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Let Q consist of 1 together with the primes ( A008578); form the lexicographically earliest infinite sequence S of distinct positive numbers with the property that a(k) is in Q if and only if k is a term in S.
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0
1, 2, 3, 5, 7, 4, 11, 9, 13, 12, 17, 19, 23, 15, 29, 18, 31, 37, 41, 21, 43, 24, 47, 53, 26, 59, 28, 61, 67, 32, 71, 73, 34, 79, 36, 83, 89, 39, 97, 42, 101, 103, 107, 45, 109, 48, 113, 127, 50, 131, 52, 137
COMMENTS
In the early 20th century, 1 was regarded as a prime (see A008578). The present sequence is therefore a 20th-century analog of A121053. That is, the sequence answers the question "Which terms are in Q?", and is the lexicographically earliest answer. See A121053 for further information. Like A121053, this is an example of a "Lexicographically Earliest Sequence" for which there is a greedy algorithm: no backtracking is needed. Theorem. Let p(k) = k-th prime, c(k) = k-th composite number. For n >= 7, if n is a prime or n = c(2*t) for some t, then a(n) = p(k) where k = floor((n+PrimePi(n)-1)/2); otherwise, n = c(2*t-1) for some t and a(n) = c(2*t).
REFERENCES
N. J. A. Sloane, The Remarkable Sequences of Éric Angelini, MS in preparation, December 2024.
EXAMPLE
1 is the smallest possible choice for a(1), and 1 is in Q, and it turns out that there is no contradiction in choosing a(1) = 1.
After a(5) = 7, 4 is the smallest number not yet in the sequence, and a(4) = 5 is in Q, so we can try a(6) = 4 (and it turns out that this does not lead to a contradiction later).
Decimal expansion of Pi*G - 7*zeta(3)/4, where G = A006752.
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0
7, 7, 3, 9, 9, 1, 2, 0, 1, 0, 7, 8, 8, 7, 1, 1, 5, 2, 3, 2, 8, 0, 3, 8, 3, 8, 3, 8, 7, 6, 5, 1, 0, 3, 1, 6, 2, 7, 6, 1, 2, 8, 3, 8, 8, 4, 5, 6, 8, 0, 6, 0, 3, 2, 6, 2, 5, 7, 2, 0, 5, 8, 0, 3, 0, 6, 6, 4, 4, 5, 7, 9, 2, 6, 5, 7, 4, 3, 0, 3, 4, 6, 7, 7, 5, 5, 8, 5, 3, 6, 4, 1, 4, 6, 9, 0, 6, 2, 9, 2
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.
FORMULA
Equals Sum_{n>=1} (-1)^(n+1)/n^2 Sum_{k=0..n-1} 1/(2*k + 1) (see Finch).
MAPLE
0.773991201078871152328038383876510316276128388...
MATHEMATICA
RealDigits[Pi Catalan-7Zeta[3]/4, 10, 100][[1]]
Decimal expansion of Pi*G - 33*zeta(3)/16, where G = A006752.
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0
3, 9, 8, 3, 4, 8, 4, 1, 8, 8, 4, 1, 4, 9, 7, 9, 3, 8, 1, 4, 0, 6, 2, 0, 2, 0, 8, 4, 0, 4, 1, 8, 2, 1, 9, 4, 1, 6, 2, 0, 7, 0, 1, 7, 2, 1, 0, 0, 4, 0, 0, 1, 3, 2, 0, 6, 5, 6, 3, 5, 7, 1, 9, 2, 6, 2, 3, 2, 0, 1, 3, 9, 9, 5, 7, 5, 2, 0, 1, 9, 3, 9, 9, 4, 2, 9, 1, 0, 7, 5, 9, 1, 1, 7, 7, 3, 3, 6, 1, 1
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.
FORMULA
Equals Sum_{n>=1} (-1)^(n+1)/n^2 Sum_{k=1..n} 1/(k + n) (see Finch).
EXAMPLE
0.3983484188414979381406202084041821941620701721...
MATHEMATICA
RealDigits[Pi Catalan-33Zeta[3]/16, 10, 100][[1]]
Decimal expansion of 1/2 - log(2)/4 - G/Pi, where G = A006752.
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0
3, 5, 1, 5, 2, 3, 0, 0, 8, 2, 9, 1, 9, 4, 8, 9, 2, 5, 0, 7, 3, 0, 7, 5, 1, 3, 1, 6, 7, 0, 6, 0, 9, 3, 9, 1, 1, 7, 0, 5, 8, 8, 1, 2, 4, 2, 4, 0, 9, 8, 9, 1, 6, 2, 0, 8, 8, 2, 8, 4, 2, 8, 5, 1, 4, 9, 0, 3, 9, 5, 7, 6, 2, 7, 1, 3, 7, 4, 5, 9, 3, 7, 1, 9, 0, 7, 3, 4, 3, 9, 4, 7, 7, 6, 1, 2, 6, 9, 2, 0
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 56.
FORMULA
Equals Sum_{n>=1} zeta(2*n)/(2^(4*n)*(2*n + 1)) (see Finch).
EXAMPLE
0.035152300829194892507307513167060939117058812424...
MATHEMATICA
RealDigits[1/2-Log[2]/4-Catalan/Pi, 10, 100][[1]]
Array read by rows (blocks). Row n contains 3n/2 elements if n is even, and (n+1)/2 elements if n is odd. Each row is a permutation of the numbers of its constituents; see Comments.
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0
1, 4, 2, 3, 6, 5, 12, 10, 8, 7, 9, 11, 15, 13, 14, 24, 22, 20, 18, 16, 17, 19, 21, 23, 28, 26, 25, 27, 40, 38, 36, 34, 32, 30, 29, 31, 33, 35, 37, 39, 45, 43, 41, 42, 44, 60, 58, 56, 54, 52, 50, 48, 46, 47, 49, 51, 53, 55, 57, 59, 66, 64, 62, 61, 63, 65, 84, 82, 80, 78, 76, 74, 72, 70, 68, 67, 69, 71, 73, 75, 77, 79, 81, 83, 91, 89, 87, 85, 86
COMMENTS
The array consists of two triangular arrays alternating row by row.
For odd n, row n consists of permutations of the integers from A001844((n-1)/2) to A265225(n-1). For even n, row n consists of permutations of the integers from A130883(n/2) to A265225(n-1). These permutations are generated by the algorithm described A130517. The sequence is an intra-block permutation of integer positive numbers.
FORMULA
Linear sequence:
a(n) = P(n) + B(L(n)-1), where L(n) = ceiling(x(n)), x(n) is largest real root of the equation B(x) - n = 0. B(n) = (n+1)*(2*n-(-1)^n+5)/4 = A265225(n). P(n) = A162630(n)/2. Array T(n,k) (see Example):
T(n, k) = P(n, k) + (n^2 - n)/2 if n is even, T(n, k) = P(n, k) + (n^2 - 1)/2 if n is odd, T(n, k) = P(n, k) + A265225(n-1). P(n, k) = |2k - 3n / 2 - 2| if n is even and if 2k <= 3n / 2 + 1, P(n, k) = |2k - 3n / 2 - 1| if n is even and if 2k > 3n / 2 + 1. P(n, k) = |2k - (n + 1) / 2 - 2| if n is odd and if 2k <= (n + 1) / 2 + 1, P(n, k) = |2k - (n + 1) / 2 - 1| if n is odd and if 2k > (n + 1) / 2 + 1. There are several special cases: P(n, 1) = 3n/2 if n is even, P(n, 1) = (n+1)/2 if n is odd. P(2, 2) = 1. P(n, n) = n/2 - 1 if n is even, P(n, n) = (n-3)/2 if n is odd.
EXAMPLE
Array begins:
k = 1 2 3 4 5 6
n=1: 1;
n=2: 4, 2, 3;
n=3: 6, 5;
n=4: 12, 10, 8, 7, 9, 11;
The triangular arrays alternate by row: n=1 and n=3 comprise one, and n=2 and n=4 comprise the other.
Subtracting (n^2 - 1)/2 if n is odd from each term in row n yields a permutation of 1 .. (n+1)/2. Subtracting (n^2 - n)/2 if n is even from each term in row n is a permutation of 1 .. 3n/2:
1,
3, 1, 2,
2, 1,
6, 4, 2, 1, 3, 5,
...
MATHEMATICA
a[n_]:=Module[{L, R, P, Result}, L=Ceiling[Max[x/.NSolve[x*(2*(x-1)-Cos[Pi*(x-1)]+5)-4*n==0, x, Reals]]]; R=n-If[EvenQ[L], (L^2-L)/2, (L^2-1)/2]; P[(L+1)*(2*L-(-1)^L+5)/4]=If[EvenQ[L], 3L/2, (L+1)/2]; P[3]=2; P= Abs[2*R-If[EvenQ[L], 3L/2, (L+1)/2]-If[2*R<=If[EvenQ[L], 3L/2, (L+1)/2]+1, 2, 1]]; Res=P+If[EvenQ[L], (L^2-L)/2, (L^2-1)/2]; Result=Res; Result] Nmax= 12; Table[a[n], {n, 1, Nmax}]
Triangle read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of row n is the (n+1)-st Fibonacci number for n > 0; see Comments.
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1, 2, 3, 4, 5, 7, 6, 8, 10, 12, 11, 9, 13, 15, 17, 19, 20, 18, 16, 14, 21, 23, 25, 27, 29, 31, 33, 32, 30, 28, 26, 24, 22, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 53, 51, 49, 47, 45, 43, 41, 39, 37, 35, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 88, 86, 84, 82, 80, 78, 76, 74, 72, 70, 68, 66, 64, 62, 60, 58, 56, 89
COMMENTS
Row n consists of permutation of the integers from F(n+2) to F(n+3) - 1, where F(n) = A000045(n). The permutation is generated using Kevin Ryde's formula from A194959. The sequence is an intra-block permutation of the positive integers.
FORMULA
T(n,k) for 1 <= k <= F(n) (see Example):
T(n,k) = P(n,k) + F(n+1)-1, T(n,k) = P(n,k) + A000045(n+1)-1, where P(n,k) = 2*k-1 if 2*k-1 <= F(n), P(n,k) = 2*(F(n)+1-k) if 2*k-1 > F(n).
EXAMPLE
Triangle begins:
k = 1 2 3 4 5 6 7 8
n=1: 1;
n=2: 2;
n=3: 3, 4;
n=4: 5, 7, 6;
n=5: 8, 10, 12, 11, 9;
n=6: 13, 15, 17, 19, 20, 18, 16, 14;
Subtracting F(n)-1 from each term in row n produces a permutation of 1 .. F(n):
1;
1;
1,2;
1,3,2;
1,3,5,4,2;
1,3,5,7,8,6,4,2;
...
MATHEMATICA
T[n_, k_]:=Module[{P, Result}, P= If[2*k-1 <=Fibonacci[n], 2*k-1, 2*(Fibonacci[n]+1-k)]; Result=P+Fibonacci[n+1]-1; Result]; Nmax=6; Table[T[n, k], {n, 1, Nmax}, {k, 1, Fibonacci[n]}]//Flatten
Number of digit patterns of length n that satisfy no divisibility rules but do not generate primes.
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1
0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 32, 9
COMMENTS
Digit patterns (or digital types) are as per A266946. The divisibility rules are per A376918 and they act to exclude patterns which always result in composite numbers, just due to the pattern. There are A376918(n) remaining patterns but not all of them actually contain primes, and a(n) is how many of them do not, so that a(n) = A376918(n) - A267013(n). We call these digital types primonumerophobic and a(n) is the number of these of length n.
It is conjectured that the next terms are a(13)=207, a(14)=362, a(15)=363, a(16)=1448. This is based on the calculated number of primonumerophobic digit patterns with only 2 or 3 distinct digits and the vanishingly small combinatorial probability for the existence of additional primonumerophobic digit patterns of this length with 4 or more distinct digits.
EXAMPLE
For n=10, the a(10) = 3 primonumerophobic patterns of length 10, which are also the smallest which exist, are
---------- ----------
AAABBBAAAB 1110001110
AABABBBBBA 1101000001
ABAAAAABBB 1011111000
These patterns have 2 distinct digits (A and B) so that there are in total 81 numbers of each pattern that all happen to be composite despite the pattern coefficients in each having no common divisors.
Cogrowth sequence for the 18-element group C6 X C3 = <S,T | S^6, T^3, [S,T]>.
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0
1, 1, 2, 85, 926, 5461, 37130, 349525, 2973350, 22369621, 174174002, 1431655765, 11582386286, 91625968981, 729520967450, 5864062014805, 47006639297270, 375299968947541, 2999857885752002, 24019198012642645, 192222214478506046, 1537228672809129301
COMMENTS
Sequence gives terms for n = 0 (mod 3), all other terms are 0.
FORMULA
G.f.: (36*x^4+99*x^3-14*x^2+6*x-1) / ((8*x-1) * (x+1) * (27*x^2+1)).
After A121053(n) has been found, a(n) is the smallest candidate for A121053(n+1) that has not been eliminated.
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0
1, 1, 1, 6, 6, 9, 9, 9, 12, 12, 12, 15, 15, 15, 18, 18, 18, 21, 21, 21, 24, 24, 24, 26, 26, 28, 28, 32, 32, 32, 32, 34, 34, 36, 36, 39, 39, 39, 42, 42, 42, 45, 45, 45, 48, 48, 48, 50, 50, 52, 52, 55
COMMENTS
[Computed by hand, should be checked]
EXAMPLE
After a(8) = 9, and A121053(9) = 10 has been determined, the smallest prime not yet used is 17 and the smallest composite not yet used or eliminated is 12 (10 is now eliminated because the terms of A121053 must be distinct), so a(9) = 12.
a(n) = b(10*n+1), with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo 10; 10th column of A378007.
+0
0
1, 4, 0, 4, 4, 0, 4, 4, 1, 0, 4, 0, 10, 4, 0, 4, 0, 0, 4, 4, 0, 4, 0, 0, 4, 4, 0, 4, 4, 0, 0, 4, 0, 4, 16, 0, 2, 0, 0, 0, 4, 0, 4, 4, 0, 16, 4, 0, 0, 4, 0, 0, 4, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0, 4, 4, 0, 4, 16, 0, 4, 4, 0, 0, 0, 0, 4, 4, 0, 16, 0, 0, 4, 4, 0, 2, 0, 0, 0, 4, 4, 0
FORMULA
a(n) = b(10*n+1), where {b(n)} is multiplicative with:
- b(2^e) = b(5^e) = 0;
- for p == 1 (mod 10), b(p^e) = binomial(e+3,3) = (e+3)*(e+2)*(e+1)/6;
- for p == 9 (mod 10), b(p^e) = e/2 + 1 if e is even, and 0 otherwise;
- for p == 3, 7 (mod 10), b(p^e) = 1 if 4 divides e, and 0 otherwise.
EXAMPLE
(1 + 1/3^s + 1/7^s + 1/9^s + ...)*(1 + i/3^s - i/7^s - 1/9^s + ...)*(1 - 1/3^s - 1/7^s + 1/9^s + ...)*(1 - i/3^s + i/7^s - 1/9^s + ...) = 1 + 4/11^s + 4/31^s + 4/41^s + 4/61^s + 4/71^s + 1/81^s + 4/101^s + ...
PROG
my(f = factor(10*n+1), res = 1); for(i=1, #f~,
if(f[i, 1] % 10 == 1, res *= binomial(f[i, 2]+3, 3));
if(f[i, 1] % 10 == 9, if(f[i, 2] % 2 == 0, res *= f[i, 2]/2+1, return(0)));
if(f[i, 1] % 10 == 3 || f[i, 1] % 10 == 7, if(f[i, 2] % 4 != 0, return(0))));
res; }
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