arXiv:1511.05645 [math.NT], 2015.
Diana Savin and Elif Tan, On Companion sequences associated with Leonardo quaternions: Applications over finite fields, arXiv:2403.01592 [math.CO], 2024. See p. 10.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

a105870 n = a105870_list !! (n-1)
a105870_list = 1 : 1 : zipWith (\u v -> (u + v) `mod` 7)
                               (tail a105870_list) a105870_list
-- Reinhard Zumkeller, Jan 15 2014

[Fibonacci(n) mod 7: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

Table[Mod[Fibonacci[n], 7], {n, 0, 100}] (* Alonso del Arte, Jul 29 2013 *)

a(n)=fibonacci(n)%7 \\ Charles R Greathouse IV, Jun 04 2013

a(n)=lift(((Mod([1,1;1,0],7))^n)[1,2]) \\ Charles R Greathouse IV, Jun 04 2013

a(n)=fibonacci(n%16)%7 \\ Charles R Greathouse IV, Jan 06 2016

A105870_list, a, b, = [], 0, 1
for _ in range(10**3):
    A105870_list.append(a)
    a, b = b, (a+b) % 7 # Chai Wah Wu, Nov 26 2015

G.f.: - x*(1 + x + 2*x^2 + 3*x^3 + 5*x^4 + x^5 + 6*x^6 + 6*x^8 + 6*x^9 + 5*x^10 + 4*x^11 + 2*x^12 + 6*x^13 + x^14)/((x - 1)*(1 + x)*(1 + x^2)*(1 + x^4)*(1 + x^8)). - R. J. Mathar, Jul 14 2012

a(1) = a(2) = 1, then a(n) = (a(n - 2) + a(n - 1)) mod 7. - Alonso del Arte, Jul 30 2013

a(0)=0 from Vincenzo Librandi, Feb 04 2014

A142151 a(n) = OR{k XOR (n-k): 0<=k<=n}.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 6, 7, 14, 13, 14, 11, 14, 13, 14, 15, 30, 29, 30, 27, 30, 29, 30, 23, 30, 29, 30, 27, 30, 29, 30, 31, 62, 61, 62, 59, 62, 61, 62, 55, 62, 61, 62, 59, 62, 61, 62, 47, 62, 61, 62, 59, 62, 61, 62, 55, 62, 61, 62, 59, 62, 61, 62, 63, 126, 125, 126, 123, 126, 125

Offset: 0

Reinhard Zumkeller, Jul 15 2008

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Logical Convolutions

Cf. A003817, A000004, A142149, A086099, A142150, A001477, A089633.

import Data.Bits (xor, (.|.))
a142151 :: Integer -> Integer
a142151 = foldl (.|.) 0 . zipWith xor [0..] . reverse . enumFromTo 1
-- Reinhard Zumkeller, Mar 31 2015

using IntegerSequences
A142151List(len) = [Bits("CIMP", n, n+1) for n in 0:len]
println(A142151List(69))  # Peter Luschny, Sep 25 2021

A142151 := n -> n + Bits:-Nor(n, n+1):
seq(A142151(n), n=0..69); # Peter Luschny, Sep 26 2019

from functools import reduce
from operator import or_
def A142151(n): return 0 if n == 0 else reduce(or_,(k^n-k for k in range(n+1))) if n % 2 else (1 << n.bit_length()-1)-1 <<1 # Chai Wah Wu, Jun 30 2022

a(2*n) = 2*(A062383(n)-1);

A023416(a(n)) <= 1.

A210838 Coordinates (x,y) of the endpoint of a structure (or curve) formed by Q-toothpicks of size = 1..n. The inflection points are the n-th nodes if n is a triangular number A000217.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 0, 6, -4, 10, 1, 15, 7, 9, 14, 2, 22, 10, 13, 19, 3, 9, -8, -2, -20, 10, -7, 23, 7, 9, -8, -6, -24, -22, -7, -39, 11, -21, -8, -2, -28, -22, -7, -43, 15, -65, -8, -88, -32, -64, -7, -39, 19, -65, -8, -92, -36, -64, -65, -35, -95, -65, -64, -96

Offset: 0

Omar E. Pol, Mar 28 2012

It appears there is an infinite family of this type of curves or structures in which the terms of a sequence of positive integers are represented as inflection points and the gaps between them are essentially represented as nodes of spirals. For example: consider a structure formed by Q-toothpicks of size = Axxxxxa connected by their endpoints in which the inflection points are the exposed endpoints at stage Axxxxxb(n), where both Axxxxxa and Axxxxxb are sequences with positive integers. Also instead of Q-toothpicks we can use semicircumferences or also 3/4 of circumferences. For the definition of Q-toothpicks see A187210.
We start at stage 0 with no Q-toothpicks.
At stage 1 we place a Q-toothpick of size 1 centered at (1,0) with its endpoints at (0,0) and (1,1). Since 1 is a positive triangular number we have that the end of the curve is also an inflection point.
At stage 2 we place a Q-toothpick of size 2 centered at (1,3) with its endpoints at (1,1) and (3,3).
At stage 3 we place a Q-toothpick of size 3 centered at (0,3) with its endpoints at (3,3) and (0,6). Since 3 is a positive triangular number we have that the end of the curve is also an inflection point.
At stage 4 we place a Q-toothpick of size 4 centered at (0,10) with its endpoints at (0,6) and (-4,10).
And so on...

			-------------------------------------
Stage n also              The end as
the size of     Pair      inflection
Q-toothpick   (x    y)      point
-------------------------------------
.    0         0,   0,        -
.    1         1,   1,       Yes
.    2         3,   3,        -
.    3         0,   6,       Yes
.    4        -4,  10,        -
.    5         1,  15,        -
.    6         7,   9,       Yes
.    7        14,   2,        -
.    8        22,  10,        -
.    9        13,  19,        -
.   10         3,   9,       Yes
.   11        -8,  -2,        -
.   12       -20,  10,        -
.   13        -7,  23,        -
.   14         7,   9,        -
.   15        -8,  -6,       Yes

Paolo Xausa, Table of n, a(n) for n = 0..9999
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Paolo Xausa, Animation of terms n = 0..41 (first 21 coordinate pairs), where orange dots are toothpick endpoints (hollow dots are inflection points) and blue dots are toothpick centers
Paolo Xausa, Animation of terms n = 0..507 (first 254 coordinate pairs)
Paolo Xausa, Scatterplot of 10^5 coordinate pairs
Index entries for sequences related to toothpick sequences

Cf. A210841 (the same idea for primes).

Cf. A000217, A005132, A187210, A210606, A211000, A211011.

A210838[nmax_]:=Module[{ep={0, 0}, angle=3/4Pi, turn=Pi/2, infl=0}, Join[{ep}, Table[If[n>1&&IntegerQ[Sqrt[8(n-1)+1]], infl++, If[Mod[infl, 2]==1, turn*=-1]; angle-=turn; infl=0]; ep=AngleVector[ep, {Sqrt[2]n, angle}], {n, nmax}]]];
A210838[100] (* Generates 101 coordinate pairs *) (* Paolo Xausa, Jan 12 2023 *)

A210838(nmax) = my(ep=vector(nmax+1), turn=1, infl=0, ep1, ep2); ep[1]=[0, 0]; if(nmax==0, return(ep)); ep[2]=[1, 1]; for(n=2, nmax, ep1=ep[n-1]; ep2=ep[n]; if(issquare((n-1)<<3+1), infl++; ep[n+1]=[ep2[1]+n*sign(ep2[1]-ep1[1]), ep2[2]+n*sign(ep2[2]-ep1[2])], if(infl%2, turn*=-1); infl=0; ep[n+1]=[ep2[1]-turn*n*sign(ep1[2]-ep2[2]), ep2[2]+turn*n*sign(ep1[1]-ep2[1])])); ep;
A210838(100) \\ Generates 101 coordinate pairs - Paolo Xausa, Jan 12 2023

from numpy import sign
from sympy import integer_nthroot
def A210838(nmax):
    ep, turn, infl = [(0, 0), (1, 1)], 1, 0
    for n in range(2, nmax + 1):
        ep1, ep2 = ep[-2], ep[-1]
        if integer_nthroot(((n - 1) << 3) + 1, 2)[1]: # Continue straight
            infl += 1
            dx = n * sign(ep2[0] - ep1[0])
            dy = n * sign(ep2[1] - ep1[1])
        else: # Turn
            if infl % 2: turn *= -1
            infl = 0
            dx = turn * n * sign(ep2[1] - ep1[1])
            dy = turn * n * sign(ep1[0] - ep2[0])
        ep.append((ep2[0] + dx, ep2[1] + dy))
    return ep[:nmax+1]
print(A210838(100)) # Generates 101 coordinate pairs - Paolo Xausa, Jan 12 2023

a(30)-a(33) corrected and more terms by Paolo Xausa, Jan 12 2023

A239690 Base 4 sum of digits of prime(n).

Original entry on oeis.org

2, 3, 2, 4, 5, 4, 2, 4, 5, 5, 7, 4, 5, 7, 8, 5, 8, 7, 4, 5, 4, 7, 5, 5, 4, 5, 7, 8, 7, 5, 10, 5, 5, 7, 5, 7, 7, 7, 8, 8, 8, 7, 11, 4, 5, 7, 7, 10, 8, 7, 8, 11, 7, 11, 2, 5, 5, 7, 4, 5, 7, 5, 7, 8, 7, 8, 7, 4, 8, 7, 5, 8, 10, 7, 10, 11, 5, 7, 5, 7, 8, 7, 11, 7

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-4 dominance order on the natural numbers.

			The sixth prime is 13, 13 in base 4 is (3,1) so a(6)=3+1=4.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.

Cf. A007605, A014499, A053737, A239619.

Cf. A000040, A007090.

a239690 = a053737 . a000040  -- Reinhard Zumkeller, Mar 20 2015

[&+Intseq(NthPrime(n),4): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 4], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

[sum(i.digits(base=4)) for i in primes_first_n(200)]

a(n) = A053737(A000040(n)).

cp's OEIS Frontend

A229818 Even bisection gives sequence a itself, n->a(2*(3*n+k)-1) gives k-th differences of a for k=1..3 with a(n)=n for n<2.

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A260643 Start a spiral of numbers on a square grid, with the initial square as a(1) = 1. a(n) is the smallest positive integer not equal to or previously adjacent (horizontally/vertically) to its neighbors. See the Comments section for a more exact definition.

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A325794 Number of divisors of n minus the sum of prime indices of n.

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A140263 Permutation of nonnegative integers obtained by interleaving A117967 and A117968.

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A240830 a(n)=1 for n <= s+k; thereafter a(n) = Sum(a(n-i-s-a(n-i-1)),i=0..k-1) where s=0, k=7.

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A051933 Triangle T(n,m) = Nim-sum (or XOR) of n and m, read by rows, 0<=m<=n.

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Julia

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A105870 Fibonacci sequence (mod 7).

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A229818 Even bisection gives sequence a itself, n->a(2(3n+k)-1) gives k-th differences of a for k=1..3 with a(n)=n for n<2.