A285729 -id:A285729 - cp's OEIS frontend

A300229 Restricted growth sequence transform of A285729, combining A032742(n) and A046523(n), the largest proper divisor and the prime signature of n.

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

Offset: 1

Antti Karttunen, Mar 01 2018

nonn

			a(10) = a(15) (= 7) because both are nonsquare semiprimes (2*5 and 3*5), and when the smallest prime factor is divided out, both yield the same quotient, 5.

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

Cf. A032742, A046523, A285729.

Cf. also A300223, A300226, A300230, A300231, A300232, A300233, A300235.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
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]); };  \\ From A046523
A285729(n) = (1/2)*(2 + ((A032742(n)+A046523(n))^2) - A032742(n) - 3*A046523(n));
write_to_bfile(1,rgs_transform(vector(65537,n,A285729(n))),"b300229.txt");

A286472 Compound filter (for counting prime gaps): a(1) = 1, a(n) = 2*A032742(n) + (1 if n is composite and spf(A032742(n)) > nextprime(spf(n)), and 0 otherwise). Here spf is the smallest prime factor, A020639.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 6, 11, 2, 12, 2, 15, 10, 16, 2, 18, 2, 20, 15, 23, 2, 24, 10, 27, 18, 28, 2, 30, 2, 32, 23, 35, 14, 36, 2, 39, 27, 40, 2, 42, 2, 44, 30, 47, 2, 48, 14, 51, 35, 52, 2, 54, 23, 56, 39, 59, 2, 60, 2, 63, 42, 64, 27, 66, 2, 68, 47, 71, 2, 72, 2, 75, 50, 76, 22, 78, 2

Offset: 1

Antti Karttunen, May 11 2017

nonn

For n > 1, a(n) is odd if and only if n is a composite with its smallest prime factor occurring only once and with a gap of at least one between the smallest and the next smallest prime factor.

For all i, j: a(i) = a(j) => A073490(i) = A073490(j). This follows because A073490(n) can be computed by recursively invoking a(n), without needing any other information.

			For n = 4 = 2*2, the two smallest prime factors (taken with multiplicity) are 2 and 2, and the difference between their indices is 0, thus a(4) = 2*A032742(4) + 0 = 2*(4/2) + 0 = 2.
For n = 6 = 2*3 = prime(1)*prime(2), the difference between the indices of two smallest prime factors is 1 (which is less than required 2), thus a(6) = 2*A032742(6) + 0 = 2*(6/2) + 0 = 6.
For n = 10 = 2*5 = prime(1)*prime(3), the difference between the indices of two smallest prime factors is 2, thus a(10) = 2*A032742(10) + 1 = 2*(10/2) + 1 = 11.

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

Cf. A032742, A285729, A286471, A286473, A286474, A286475, A286476.
Cf. A000040 (primes give the positions of 2's).
Cf. A073490 (one of the matched sequences).

Table[Function[{p, d}, 2 d + If[And[CompositeQ@ n, FactorInteger[d][[1, 1]] > NextPrime[p]], 1, 0] - Boole[n == 1]] @@ {#, n/#} &@ FactorInteger[n][[1, 1]], {n, 98}] (* Michael De Vlieger, May 12 2017 *)

from sympy import primefactors, divisors, nextprime
def ok(n): return 1 if isprime(n)==0 and min(primefactors(divisors(n)[-2])) > nextprime(min(primefactors(n))) else 0
def a(n): return 1 if n==1 else 2*divisors(n)[-2] + ok(n) # Indranil Ghosh, May 12 2017

(define (A286472 n) (if (= 1 n) n (+ (* 2 (A032742 n)) (if (> (A286471 n) 2) 1 0))))

a(n) = 2*A032742(n) + [A286471(n) > 2], a(1) = 1.

A286473 Compound filter (for counting primes of form 4k+1, 4k+2 and 4k+3): a(n) = 4*A032742(n) + (A020639(n) mod 4), a(1) = 1.

Original entry on oeis.org

1, 6, 7, 10, 5, 14, 7, 18, 15, 22, 7, 26, 5, 30, 23, 34, 5, 38, 7, 42, 31, 46, 7, 50, 21, 54, 39, 58, 5, 62, 7, 66, 47, 70, 29, 74, 5, 78, 55, 82, 5, 86, 7, 90, 63, 94, 7, 98, 31, 102, 71, 106, 5, 110, 45, 114, 79, 118, 7, 122, 5, 126, 87, 130, 53, 134, 7, 138, 95, 142, 7, 146, 5, 150, 103, 154, 47, 158, 7, 162, 111, 166, 7, 170, 69, 174, 119, 178, 5, 182, 55

Offset: 1

Antti Karttunen, May 11 2017

nonn

For all i, j: a(i) = a(j) => A079635(i) = A079635(j). This follows because A079635(n) can be computed by recursively invoking a(n), without needing any other information.

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

Cf. A020639, A032742, A285729, A286472, A286474, A286475, A286476.

Cf. A001511, A007814, A065339, A079635, A083025 (some of the matched sequences).

With[{k = 4}, Table[Function[{p, d}, k d + Mod[p, k] - k Boole[n == 1]] @@ {#, n/#} &@ FactorInteger[n][[1, 1]], {n, 91}]] (* Michael De Vlieger, May 12 2017 *)

from sympy import divisors, primefactors
def a(n): return 1 if n==1 else 4*divisors(n)[-2] + (min(primefactors(n))%4) # Indranil Ghosh, May 12 2017

(define (A286473 n) (if (= 1 n) n (+ (* 4 (A032742 n)) (modulo (A020639 n) 4))))

a(1) = 1, for n > 1, a(n) = 4*A032742(n) + (A020639(n) mod 4).

A286474 Compound filter: a(n) = 4*A032742(n) + (n mod 4), a(1) = 1.

Original entry on oeis.org

1, 6, 7, 8, 5, 14, 7, 16, 13, 22, 7, 24, 5, 30, 23, 32, 5, 38, 7, 40, 29, 46, 7, 48, 21, 54, 39, 56, 5, 62, 7, 64, 45, 70, 31, 72, 5, 78, 55, 80, 5, 86, 7, 88, 61, 94, 7, 96, 29, 102, 71, 104, 5, 110, 47, 112, 77, 118, 7, 120, 5, 126, 87, 128, 53, 134, 7, 136, 93, 142, 7, 144, 5, 150, 103, 152, 45, 158, 7, 160, 109, 166, 7, 168, 69, 174, 119, 176, 5, 182, 55

Offset: 1

Antti Karttunen, May 11 2017

nonn

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

Cf. A032742, A285729, A286472, A286473, A286475, A286476.

Table[If[n == 1, 1, 4 (Divisors[n][[-2]]) + Mod[n, 4]], {n, 91}] (* Michael De Vlieger, May 12 2017 *)

from sympy import divisors, primefactors
def a(n): return 1 if n==1 else 4*divisors(n)[-2] + n%4 # Indranil Ghosh, May 12 2017

(define (A286474 n) (if (= 1 n) n (+ (* 4 (A032742 n)) (modulo n 4))))

a(1) = 1, for n > 1, a(n) = 4*A032742(n) + (n mod 4).

A286476 Compound filter: a(n) = 6*A032742(n) + (n mod 6), a(1) = 1.

Original entry on oeis.org

1, 8, 9, 16, 11, 18, 7, 26, 21, 34, 11, 36, 7, 44, 33, 52, 11, 54, 7, 62, 45, 70, 11, 72, 31, 80, 57, 88, 11, 90, 7, 98, 69, 106, 47, 108, 7, 116, 81, 124, 11, 126, 7, 134, 93, 142, 11, 144, 43, 152, 105, 160, 11, 162, 67, 170, 117, 178, 11, 180, 7, 188, 129, 196, 83, 198, 7, 206, 141, 214, 11, 216, 7, 224, 153, 232, 71, 234, 7, 242, 165, 250, 11, 252, 103

Offset: 1

Antti Karttunen, May 11 2017

nonn

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

Cf. A032742, A285729, A286472, A286473, A286474, A286475.

With[{k = 6}, Table[If[n == 1, 1, k (Divisors[n][[-2]]) + Mod[n, k]], {n, 85}]] (* Michael De Vlieger, May 12 2017 *)

from sympy import divisors
def a(n): return 1 if n==1 else 6*divisors(n)[-2] + n%6 # Indranil Ghosh, May 12 2017

(define (A286476 n) (if (= 1 n) n (+ (* 6 (A032742 n)) (modulo n 6))))

a(1) = 1, for n > 1, a(n) = 6*A032742(n) + (n mod 6).

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)).

A286475 Compound filter (for counting primes of form 6k+1, 6k+2, 6k+3 and 6k+5): a(n) = 6*A032742(n) + (A020639(n) mod 6), a(1) = 1.

Original entry on oeis.org

1, 8, 9, 14, 11, 20, 7, 26, 21, 32, 11, 38, 7, 44, 33, 50, 11, 56, 7, 62, 45, 68, 11, 74, 35, 80, 57, 86, 11, 92, 7, 98, 69, 104, 47, 110, 7, 116, 81, 122, 11, 128, 7, 134, 93, 140, 11, 146, 43, 152, 105, 158, 11, 164, 71, 170, 117, 176, 11, 182, 7, 188, 129, 194, 83, 200, 7, 206, 141, 212, 11, 218, 7, 224, 153, 230, 67, 236, 7, 242, 165, 248, 11, 254, 107

Offset: 1

Antti Karttunen, May 11 2017

nonn

			For n = 55 = 5*11, a(n) = 6*A032742(55) + (5 modulo 6) = 6*11 + 5 = 71.
For n = 121 = 11*11, a(n) = 6*A032742(121) + (11 modulo 6) = 6*11 + 1 = 71.
For n = 91 = 7*13, a(n) = 6*A032742(91) + (7 modulo 6) = 6*13 + 1 = 79.
For n = 169 = 13*13, a(n) = 6*A032742(169) + (13 modulo 6) = 6*13 + 1 = 79.

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

Cf. A020639, A032742, A285729, A286472, A286473, A286474, A286476.

With[{k = 6}, Table[Function[{p, d}, k d + Mod[p, k] - k Boole[n == 1]] @@ {#, n/#} &@ FactorInteger[n][[1, 1]], {n, 85}]] (* Michael De Vlieger, May 12 2017 *)

from sympy import divisors, primefactors
def a(n): return 1 if n==1 else 6*divisors(n)[-2] +(min(primefactors(n))%6) # Indranil Ghosh, May 12 2017

(define (A286475 n) (if (= 1 n) n (+ (* 6 (A032742 n)) (modulo (A020639 n) 6))))

a(1) = 1, for n > 1, a(n) = 6*A032742(n) + (A020639(n) mod 6).

A286149 Compound filter: a(n) = T(A046523(n), A109395(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 8, 14, 17, 34, 30, 44, 19, 51, 68, 103, 93, 72, 196, 152, 155, 103, 192, 132, 72, 126, 278, 349, 32, 159, 53, 165, 437, 976, 498, 560, 709, 237, 786, 739, 705, 282, 159, 402, 863, 660, 948, 243, 337, 384, 1130, 1273, 49, 132, 1546, 288, 1433, 349, 126, 459, 282, 567, 1772, 2761, 1893, 636, 165, 2144, 2421, 1921, 2280, 390, 2707, 2046, 2558, 2773, 2703

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, A109395, A285729, A286142, A286143, A286144, A286152, A286154, A286160, A286161, A286162, A286163, A286164.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1], Denominator[EulerPhi[n]/n]}, {n, 73}] (* Michael De Vlieger, May 04 2017 *)

A109395(n) = n/gcd(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
A286149(n) = (1/2)*(2 + ((A046523(n)+A109395(n))^2) - A046523(n) - 3*A109395(n));
for(n=1, 10000, write("b286149.txt", n, " ", A286149(n)));

from sympy import factorint, totient, 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 a(n): return T(a046523(n), n/gcd(n, totient(n))) # Indranil Ghosh, May 05 2017

(define (A286149 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A109395 n)) 2) (- (A046523 n)) (- (* 3 (A109395 n))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A109395(n))^2) - A046523(n) - 3*A109395(n)).

A286379 Compound filter ("discard the smallest prime factor" & "signature for 1-runs in base-2"): a(n) = P(A032742(n), A278222(n)), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = 1.

Original entry on oeis.org

1, 2, 7, 5, 16, 18, 29, 14, 31, 50, 67, 42, 67, 98, 195, 44, 16, 100, 67, 115, 637, 242, 277, 117, 125, 289, 955, 224, 277, 450, 497, 152, 131, 248, 160, 271, 436, 454, 643, 320, 436, 1246, 1771, 550, 2716, 1058, 1129, 375, 160, 655, 1343, 692, 1771, 1918, 3332, 623, 880, 1355, 2557, 1020, 1129, 1922, 3507, 560, 166, 736, 67, 775, 1349, 1070, 277, 856, 436

Offset: 1

Antti Karttunen, May 13 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A032742, A278222, A285729.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
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
A278222(n) = A046523(A005940(1+n));
A286379(n) = if(1==n,n,(1/2)*(2 + ((A032742(n)+A278222(n))^2) - A032742(n) - 3*A278222(n)));
for(n=1, 16384, write("b286379.txt", n, " ", A286379(n)));

from sympy import factorint, divisors
import math
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
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 a278222(n): return a046523(a005940(n + 1))
def a(n): return 1 if n==1 else T(divisors(n)[-2], a278222(n)) # Indranil Ghosh, May 13 2017

(define (A286379 n) (if (= 1 n) n (* (/ 1 2) (+ (expt (+ (A032742 n) (A278222 n)) 2) (- (A032742 n)) (- (* 3 (A278222 n))) 2))))

a(1) = 1, for n > 1, a(n) = (1/2)*(2 + ((A032742(n)+A278222(n))^2) - A032742(n) - 3*A278222(n)).

cp's OEIS Frontend

A300229 Restricted growth sequence transform of A285729, combining A032742(n) and A046523(n), the largest proper divisor and the prime signature of n.

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A286472 Compound filter (for counting prime gaps): a(1) = 1, a(n) = 2*A032742(n) + (1 if n is composite and spf(A032742(n)) > nextprime(spf(n)), and 0 otherwise). Here spf is the smallest prime factor, A020639.

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A286473 Compound filter (for counting primes of form 4k+1, 4k+2 and 4k+3): a(n) = 4*A032742(n) + (A020639(n) mod 4), a(1) = 1.

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A286474 Compound filter: a(n) = 4*A032742(n) + (n mod 4), a(1) = 1.

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A286476 Compound filter: a(n) = 6*A032742(n) + (n mod 6), a(1) = 1.

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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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A286475 Compound filter (for counting primes of form 6k+1, 6k+2, 6k+3 and 6k+5): a(n) = 6*A032742(n) + (A020639(n) mod 6), a(1) = 1.

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