A286356 -id:A286356 - cp's OEIS frontend

A360398 a(n) = A026430(1 + A360392(n)).

5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, 42, 44, 45, 48, 50, 52, 55, 57, 59, 61, 65, 66, 69, 70, 72, 75, 78, 80, 81, 84, 86, 88, 91, 93, 95, 98, 100, 102, 105, 107, 108, 111, 113, 116, 118, 120, 123, 125, 126, 129, 132, 134, 135, 138

Offset: 1

Clark Kimberling, Feb 10 2023

This is the first of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n)  = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360399, A286355, A286356, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];    (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A286454 Compound filter (prime signature & prime signature of conjugated prime factorization): a(n) = P(A101296(n), A286621(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 8, 9, 12, 32, 23, 20, 13, 49, 38, 51, 47, 82, 49, 35, 68, 51, 80, 72, 124, 140, 122, 74, 18, 175, 26, 111, 155, 334, 192, 65, 257, 280, 82, 116, 255, 329, 355, 99, 327, 570, 380, 177, 72, 469, 437, 132, 31, 72, 532, 216, 498, 74, 257, 144, 599, 634, 597, 448, 632, 745, 159, 119, 784, 1044, 782, 331, 907, 570, 863, 186, 905, 1039, 72, 384, 140, 1335, 1037

Offset: 1

Antti Karttunen, May 14 2017

nonn

Here, instead of A046523 and A278221 we use as the components of a(n) their rgs-versions A101296 and A286621 because of the latter sequence's moderate growth rates.

For all i, j: a(i) = a(j) => A286356(i) = A286356(j).

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

Cf. A000027, A046523, A101296, A122111, A278221, A285334, A286621, A286356, A286455, A286456.

(define (A286454 n) (* (/ 1 2) (+ (expt (+ (A101296 n) (A286621 n)) 2) (- (A101296 n)) (- (* 3 (A286621 n))) 2)))

a(n) = (1/2)*(2 + ((A101296(n)+A286621(n))^2) - A101296(n) - 3*A286621(n)).

A318891 Filter sequence combining the prime signature of n (A046523) with the largest prime factor of n (A006530).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 16 2018

nonn

Restricted growth sequence transform of A286356.

For all i, j: a(i) = a(j) => A297112(i) = A297112(j). (Also, equivalently, A297113 or A297167 in place of A297112.)

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A006530, A046523, A061395, A286356, A318890.

Cf. also A297112, A297113, A297167.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A318891aux(n) = [A046523(n), A061395(n)];
v318891 = rgs_transform(vector(up_to,n,A318891aux(n)));
A318891(n) = v318891[n];

A360399 a(n) = A026430(1 + A360393(n)).

Original entry on oeis.org

1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, 73, 77, 82, 87, 90, 96, 99, 103, 109, 114, 117, 121, 128, 130, 136, 141, 144, 149, 154, 159, 162, 168, 171, 175, 181, 186, 189, 194, 199, 203, 209, 213, 216, 222, 225, 230, 235, 239, 245, 249

Offset: 1

Clark Kimberling, Feb 10 2023

This is the second of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n)  = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360398, A286355, A286356, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];    (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A360400 a(n) = A356133(A360392(n)).

Original entry on oeis.org

7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, 83, 85, 89, 94, 97, 104, 110, 112, 115, 122, 127, 131, 137, 140, 142, 148, 155, 157, 161, 166, 169, 176, 182, 184, 187, 193, 200, 202, 208, 211, 215, 220, 223, 229, 236, 238, 244, 247, 251, 257

Offset: 1

Clark Kimberling, Mar 11 2023

This is the third of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n)  = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360398, A286354, A286356, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A286452 Compound filter (largest prime factor of n & prime signature of 2n-1): a(n) = P(A061395(n), A046523(2n-1)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 5, 2, 18, 5, 14, 16, 5, 9, 50, 5, 42, 59, 9, 2, 73, 23, 44, 31, 14, 20, 199, 5, 18, 61, 5, 40, 115, 9, 77, 67, 50, 35, 40, 5, 90, 179, 61, 9, 391, 14, 185, 50, 9, 100, 205, 23, 14, 94, 35, 27, 1006, 5, 20, 40, 44, 115, 395, 31, 228, 131, 59, 2, 61, 20, 295, 442, 54, 14, 320, 23, 346, 265, 9, 44, 125, 61, 275, 31, 23, 104, 1349, 14, 52, 314, 65, 125, 430

Offset: 1

Antti Karttunen, May 14 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, A278223, A285334, A286356, A286372, A286453.

from sympy import primepi, primefactors, 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 a061395(n): return 0 if n==1 else primepi(primefactors(n)[-1])
def a(n): return T(a061395(n), a046523(2*n - 1)) # Indranil Ghosh, May 14 2017

(define (A286452 n) (* (/ 1 2) (+ (expt (+ (A061395 n) (A046523 (+ n n -1))) 2) (- (A061395 n)) (- (* 3 (A046523 (+ n n -1)))) 2)))

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

A286453 Compound filter: a(n) = P(A061395(n), A286465(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 5, 11, 94, 5, 14, 254, 17, 9, 195, 47, 259, 500, 9, 11, 413, 138, 44, 303, 32, 20, 2784, 47, 354, 216, 5, 329, 506, 9, 77, 3161, 356, 35, 175, 107, 202, 2709, 216, 24, 11188, 14, 420, 356, 24, 285, 450, 498, 70, 2349, 35, 51, 115937, 5, 20, 329, 74, 310, 3420, 864, 1243, 336, 500, 11, 384, 20, 580, 47285, 87, 14, 615, 498, 1296, 3015, 9, 74, 3491, 216

Offset: 1

Antti Karttunen, May 14 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python program to generate the sequence

Cf. A000027, A046523, A061395, A112049, A278223, A286356, A286372, A286452.

(define (A286453 n) (* (/ 1 2) (+ (expt (+ (A061395 n) (A286465 n)) 2) (- (A061395 n)) (- (* 3 (A286465 n))) 2)))

a(n) = (1/2)*(2 + ((A061395(n)+A286465(n))^2) - A061395(n) - 3*A286465(n)).

cp's OEIS Frontend

A360398 a(n) = A026430(1 + A360392(n)).

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A286454 Compound filter (prime signature & prime signature of conjugated prime factorization): a(n) = P(A101296(n), A286621(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A318891 Filter sequence combining the prime signature of n (A046523) with the largest prime factor of n (A006530).

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PARI

A360399 a(n) = A026430(1 + A360393(n)).

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Mathematica

A360400 a(n) = A356133(A360392(n)).

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Mathematica

A286452 Compound filter (largest prime factor of n & prime signature of 2n-1): a(n) = P(A061395(n), A046523(2n-1)), where P(n,k) is sequence A000027 used as a pairing function.

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Python

Scheme

Formula

A286453 Compound filter: a(n) = P(A061395(n), A286465(n)), where P(n,k) is sequence A000027 used as a pairing function.

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