A000027 -id:A000027 - cp's OEIS frontend

A364558 a(n) = A364557(n) - A000010(n), where A364557 is the Möbius transform of A005941, and A000010 (Euler phi) is the Möbius transform of A000027.

0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 6, 0, 20, 2, -4, 0, 48, -2, 110, 0, -4, 6, 234, 0, -12, 20, -10, 4, 484, -4, 994, 0, -4, 48, -16, -4, 2012, 110, 8, 0, 4056, -4, 8150, 12, -16, 234, 16338, 0, -26, -12, 32, 40, 32716, -10, -24, 8, 92, 484, 65478, -8, 131012, 994, -20, 0, -16, -4, 262078, 96, 212, -16, 524218, -8, 1048504

Offset: 1

Antti Karttunen, Jul 28 2023

sign

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

Cf. A000010, A005941, A364557, A364559 (inverse Möbius transform), A364565 (positions of 0's), A364566 (of terms < 0).

PARI
```
A364558(n) = (A364557(n)-eulerphi(n));
```

from math import prod
from sympy import factorint, primepi
def A364558(n): return (1<1 else 0 # Chai Wah Wu, Jul 29 2023

A131042 Natural numbers A000027 with 6n+3 and 6n+4 terms swapped.

Original entry on oeis.org

1, 2, 4, 3, 5, 6, 7, 8, 10, 9, 11, 12, 13, 14, 16, 15, 17, 18, 19, 20, 22, 21, 23, 24, 25, 26, 28, 27, 29, 30, 31, 32, 34, 33, 35, 36, 37, 38, 40, 39, 41, 42, 43, 44, 46, 45, 47, 48, 49, 50, 52, 51, 53, 54, 55, 56, 58, 57, 59, 60, 61, 62, 64, 63, 65, 66, 67, 68, 70, 69, 71, 72

Offset: 1

Paul Curtz, Sep 23 2007

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

LinearRecurrence[{1,0,0,0,0,1,-1},{1,2,4,3,5,6,7},80] (* Harvey P. Dale, Aug 26 2024 *)

Vec(x*(1 + x + 2*x^2 - x^3 + 2*x^4 + x^5) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) + O(x^100)) \\ Colin Barker, Apr 08 2017

a(n) = (24*floor(n/6)+3*n^2-3*n+8+9*floor(n/3)*(3*floor(n/3)-2*n+1)-(3*n^2-7*n+8+3*floor(n/3)*(9*floor(n/3)-6*n+7))*(-1)^floor(n/3))/4. - Luce ETIENNE, Apr 08 2017
From Colin Barker, Apr 08 2017: (Start)
G.f.: x*(1 + x + 2*x^2 - x^3 + 2*x^4 + x^5) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.
(End)

A185508 Third accumulation array, T, of the natural number array A000027, by antidiagonals.

Original entry on oeis.org

1, 5, 6, 16, 29, 21, 41, 89, 99, 56, 91, 219, 295, 259, 126, 182, 469, 705, 755, 574, 252, 336, 910, 1470, 1765, 1645, 1134, 462, 582, 1638, 2786, 3605, 3780, 3206, 2058, 792, 957, 2778, 4914, 6706, 7595, 7266, 5754, 3498, 1287, 1507, 4488, 8190, 11634, 13916, 14406, 12894, 9690, 5643, 2002, 2288, 6963, 13035, 19110, 23814, 26068, 25284, 21510, 15510, 8723, 3003, 3367, 10439, 19965, 30030, 38640, 44100

Offset: 1

Clark Kimberling, Jan 29 2011

See A144112 (and A185506) for the definition of accumulation array (aa).

Sequence is aa(aa(aa(A000027))).

			Northwest corner:
   1    5   16   41   91  182
   6   29   89  219  469  910
  21   99  295  705 1470 2786
  56  259  755 1765 3605 6706

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000027, A185506, A185507, A185509.

Cf. A000389 (column 1), A257199 (row 1).

h[n_,k_]:=k(k+1)(k+2)n(n+1)(n+2)*(4n^2+(5k+23)n+4k^2+3k+41)/2880;
TableForm[Table[h[n,k],{n,1,10},{k,1,15}]]
Table[h[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

{h(n,k) = k*(k+1)*(k+2)*n*(n+1)*(n+2)*(4*n^2+(5*k+23)*n +4*k^2 +3*k + 41)/2880}; for(n=1,10, for(k=1,n, print1(h(k, n-k+1), ", "))) \\ G. C. Greubel, Nov 23 2017

T(n,k) = F*(4n^2 + (5k+23)n + 4k^2 + 3k+41), where F = k(k+1)(k+2)n(n+1)(n+2)/2880.

A286034 Compound filter: a(n) = P(A046523(n), A161942(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 8, 3, 49, 8, 34, 3, 239, 124, 97, 8, 165, 30, 34, 34, 1051, 47, 1237, 17, 508, 21, 97, 8, 727, 565, 331, 74, 165, 122, 733, 3, 4403, 34, 502, 34, 7911, 192, 196, 72, 2302, 233, 526, 68, 508, 1237, 97, 8, 3051, 1774, 5368, 97, 1782, 380, 727, 97, 727, 51, 1231, 122, 3220, 498, 34, 288, 18019, 331, 733, 155, 2713, 34, 733, 47, 35317, 705, 1897, 873, 1047, 34

Offset: 1

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
MathWorld, Pairing Function

Cf. A000027, A046523, A161942, A286260.

A000265(n) = (n >> valuation(n, 2));
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
A161942(n) = A000265(sigma(n));
A286034(n) = (2 + ((A046523(n)+A161942(n))^2) - A046523(n) - 3*A161942(n))/2;
for(n=1, 16384, write("b286034.txt", n, " ", A286034(n)));

from sympy import factorint, divisors, divisor_sigma
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
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 a000265(n): return max(list(filter(lambda i: i%2 == 1, divisors(n))))
def a161942(n): return a000265(divisor_sigma(n))
def a(n): return T(a046523(n), a161942(n)) # Indranil Ghosh, May 07 2017

(define (A286034 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A161942 n)) 2) (- (A046523 n)) (- (* 3 (A161942 n))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A161942(n))^2) - A046523(n) - 3*A161942(n)).

A286102 Square array A(n,k) read by antidiagonals: A(n,k) = T(lcm(n,k), gcd(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

Original entry on oeis.org

1, 3, 3, 6, 5, 6, 10, 21, 21, 10, 15, 14, 13, 14, 15, 21, 55, 78, 78, 55, 21, 28, 27, 120, 25, 120, 27, 28, 36, 105, 34, 210, 210, 34, 105, 36, 45, 44, 231, 90, 41, 90, 231, 44, 45, 55, 171, 300, 406, 465, 465, 406, 300, 171, 55, 66, 65, 64, 63, 630, 61, 630, 63, 64, 65, 66, 78, 253, 465, 666, 820, 903, 903, 820, 666, 465, 253, 78, 91, 90, 561, 230, 1035, 324, 85, 324, 1035, 230, 561, 90, 91

Offset: 1

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

			The top left 12 X 12 corner of the array:
   1,   3,   6,  10,   15,   21,   28,   36,   45,   55,   66,   78
   3,   5,  21,  14,   55,   27,  105,   44,  171,   65,  253,   90
   6,  21,  13,  78,  120,   34,  231,  300,   64,  465,  561,  103
  10,  14,  78,  25,  210,   90,  406,   63,  666,  230,  990,  117
  15,  55, 120, 210,   41,  465,  630,  820, 1035,  101, 1540, 1830
  21,  27,  34,  90,  465,   61,  903,  324,  208,  495, 2211,  148
  28, 105, 231, 406,  630,  903,   85, 1596, 2016, 2485, 3003, 3570
  36,  44, 300,  63,  820,  324, 1596,  113, 2628,  860, 3916,  375
  45, 171,  64, 666, 1035,  208, 2016, 2628,  145, 4095, 4950,  739
  55,  65, 465, 230,  101,  495, 2485,  860, 4095,  181, 6105, 1890
  66, 253, 561, 990, 1540, 2211, 3003, 3916, 4950, 6105,  221, 8778
  78,  90, 103, 117, 1830,  148, 3570,  375,  739, 1890, 8778,  265

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array
MathWorld, Pairing Function

Cf. A000027, A003989, A003990, A003991, A038722, A285722, A285732, A286098, A286099, A286101, A285724.

Cf. A000217 (row 1 and column 1), A001844 (main diagonal).

(define (A286102 n) (A286102bi (A002260 n) (A004736 n)))
(define (A286102bi row col) (A000027bi (lcm row col) (gcd row col)))
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

A(n,k) = T(lcm(n,k), gcd(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table, that is, as a pairing function from N x N to N.

A(n,k) = A(k,n), or equivalently, a(A038722(n)) = a(n). [Array is symmetric.]

A286152 Compound filter: a(n) = T(A051953(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 2, 12, 2, 40, 2, 59, 18, 61, 2, 179, 2, 86, 73, 261, 2, 265, 2, 265, 100, 148, 2, 757, 33, 185, 129, 367, 2, 1297, 2, 1097, 166, 271, 131, 1735, 2, 320, 205, 1105, 2, 1741, 2, 619, 517, 430, 2, 3113, 52, 850, 295, 769, 2, 1747, 205, 1517, 346, 625, 2, 5297, 2, 698, 730, 4497, 248, 2821, 2, 1117, 460, 2821, 2, 7069, 2, 941, 1070, 1315, 248, 3457, 2, 4513

Offset: 1

Antti Karttunen, May 04 2017

nonn

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

Cf. A000027, A046523, A051953, A285729, A286142, A286143, A286144, A286149, A286154, A286160, A286161, A286162, A286163, A286164.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {n - EulerPhi@ n, Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1]}, {n, 80}] (* Michael De Vlieger, May 04 2017 *)

A051953(n) = (n - eulerphi(n));
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
A286152(n) = (2 + ((A051953(n)+A046523(n))^2) - A051953(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286152.txt", n, " ", A286152(n)));

from sympy import factorint, totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
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 a(n): return T(n - totient(n), a046523(n)) # Indranil Ghosh, May 05 2017

(define (A286152 n) (* (/ 1 2) (+ (expt (+ (A051953 n) (A046523 n)) 2) (- (A051953 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A051953(n)+A046523(n))^2) - A051953(n) - 3*A046523(n)).

A286258 Compound filter: a(n) = P(A046523(n), A046523(2n+1)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 5, 5, 25, 5, 27, 23, 44, 14, 61, 5, 117, 38, 27, 27, 226, 23, 90, 23, 90, 27, 142, 5, 375, 40, 27, 86, 148, 5, 495, 80, 698, 27, 61, 27, 702, 80, 61, 27, 765, 5, 625, 23, 90, 148, 61, 23, 1224, 109, 90, 27, 832, 5, 324, 61, 324, 61, 142, 23, 2013, 23, 84, 90, 2410, 27, 625, 302, 90, 27, 625, 23, 2998, 80, 27, 90, 265, 61, 495, 23, 1426, 152, 601, 5, 2013, 142, 27, 142

Offset: 1

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A046523, A286255, A286256, A286257.

Cf. A005384 (gives the positions of 5's), A234095 (of 23's).

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
A286258(n) = (1/2)*(2 + ((A046523(n)+A046523((2*n)+1))^2) - A046523(n) - 3*A046523((2*n)+1));
for(n=1, 10000, write("b286258.txt", n, " ", A286258(n)));

from sympy import factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
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 a(n): return T(a046523(n), a046523(2*n + 1)) # Indranil Ghosh, May 07 2017

(define (A286258 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A046523 (+ 1 n n))) 2) (- (A046523 n)) (- (* 3 (A046523 (+ 1 n n)))) 2)))

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

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)).

A286573 Compound filter: a(n) = P(A007733(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 7, 14, 23, 9, 29, 42, 40, 65, 80, 90, 31, 40, 121, 44, 142, 189, 109, 61, 115, 77, 302, 273, 148, 318, 94, 434, 532, 20, 497, 115, 86, 148, 826, 702, 271, 148, 355, 230, 601, 119, 220, 265, 131, 299, 1178, 297, 485, 86, 265, 1430, 838, 320, 328, 271, 556, 1769, 1957, 1890, 50, 142, 2017, 148, 751, 2277, 179, 373, 832, 665, 2932, 54, 856, 485

Offset: 1

Antti Karttunen, May 26 2017

nonn

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

Cf. A000027, A007733, A046523, A286160, A286161.

A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ This function from Michel Marcus, Apr 11 2015
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
A286573(n) = (1/2)*(2 + ((A007733(n)+A046523(n))^2) - A007733(n) - 3*A046523(n));

from sympy import divisors, factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a002326(n):
    m=1
    while True:
        if (2**m - 1)%(2*n + 1)==0: return m
        else: m+=1
def a000265(n): return max(list(filter(lambda i: i%2 == 1, divisors(n))))
def a007733(n): return a002326((a000265(n) - 1)/2)
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 a(n): return T(a007733(n), a046523(n)) # Indranil Ghosh, May 26 2017

a(n) = (1/2)*(2 + ((A007733(n)+A046523(n))^2) - A007733(n) - 3*A046523(n)).

A290990 p-INVERT of the nonnegative integers (A000027), where p(S) = 1 - S - S^2.

Original entry on oeis.org

0, 1, 2, 5, 12, 28, 64, 145, 328, 743, 1686, 3830, 8704, 19781, 44950, 102133, 232048, 527208, 1197808, 2721421, 6183108, 14048151, 31917714, 72517738, 164761792, 374342057, 850512458, 1932380869, 4390407092, 9975090996, 22663602720, 51492150953

Offset: 0

Clark Kimberling, Aug 21 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A290890 for a guide to related sequences.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2,1).

Cf. A000027, A033453, A289780, A290890, A290991.

I:=[0,1,2,5]; [n le 4 select I[n] else 4*Self(n-1) -5*Self(n-2) +2*Self(n-3) +Self(n-4): n in [1..50]]; // G. C. Greubel, Apr 12 2023

z = 60; s = x^2/(1-x)^2; p = 1 -s -s^2;
Drop[CoefficientList[Series[s, {x,0,z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x,0,z}], x], 1]  (* A290990 *)
LinearRecurrence[{4,-5,2,1}, {0,1,2,5}, 50] (* G. C. Greubel, Apr 12 2023 *)

concat(0, Vec(x*(1-2*x+2*x^2)/(1-4*x+5*x^2-2*x^3-x^4) + O(x^50))) \\ Colin Barker, Aug 24 2017

@CachedFunction
def a(n): # a = A290990
    if (n<4): return (0,1,2,5)[n]
    else: return 4*a(n-1) -5*a(n-2) +2*a(n-3) +a(n-4)
[a(n) for n in range(51)] # G. C. Greubel, Apr 12 2023

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) + a(n-4).

G.f.: x*(1 - 2*x + 2*x^2) / (1 - 4*x + 5*x^2 - 2*x^3 - x^4). - Colin Barker, Aug 24 2017

cp's OEIS Frontend

A364558 a(n) = A364557(n) - A000010(n), where A364557 is the Möbius transform of A005941, and A000010 (Euler phi) is the Möbius transform of A000027.

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A131042 Natural numbers A000027 with 6n+3 and 6n+4 terms swapped.

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A185508 Third accumulation array, T, of the natural number array A000027, by antidiagonals.

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A286034 Compound filter: a(n) = P(A046523(n), A161942(n)), where P(n,k) is sequence A000027 used as a pairing function.

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PARI

Python

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Formula

A286102 Square array A(n,k) read by antidiagonals: A(n,k) = T(lcm(n,k), gcd(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

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Examples

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Scheme

Formula

A286152 Compound filter: a(n) = T(A051953(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

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Mathematica

PARI

Python

Scheme

Formula

A286258 Compound filter: a(n) = P(A046523(n), A046523(2n+1)), where P(n,k) is sequence A000027 used as a pairing function.

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PARI

Python

Scheme

Formula

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.

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