A112049 -id:A112049 - cp's OEIS frontend

A112060 Square array A(x,y) = y-th natural number k for which A112049(k)=x and 0 if no such k exists; read by antidiagonals A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), ...

1, 2, 3, 5, 4, 11, 6, 7, 12, 24, 9, 8, 23, 35, 60, 10, 15, 36, 59, 155, 84, 13, 16, 47, 95, 275, 239, 144, 14, 19, 48, 119, 335, 575, 779, 180, 17, 20, 71, 120, 359, 659, 1499, 2855, 264, 18, 27, 72, 179, 419, 839, 1535, 4199, 5279, 420, 21, 28, 83, 204, 504

Offset: 1

Antti Karttunen, Aug 27 2005

This is a permutation of natural numbers provided that the sequence A112046 contains only prime values [which is true] and every prime occurs infinitely many times there.

			The top left corner of the array:
   1,  2,  5,  6,  9, 10, ...
   3,  4,  7,  8, 15, 16, ...
  11, 12, 23, 36, 47, ...

Index entries for sequences that are permutations of the natural numbers

A112070(x, y) = 2*A(X, Y)+1. Transpose: A112061. Column 1: A112051. Row 1: A042963, Row 2: A112062, Row 3: A112063, Row 4: A112064, Row 5: A112065, Row 6: A112066, Row 7: A112067, Row 8: A112068, Row 9: A112069.

Cf. also A227196.

A112070 Square array A(x,y) = y-th odd number 2i+1 (i>=1) for which A112049(2i+1)=x, or 0 if no such i exists; read by descending antidiagonals.

Original entry on oeis.org

3, 5, 7, 11, 9, 23, 13, 15, 25, 49, 19, 17, 47, 71, 121, 21, 31, 73, 119, 311, 169, 27, 33, 95, 191, 551, 479, 289, 29, 39, 97, 239, 671, 1151, 1559, 361, 35, 41, 143, 241, 719, 1319, 2999, 5711, 529, 37, 55, 145, 359, 839, 1679, 3071, 8399, 10559, 841, 43, 57

Offset: 1

Antti Karttunen, Aug 27 2005

This is a permutation of odd numbers greater than unity provided that the sequence A112046 contains only prime values and every prime occurs infinitely many times there. Because the Jacobi symbol is multiplicative with respect to its modulus, it follows that if n occurs on row i and m occurs on row j, then n*m cannot occur before row min(i,j).

			The top left corner of the array:
3,5,11,13,19,21,...
7,9,15,17,31,33,...
23,25,47,73,95,...

A(x, y) = 2*A112060(x, y)+1. Transpose: A112071. Column 1: A112052. Row 1: A047621, Row 2: A112072 Row 3: A112073, Row 4: A112074, Row 5: A112075, Row 6: A112076, Row 7: A112077, Row 8: A112078, Row 9: A112079.

A286465 Compound filter: a(1) = 1, a(n) = P(A112049(n-1), A278223(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 2, 5, 12, 2, 2, 23, 5, 2, 16, 9, 18, 29, 2, 5, 23, 16, 2, 23, 5, 2, 67, 9, 25, 16, 2, 23, 23, 2, 2, 80, 23, 2, 16, 14, 9, 67, 16, 5, 138, 2, 16, 23, 5, 16, 16, 31, 9, 67, 2, 5, 467, 2, 2, 23, 5, 16, 67, 40, 33, 16, 29, 5, 23, 2, 16, 302, 5, 2, 16, 31, 31, 67, 2, 5, 80, 16, 2, 23, 23, 2, 436, 9, 42, 67, 2, 80, 23, 2, 2, 23, 23, 16, 277, 14, 9, 436, 2, 5

Offset: 1

Antti Karttunen, May 10 2017

nonn

After a(1) = 1, the information combined together to a(n) consists of A046523(2n-1), giving essentially the prime signature of 2n-1, and the index of the first prime p >= 1 for which the Jacobi symbol J(p,2n-1) is not +1 (i.e. is either 0 or -1), the value which is returned by A112049(n-1).

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

Cf. A000027, A046523, A112049, A278223, A286461, A286466.

A112049(n) = for(i=1,(2*n),if((kronecker(i,(n+n+1)) < 1),return(primepi(i))));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286465(n) = if(1==n,n,(1/2)*(2 + ((A112049(n-1)+A046523((2*n)-1))^2) - A112049(n-1) - 3*A046523((2*n)-1)));
for(n=1, 10000, write("b286465.txt", n, " ", A286465(n)));

from sympy import jacobi_symbol as J, factorint, isprime, primepi
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a278223(n): return a046523(2*n - 1)
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a049084(n): return primepi(n) if isprime(n) else 0
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a112049(n): return a049084(a112046(n))
def a(n): return 1 if n==1 else T(a112049(n - 1), a278223(n)) # Indranil Ghosh, May 11 2017

(define (A286465 n) (if (= 1 n) n (* (/ 1 2) (+ (expt (+ (A112049 (- n 1)) (A046523 (+ -1 n n))) 2) (- (A112049 (- n 1))) (- (* 3 (A046523 (+ -1 n n)))) 2))))

a(1) = 1; for n > 1, a(n) = (1/2)*(2 + ((A112049(n-1)+A046523((2*n)-1))^2) - A112049(n-1) - 3*A046523((2*n)-1)).

A286466 Compound filter: a(n) = P(A112049(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 12, 2, 16, 5, 38, 7, 16, 9, 94, 2, 16, 23, 138, 2, 67, 5, 80, 16, 16, 9, 355, 7, 16, 38, 80, 2, 436, 5, 530, 16, 16, 40, 706, 2, 16, 23, 302, 2, 436, 5, 80, 67, 16, 9, 1228, 7, 67, 23, 80, 2, 277, 23, 302, 16, 16, 14, 2021, 2, 16, 80, 2082, 16, 436, 5, 80, 16, 436, 9, 2704, 2, 16, 80, 80, 16, 436, 5, 1178, 121, 16, 9, 2086, 16, 16, 23, 302, 2, 1771

Offset: 1

Antti Karttunen, May 10 2017

nonn

Here the information combined together to a(n) consists of A046523(n), giving essentially the prime signature of n, and the index of the first prime p >= 1 for which the Jacobi symbol J(p,2n+1) is not +1 (i.e. is either 0 or -1), the value which is returned by A112049(n).

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

Cf. A000027, A046523, A112049, A286258, A286465.

A112049(n) = for(i=1,(2*n),if((kronecker(i,(n+n+1)) < 1),return(primepi(i))));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286466(n) = (1/2)*(2 + ((A112049(n)+A046523(n))^2) - A112049(n) - 3*A046523(n));
for(n=1, 10000, write("b286466.txt", n, " ", A286466(n)));

from sympy import jacobi_symbol as J, factorint, isprime, primepi
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a049084(n): return primepi(n) if isprime(n) else 0
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a112049(n): return a049084(a112046(n))
def a(n): return T(a112049(n), a046523(n)) # Indranil Ghosh, May 11 2017

(define (A286466 n) (* (/ 1 2) (+ (expt (+ (A112049 n) (A046523 n)) 2) (- (A112049 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A112049(n)+A046523(n))^2) - A112049(n) - 3*A046523(n)).

A112062 Positive integers i for which A112049(i) == 2.

Original entry on oeis.org

3, 4, 7, 8, 15, 16, 19, 20, 27, 28, 31, 32, 39, 40, 43, 44, 51, 52, 55, 56, 63, 64, 67, 68, 75, 76, 79, 80, 87, 88, 91, 92, 99, 100, 103, 104, 111, 112, 115, 116, 123, 124, 127, 128, 135, 136, 139, 140, 147, 148, 151, 152, 159, 160, 163, 164, 171, 172, 175, 176

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Row 2 of A112060.

A112063 Positive integers i for which A112049(i) == 3.

Original entry on oeis.org

11, 12, 23, 36, 47, 48, 71, 72, 83, 96, 107, 108, 131, 132, 143, 156, 167, 168, 191, 192, 203, 216, 227, 228, 251, 252, 263, 276, 287, 288, 311, 312, 323, 336, 347, 348, 371, 372, 383, 396, 407, 408, 431, 432, 443, 456, 467, 468, 491, 492, 503, 516, 527

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Row 3 of A112060.

A112064 Positive integers n for which A112049(n) == 4.

Original entry on oeis.org

24, 35, 59, 95, 119, 120, 179, 204, 215, 240, 299, 300, 324, 360, 384, 395, 444, 455, 479, 515, 539, 540, 599, 624, 635, 660, 719, 720, 744, 780, 804, 815, 864, 875, 899, 935, 959, 960, 1019, 1044, 1055, 1080, 1139, 1140, 1164, 1200, 1224, 1235, 1284

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Row 4 of A112060.

A112065 Positive integers i for which A112049(i) == 5.

Original entry on oeis.org

60, 155, 275, 335, 359, 419, 504, 564, 600, 695, 755, 900, 984, 995, 1020, 1079, 1115, 1199, 1259, 1260, 1415, 1440, 1524, 1595, 1619, 1679, 1680, 1824, 1919, 1944, 2015, 2039, 2100, 2184, 2255, 2280, 2339, 2364, 2435, 2519, 2580, 2604

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Row 5 of A112060.

A112066 Positive integers i for which A112049(i) == 6.

Original entry on oeis.org

84, 239, 575, 659, 839, 1175, 1320, 1344, 1764, 1835, 1955, 2099, 2160, 2375, 2459, 2759, 2784, 2879, 3084, 3299, 3515, 3695, 3779, 3780, 3840, 3935, 4139, 4439, 4475, 4620, 4764, 4800, 4859, 4884, 5040, 5544, 5795, 5964, 6024, 6119, 6155

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Row 6 of A112060.

A112067 Positive integers i for which A112049(i) == 7.

Original entry on oeis.org

144, 779, 1499, 1535, 3215, 3275, 4044, 4355, 4380, 4535, 5459, 5460, 6864, 6995, 7284, 7379, 7499, 7704, 8315, 8399, 9059, 9420, 9504, 10079, 10164, 10560, 10775, 10980, 11075, 11699, 11760, 11999, 12024, 12455, 12935, 13715, 13775, 14040

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Row 7 of A112060.

cp's OEIS Frontend

A112060 Square array A(x,y) = y-th natural number k for which A112049(k)=x and 0 if no such k exists; read by antidiagonals A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), ...

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A112070 Square array A(x,y) = y-th odd number 2i+1 (i>=1) for which A112049(2i+1)=x, or 0 if no such i exists; read by descending antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

A286465 Compound filter: a(1) = 1, a(n) = P(A112049(n-1), A278223(n)), where P(n,k) is sequence A000027 used as a pairing function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Scheme

Formula

A286466 Compound filter: a(n) = P(A112049(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Scheme

Formula

A112062 Positive integers i for which A112049(i) == 2.

Views

Author

Keywords

Crossrefs

A112063 Positive integers i for which A112049(i) == 3.

Views

Author

Keywords

Crossrefs

A112064 Positive integers n for which A112049(n) == 4.

Views

Author

Keywords

Crossrefs

A112065 Positive integers i for which A112049(i) == 5.

Views

Author

Keywords

Crossrefs

A112066 Positive integers i for which A112049(i) == 6.

Views

Author

Keywords

Crossrefs

A112067 Positive integers i for which A112049(i) == 7.

Views

Author

Keywords

Crossrefs