A000027 -id:A000027 - cp's OEIS frontend

A285729 Compound filter: a(n) = T(A032742(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

1, 2, 2, 12, 2, 31, 2, 59, 18, 50, 2, 142, 2, 73, 50, 261, 2, 199, 2, 220, 73, 131, 2, 607, 33, 166, 129, 314, 2, 961, 2, 1097, 131, 248, 73, 1396, 2, 295, 166, 923, 2, 1246, 2, 550, 340, 401, 2, 2509, 52, 655, 248, 692, 2, 1252, 131, 1303, 295, 590, 2, 3946, 2, 661, 517, 4497, 166, 1924, 2, 1024, 401, 2051, 2, 5707, 2, 898, 655, 1214, 131, 2317, 2, 3781, 888

Offset: 1

Antti Karttunen, May 04 2017

nonn

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

Cf. A000027, A032742, A046523, A286142, A286143, A286144, A286149, A286152, A286154, A286160, A286161, A286162, A286163, A286164.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {Sort[Flatten@ Apply[ TensorProduct, # /. {p_, e_} /; p > 1 :> p^Range[0, e]]][[-2]], Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[#[[All, -1]], Greater]] - Boole[n == 1]} &@ FactorInteger@ n, {n, 81}] (* Michael De Vlieger, May 04 2017 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
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
A285729(n) = (1/2)*(2 + ((A032742(n)+A046523(n))^2) - A032742(n) - 3*A046523(n));
for(n=1, 10000, write("b285729.txt", n, " ", A285729(n)));

from sympy import divisors, factorint
def a032742(n): return 1 if n==1 else max(divisors(n)[:-1])
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(a032742(n), a046523(n)) # Indranil Ghosh, May 05 2017

(define (A285729 n) (* (/ 1 2) (+ (expt (+ (A032742 n) (A046523 n)) 2) (- (A032742 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A032742(n)+A046523(n))^2) - A032742(n) - 3*A046523(n)).

A136302 Transform of A000027 by the T_{1,1} transformation (see link).

Original entry on oeis.org

2, 6, 15, 35, 81, 188, 437, 1016, 2362, 5491, 12765, 29675, 68986, 160373, 372822, 866706, 2014847, 4683951, 10888865, 25313540, 58846841, 136802308, 318026782, 739322571, 1718716457, 3995531011, 9288482690, 21593102505, 50197873146, 116695897118, 271285047567

Offset: 1

Richard Choulet, Mar 22 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
Richard Choulet, Curtz-like transformation.
Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

Cf. A095263, A136303, A136304, A136305.

I:=[2,6,15]; [n le 3 select I[n] else 3*Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Apr 12 2021

a:= n-> (<<6|2|1>>. <<3|1|0>, <-2|0|1>, <1|0|0>>^n)[1, 3]:
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 14 2008

LinearRecurrence[{3,-2,1}, {2,6,15}, 41] (* G. C. Greubel, Apr 12 2021 *)

def A136302_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(2+x^2)/(1-3*x+2*x^2-x^3) ).list()
a=A136302_list(41); a[1:] # G. C. Greubel, Apr 12 2021

G.f.: z*(2 + z^2)/(1 - 3*z + 2*z^2 - z^3).
a(n+3) = 3*a(n+2) - 2*a(n+1) + a(n) (n>=0). - Richard Choulet, Apr 07 2009
a(n) = 2*A095263(n) + A095263(n-2). - R. J. Mathar, Feb 29 2016

More terms from Alois P. Heinz, Aug 14 2008

A285722 Square array A(n,k) read by antidiagonals, A(n,n) = 0, otherwise, if n > k, A(n,k) = T(n-k,k), else A(n,k) = T(n,k-n), where T(n,k) is sequence A000027 considered as a two-dimensional table.

Original entry on oeis.org

0, 1, 1, 2, 0, 3, 4, 3, 2, 6, 7, 5, 0, 5, 10, 11, 8, 6, 4, 9, 15, 16, 12, 9, 0, 8, 14, 21, 22, 17, 13, 10, 7, 13, 20, 28, 29, 23, 18, 14, 0, 12, 19, 27, 36, 37, 30, 24, 19, 15, 11, 18, 26, 35, 45, 46, 38, 31, 25, 20, 0, 17, 25, 34, 44, 55, 56, 47, 39, 32, 26, 21, 16, 24, 33, 43, 54, 66, 67, 57, 48, 40, 33, 27, 0, 23, 32, 42, 53, 65, 78, 79, 68, 58, 49, 41, 34, 28, 22, 31, 41, 52, 64, 77, 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 14 X 14 corner of the array:
   0,  1,  2,  4,  7, 11, 16, 22, 29, 37, 46, 56, 67, 79
   1,  0,  3,  5,  8, 12, 17, 23, 30, 38, 47, 57, 68, 80
   3,  2,  0,  6,  9, 13, 18, 24, 31, 39, 48, 58, 69, 81
   6,  5,  4,  0, 10, 14, 19, 25, 32, 40, 49, 59, 70, 82
  10,  9,  8,  7,  0, 15, 20, 26, 33, 41, 50, 60, 71, 83
  15, 14, 13, 12, 11,  0, 21, 27, 34, 42, 51, 61, 72, 84
  21, 20, 19, 18, 17, 16,  0, 28, 35, 43, 52, 62, 73, 85
  28, 27, 26, 25, 24, 23, 22,  0, 36, 44, 53, 63, 74, 86
  36, 35, 34, 33, 32, 31, 30, 29,  0, 45, 54, 64, 75, 87
  45, 44, 43, 42, 41, 40, 39, 38, 37,  0, 55, 65, 76, 88
  55, 54, 53, 52, 51, 50, 49, 48, 47, 46,  0, 66, 77, 89
  66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56,  0, 78, 90
  78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67,  0, 91
  91, 90, 89, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79,  0

Antti Karttunen, Table of n, a(n) for n = 1..10585
MathWorld, Pairing Function

Transpose: A285723.
Cf. A000124 (row 1, from 1 onward), A000217 (column 1).
Cf. also A000027, A003989, A072030, A285721, A285732.

A[n_, n_] = 0;
A[n_, k_] /; k == n-1 := (k^2 - k + 2)/2;
A[1, k_] := (k^2 - 3k + 4)/2;
A[n_, k_] /; 1 <= k <= n-2 := A[n, k] = A[n, k+1] + 1;
A[n_, k_] /; k > n := A[n, k] = A[n-1, k] + 1;
Table[A[n-k+1, k], {n, 1, 14}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 19 2019 *)

def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def A(n, k): return 0 if n == k else T(n - k, k) if n>k else T(n, k - n)
for n in range(1, 21): print([A(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, May 03 2017

(define (A285722 n) (A285722bi (A002260 n) (A004736 n)))
(define (A285722bi row col) (cond ((= row col) 0) ((> row col) (A000027bi (- row col) col)) (else (A000027bi row (- col row)))))
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

If n = k, A(n,k) = 0, if n > k, A(n,k) = T(n-k,k), otherwise [when n < k], A(n,k) = T(n,k-n), 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.

A285732 Square array A(n,k) read by antidiagonals, A(n,n) = -n, otherwise, if n > k, A(n,k) = T(n-k,k), else A(n,k) = T(n,k-n), where T(n,k) is sequence A000027 considered as a two-dimensional table.

Original entry on oeis.org

-1, 1, 1, 2, -2, 3, 4, 3, 2, 6, 7, 5, -3, 5, 10, 11, 8, 6, 4, 9, 15, 16, 12, 9, -4, 8, 14, 21, 22, 17, 13, 10, 7, 13, 20, 28, 29, 23, 18, 14, -5, 12, 19, 27, 36, 37, 30, 24, 19, 15, 11, 18, 26, 35, 45, 46, 38, 31, 25, 20, -6, 17, 25, 34, 44, 55, 56, 47, 39, 32, 26, 21, 16, 24, 33, 43, 54, 66, 67, 57, 48, 40, 33, 27, -7, 23, 32, 42, 53, 65, 78

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 14 X 14 corner of the array:
  -1,  1,  2,  4,  7, 11, 16, 22, 29,  37,  46,  56,  67,  79
   1, -2,  3,  5,  8, 12, 17, 23, 30,  38,  47,  57,  68,  80
   3,  2, -3,  6,  9, 13, 18, 24, 31,  39,  48,  58,  69,  81
   6,  5,  4, -4, 10, 14, 19, 25, 32,  40,  49,  59,  70,  82
  10,  9,  8,  7, -5, 15, 20, 26, 33,  41,  50,  60,  71,  83
  15, 14, 13, 12, 11, -6, 21, 27, 34,  42,  51,  61,  72,  84
  21, 20, 19, 18, 17, 16, -7, 28, 35,  43,  52,  62,  73,  85
  28, 27, 26, 25, 24, 23, 22, -8, 36,  44,  53,  63,  74,  86
  36, 35, 34, 33, 32, 31, 30, 29, -9,  45,  54,  64,  75,  87
  45, 44, 43, 42, 41, 40, 39, 38, 37, -10,  55,  65,  76,  88
  55, 54, 53, 52, 51, 50, 49, 48, 47,  46, -11,  66,  77,  89
  66, 65, 64, 63, 62, 61, 60, 59, 58,  57,  56, -12,  78,  90
  78, 77, 76, 75, 74, 73, 72, 71, 70,  69,  68,  67, -13,  91
  91, 90, 89, 88, 87, 86, 85, 84, 83,  82,  81,  80,  79, -14

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

Transpose: A285733.
Cf. A000124 (row 1, after -1), A000217 (column 1, after -1).
Cf. also A000027, A003989, A072030, A285721, A285722, A286100.

def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def A(n, k): return -n if n == k else T(n - k, k) if n>k else T(n, k - n)
for n in range(1, 21): print([A(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, May 03 2017

(define (A285732 n) (A285732bi (A002260 n) (A004736 n)))
(define (A285732bi row col) (cond ((= row col) (- row)) ((> row col) (A000027bi (- row col) col)) (else (A000027bi row (- col row)))))

If n = k, A(n,k) = -n, if n > k, A(n,k) = T(n-k,k), otherwise [when n < k], A(n,k) = T(n,k-n), 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) = A285722(n,k) - A286100(n,k).

A286142 Compound filter: a(n) = T(A257993(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 2, 12, 2, 31, 2, 38, 7, 23, 2, 94, 2, 23, 16, 138, 2, 94, 2, 80, 16, 23, 2, 328, 7, 23, 29, 80, 2, 532, 2, 530, 16, 23, 16, 706, 2, 23, 16, 302, 2, 499, 2, 80, 67, 23, 2, 1228, 7, 80, 16, 80, 2, 328, 16, 302, 16, 23, 2, 1957, 2, 23, 67, 2082, 16, 499, 2, 80, 16, 467, 2, 2704, 2, 23, 67, 80, 16, 499, 2, 1178, 121, 23, 2, 1894, 16, 23, 16, 302, 2, 1957, 16

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, A257993, A286144, A286160, A286161, A286162, A286163, A286164.

Differs from A286143 for the first time at n=24, where a(24) = 328, while A286143(24) = 355.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 - Boole[n == 1] & @@ {Module[{i = 1}, While[! CoprimeQ[Prime@ i, n], i++]; i], Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]}, {n, 92}] (* Michael De Vlieger, May 04 2017 *)

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
A257993(n) = { for(i=1,n,if(n%prime(i),return(i))); }
A286142(n) = (1/2)*(2 + ((A257993(n)+A046523(n))^2) - A257993(n) - 3*A046523(n));
for(n=1, 10000, write("b286142.txt", n, " ", A286142(n)));

from sympy import factorint, prime, primepi, gcd
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 a053669(n):
    x=1
    while True:
        if gcd(prime(x), n) == 1: return prime(x)
        else: x+=1
def a257993(n): return primepi(a053669(n))
def a(n): return T(a257993(n), a046523(n)) # Indranil Ghosh, May 05 2017

(define (A286142 n) (* (/ 1 2) (+ (expt (+ (A257993 n) (A046523 n)) 2) (- (A257993 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A257993(n)+A046523(n))^2) - A257993(n) - 3*A046523(n)).

A190405 Decimal expansion of Sum_{k>=1} (1/2)^T(k), where T=A000217 (triangular numbers); based on column 1 of the natural number array, A000027.

Original entry on oeis.org

6, 4, 1, 6, 3, 2, 5, 6, 0, 6, 5, 5, 1, 5, 3, 8, 6, 6, 2, 9, 3, 8, 4, 2, 7, 7, 0, 2, 2, 5, 4, 2, 9, 4, 3, 4, 2, 2, 6, 0, 6, 1, 5, 3, 7, 9, 5, 6, 7, 3, 9, 7, 4, 7, 8, 0, 4, 6, 5, 1, 6, 2, 2, 3, 8, 0, 1, 4, 4, 6, 0, 3, 7, 3, 3, 3, 5, 1, 7, 7, 5, 6, 0, 0, 3, 6, 4, 1, 7, 1, 6, 2, 3, 3, 5, 9, 1, 3, 3, 0, 8, 6, 0, 8, 9, 7, 3, 5, 3, 1, 6, 3, 4, 3, 6, 1, 9, 4, 6, 1

Offset: 0

Clark Kimberling, May 10 2011

See A190404.
Binary expansion is .1010010001... (A023531). - Rick L. Shepherd, Jan 05 2014
From Amiram Eldar, Dec 07 2020: (Start)
This constant is not a quadratic irrational (Duverney, 1995).
The Engel expansion of this constant are the powers of 2 (A000079) above 1. (End)

			0.64163256065515386629...

Danny Rorabaugh, Table of n, a(n) for n = 0..10000
Daniel Duverney, Sommes de deux carrés et irrationalité de valeurs de fonctions thêta, Comptes rendus de l'Académie des sciences, Série 1, Mathématique, Vol. 320, No. 9 (1995), pp. 1041-1044.

A190404: (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T = A000217 (triangular numbers).
A190405: Sum_{k>=1} (1/2)^T(k), where T = A000217 (triangular numbers).
A190406: Sum_{k>=1} (1/2)^S(k-1), where S = A001844 (centered square numbers).
A190407: Sum_{k>=1} (1/2)^V(k), where V = A058331 (1 + 2*k^2).
Cf. A000079.

RealDigits[EllipticTheta[2, 0, 1/Sqrt[2]]/2^(7/8) - 1, 10, 120] // First (* Jean-François Alcover, Feb 12 2013 *)
RealDigits[Total[(1/2)^Accumulate[Range[50]]],10,120][[1]] (* Harvey P. Dale, Oct 18 2013 *)
(* See also A190404 *)

th2(x)=2*x^.25 + 2*suminf(n=1,x^(n+1/2)^2)
th2(sqrt(.5))/2^(7/8)-1 \\ Charles R Greathouse IV, Jun 06 2016

def A190405(b):  # Generate the constant with b bits of precision
    return N(sum([(1/2)^(j*(j+1)/2) for j in range(1,b)]),b)
A190405(409) # Danny Rorabaugh, Mar 25 2015

A286101 Square array A(n,k) read by antidiagonals: A(n,k) = T(gcd(n,k), lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 7, 16, 16, 7, 11, 12, 13, 12, 11, 16, 46, 67, 67, 46, 16, 22, 23, 106, 25, 106, 23, 22, 29, 92, 31, 191, 191, 31, 92, 29, 37, 38, 211, 80, 41, 80, 211, 38, 37, 46, 154, 277, 379, 436, 436, 379, 277, 154, 46, 56, 57, 58, 59, 596, 61, 596, 59, 58, 57, 56, 67, 232, 436, 631, 781, 862, 862, 781, 631, 436, 232, 67, 79, 80, 529, 212, 991, 302, 85, 302, 991, 212, 529, 80, 79

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,   2,   4,   7,   11,   16,   22,   29,   37,   46,   56,   67
   2,   5,  16,  12,   46,   23,   92,   38,  154,   57,  232,   80
   4,  16,  13,  67,  106,   31,  211,  277,   58,  436,  529,   94
   7,  12,  67,  25,  191,   80,  379,   59,  631,  212,  947,  109
  11,  46, 106, 191,   41,  436,  596,  781,  991,   96, 1486, 1771
  16,  23,  31,  80,  436,   61,  862,  302,  193,  467, 2146,  142
  22,  92, 211, 379,  596,  862,   85, 1541, 1954, 2416, 2927, 3487
  29,  38, 277,  59,  781,  302, 1541,  113, 2557,  822, 3829,  355
  37, 154,  58, 631,  991,  193, 1954, 2557,  145, 4006, 4852,  706
  46,  57, 436, 212,   96,  467, 2416,  822, 4006,  181, 5996, 1832
  56, 232, 529, 947, 1486, 2146, 2927, 3829, 4852, 5996,  221, 8647
  67,  80,  94, 109, 1771,  142, 3487,  355,  706, 1832, 8647,  265

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

Cf. A000124 (row 1 and column 1), A001844 (main diagonal).

Cf. A000027, A003989, A003990, A003991, A038722, A285722, A285732, A286098, A286099, A286102, A285724.

(define (A286101 n) (A286101bi (A002260 n) (A004736 n)))
(define (A286101bi row col) (A000027bi (gcd row col) (lcm row col)))
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

A(n,k) = T(gcd(n,k), lcm(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.]

A286143 Compound filter: a(n) = T(A055881(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 2, 12, 2, 31, 2, 38, 7, 23, 2, 94, 2, 23, 16, 138, 2, 94, 2, 80, 16, 23, 2, 355, 7, 23, 29, 80, 2, 499, 2, 530, 16, 23, 16, 706, 2, 23, 16, 302, 2, 499, 2, 80, 67, 23, 2, 1279, 7, 80, 16, 80, 2, 328, 16, 302, 16, 23, 2, 1894, 2, 23, 67, 2082, 16, 499, 2, 80, 16, 467, 2, 2779, 2, 23, 67, 80, 16, 499, 2, 1178, 121, 23, 2, 1894, 16, 23, 16, 302, 2, 1894, 16

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, A055881, A286144, A286160, A286161, A286162, A286163, A286164.

Differs from A286142 for the first time at n=24, where a(24) = 355, while A286142(24) = 328.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 - Boole[n == 1] & @@ {Module[{m = 1}, While[Mod[n, m!] == 0, m++]; m - 1], Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]}, {n, 92}] (* Michael De Vlieger, May 04 2017, after Robert G. Wilson v at A055881 *)

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
A055881(n) = { my(i); i=2; while((0 == (n%i)), n = n/i; i++); return(i-1); }
A286143(n) = (1/2)*(2 + ((A055881(n)+A046523(n))^2) - A055881(n) - 3*A046523(n));
for(n=1, 10000, write("b286143.txt", n, " ", A286143(n)));

from sympy import factorial, 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 a055881(n):
    m = 1
    while n%factorial(m)==0:
        m+=1
    return m - 1
def a(n): return T(a055881(n), a046523(n)) # Indranil Ghosh, May 05 2017

(define (A286143 n) (* (/ 1 2) (+ (expt (+ (A055881 n) (A046523 n)) 2) (- (A055881 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A055881(n)+A046523(n))^2) - A055881(n) - 3*A046523(n)).

A286356 Compound filter: a(n) = P(A061395(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 5, 7, 9, 23, 14, 29, 12, 31, 20, 80, 27, 40, 31, 121, 35, 80, 44, 94, 40, 50, 54, 302, 18, 61, 38, 109, 65, 499, 77, 497, 50, 73, 40, 668, 90, 86, 61, 328, 104, 532, 119, 125, 94, 100, 135, 1178, 25, 94, 73, 142, 152, 302, 50, 355, 86, 115, 170, 1894, 189, 131, 109, 2017, 61, 566, 209, 160, 100, 532, 230, 2630, 252, 148, 94, 179, 50, 601, 275, 1228, 138

Offset: 1

Antti Karttunen, May 09 2017

nonn

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

Cf. A000027, A046523, A061395, A286164.

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
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A286356(n) = (2 + ((A061395(n)+A046523(n))^2) - A061395(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286356.txt", n, " ", A286356(n)));

from sympy import factorint
from operator import mul
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 a061395(n): return 0 if n == 1 else primepi(max(primefactors(n)))
def a(n): return T(a061395(n), a046523(n)) # Indranil Ghosh, May 09 2017

(define (A286356 n) (* (/ 1 2) (+ (expt (+ (A061395 n) (A046523 n)) 2) (- (A061395 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A061395(n)+A046523(n))^2) - A061395(n) - 3*A046523(n)).

A286364 Compound filter: a(n) = P(A286361(n), A286363(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 2, 1, 7, 3, 2, 2, 3, 2, 5, 1, 3, 7, 2, 3, 16, 2, 2, 2, 10, 3, 29, 2, 3, 5, 2, 1, 16, 3, 5, 7, 3, 2, 5, 3, 3, 16, 2, 2, 12, 2, 2, 2, 7, 10, 5, 3, 3, 29, 5, 2, 16, 3, 2, 5, 3, 2, 67, 1, 21, 16, 2, 3, 16, 5, 2, 7, 3, 3, 14, 2, 16, 5, 2, 3, 121, 3, 2, 16, 21, 2, 5, 2, 3, 12, 5, 2, 16, 2, 5, 2, 3, 7, 67, 10, 3, 5, 2, 3, 23, 3, 2, 29, 3, 5, 5, 2, 3

Offset: 1

Antti Karttunen, May 08 2017

nonn

This sequence packs the values of A286361(n) and A286363(n) to a single value with the pairing function A000027. These two components essentially give the prime signature of 4k+1 part and the prime signature of 4k+3 part, and they can be accessed from a(n) with functions A002260 and A004736. For example, A004431 lists all such numbers that the first component is larger than one and the second component is a perfect square.

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

Cf. A000027, A002260, A004431, A004736, A038722, A267099, A286361, A286363, A286365.

from sympy import factorint
from operator import mul
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, k):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==k else i**f[i] for i in f])
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a(n): return T(a046523(n/A(n, 1)), a046523(n/A(n, 3))) # Indranil Ghosh, May 09 2017

(define (A286364 n) (* (/ 1 2) (+ (expt (+ (A286361 n) (A286363 n)) 2) (- (A286361 n)) (- (* 3 (A286363 n))) 2)))

a(n) = (1/2)*(2+((A286361(n)+A286363(n))^2) - A286361(n) - 3*A286363(n)).
Other identities. For all n >= 1:
a(A267099(n)) = A038722(a(n)).

cp's OEIS Frontend

A285729 Compound filter: a(n) = T(A032742(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A136302 Transform of A000027 by the T_{1,1} transformation (see link).

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A285722 Square array A(n,k) read by antidiagonals, A(n,n) = 0, otherwise, if n > k, A(n,k) = T(n-k,k), else A(n,k) = T(n,k-n), where T(n,k) is sequence A000027 considered as a two-dimensional table.

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A285732 Square array A(n,k) read by antidiagonals, A(n,n) = -n, otherwise, if n > k, A(n,k) = T(n-k,k), else A(n,k) = T(n,k-n), where T(n,k) is sequence A000027 considered as a two-dimensional table.

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A286142 Compound filter: a(n) = T(A257993(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A190405 Decimal expansion of Sum_{k>=1} (1/2)^T(k), where T=A000217 (triangular numbers); based on column 1 of the natural number array, A000027.

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A286101 Square array A(n,k) read by antidiagonals: A(n,k) = T(gcd(n,k), lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

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