A278223 -id:A278223 - cp's OEIS frontend

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.

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)).

A291769 Restricted growth sequence transform of A292249; filter combining multiplicative order of 2 mod 2n+1 & prime signature of 2n+1 (A002326 & A278223).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 18, 32, 33, 34, 35, 36, 37, 38, 39, 40, 25, 41, 12, 18, 17, 42, 43, 44, 45, 46, 47, 48, 19, 42, 15, 49, 22, 50, 51, 27, 52, 53, 54, 55, 28, 56, 57, 58, 59, 60, 41, 61, 62, 63, 64, 27, 26, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 60, 42

Offset: 0

Antti Karttunen, Oct 02 2017

nonn

Also restricted growth sequence transform of the odd bisection of A286573.

Antti Karttunen, Table of n, a(n) for n = 0..32768

Cf. A002326, A046523, A278223, A286573, A291766, A292249.

Cf. A291766, A292267 for related filters.

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])); }
A002326(n) = if(n<0, 0, znorder(Mod(2, 2*n+1))); \\ This function from Michael Somos, Mar 31 2005
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
A292249(n) = (1/2)*(2 + ((A002326(n)+A046523(n+n+1))^2) - A002326(n) - 3*A046523(n+n+1));
write_to_bfile(0,rgs_transform(vector(32769,n,A292249(n-1))),"b291769_upto32768.txt");

A286250 Filter-sequence: a(n) = A278223(A064216(n)) = A046523((2*A064216(n))-1).

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 6, 2, 4, 6, 2, 2, 2, 6, 12, 6, 8, 2, 2, 2, 2, 16, 2, 6, 4, 6, 6, 2, 2, 30, 12, 6, 6, 4, 12, 6, 6, 6, 6, 6, 2, 2, 6, 6, 30, 2, 6, 2, 6, 6, 2, 6, 2, 6, 6, 6, 6, 2, 6, 6, 2, 12, 2, 36, 2, 6, 4, 2, 12, 30, 12, 12, 2, 12, 2, 24, 2, 2, 6, 6, 24, 2, 2, 12, 2, 24, 12, 2, 2, 30, 30, 6, 6, 2, 2, 4, 6, 2, 30, 6, 32, 2, 6, 2, 6, 2, 6, 12, 4, 2, 30, 2, 2

Offset: 1

Antti Karttunen, May 07 2017

nonn

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

Cf. A046523, A064216, A245448, A278223, A286243.

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
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
A286250(n) = A046523(-1+(2*A064216(n)));
for(n=1, 10000, write("b286250.txt", n, " ", A286250(n)));

from sympy import factorint, prevprime
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 a064216(n):
    f=factorint(2*n - 1)
    return 1 if n==1 else reduce(mul, [prevprime(i)**f[i] for i in f])
def a(n): return a046523((2*a064216(n)) - 1) # Indranil Ghosh, May 13 2017

(define (A286250 n) (A046523 (+ -1 (* 2 (A064216 n)))))

a(n) = A046523(A245448(n)) = A278223(A064216(n)) = A046523((2*A064216(n))-1).

A278224 a(n) = A046523(A048673(n)).

Original entry on oeis.org

1, 2, 2, 2, 4, 8, 6, 6, 2, 2, 2, 2, 4, 2, 12, 2, 6, 6, 12, 32, 12, 12, 6, 12, 4, 6, 12, 12, 16, 2, 2, 6, 6, 2, 6, 2, 6, 6, 2, 6, 6, 2, 24, 2, 24, 12, 8, 6, 2, 6, 48, 6, 30, 12, 6, 2, 6, 2, 2, 6, 6, 24, 30, 6, 60, 12, 36, 6, 2, 12, 2, 12, 24, 6, 6, 24, 72, 128, 30, 12, 2, 6, 12, 24, 2, 2, 30, 48, 4, 2, 6, 2, 6, 48, 16, 96, 6, 30, 2, 6, 12, 6, 24, 30, 2, 2, 6

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

This sequence works as a "sentinel" for sequence A048673 by matching to any other sequence that is obtained as f(A048673(n)), where f(n) is any function that depends only on the prime signature of n (see the index entry for "sequences computed from exponents in ..."). As of Nov 11 2016 no such sequences were present in the database.

Antti Karttunen, Table of n, a(n) for n = 1..10500
Index entries for sequences computed from exponents in factorization of n

Cf. A046523, A048673, A278223.

from sympy import factorint, nextprime
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 a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))//2
def a(n): return a046523(a048673(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 12 2017

(define (A278224 n) (A046523 (A048673 n)))

a(n) = A046523(A048673(n)).

A091304 a(n) = Omega(2n-1) (number of prime factors of the n-th odd number, counted with multiplicity).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 4, 1, 2, 2, 1, 2, 2, 2, 1, 3, 1, 1, 3, 1, 1, 2, 1, 2, 3, 2, 2, 2, 3, 1, 2, 1, 2, 4, 1, 1, 2, 2, 2, 3, 1, 1, 3, 2, 1, 2, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 1, 2, 2, 2, 4, 1, 1, 3, 1, 1, 2, 2, 2, 3, 2

Offset: 1

Andrew S. Plewe, Feb 20 2004

Omega(n) of the odd integers follows a pattern similar to A001222, with 4 maxima instead of 2 - i.e. between 2^n and (2^(n+1) - 1) there are two numbers with exactly n factors (2^n and 2^(n-1) * 3) while the odd integers have 4 maxima (3^n, 3^(n-1) * 5, 3^(n-1) * 7, 5^2*3^(n-2)) between 3^n and 3^(n+1) - 1.

			Omega(1) = 0, Omega(9) = 2 (3 * 3 = 9), Omega (243) = 5 (3 * 3 * 3 * 3 * 3 = 243), Omega(51) = 2 (3 * 17 = 51).
For n = 92, A001222(2*92 - 1) = A001222(183) = 2 as 183 = 3*61, thus a(92) = 2. - _Antti Karttunen_, May 31 2017

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

One more than A285716 (after the initial term).
Cf. A001222, A000120, A092523, A099774, A099990, A244153, A278223, A286582.
Cf. A006254 (positions of ones).

a[n_] := PrimeOmega[2*n - 1]; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)

a(n) = bigomega(2*n-1) \\ Michel Marcus, Jul 26 2013, edited to reflect the changed starting offset by Antti Karttunen, May 31 2017

a(n) = Omega(2n-1). [Odd bisection of A001222.]
From Antti Karttunen, May 31 2017: (Start)
For n >= 1, a(n) = A000120(A244153(n)).
For n >= 2, a(n) = 1+A285716(n).
(End)

Starting offset changed to 1 and the definition modified respectively. Also values of the initial term and of term a(92) (= 2, previously a(91) = 1) corrected by Antti Karttunen, May 31 2017

A286372 a(n) = A286366(A064216(n)).

Original entry on oeis.org

4, 6, 8, 13, 4, 9, 8, 11, 13, 12, 14, 8, 28, 6, 9, 13, 11, 21, 9, 11, 13, 12, 8, 8, 40, 14, 9, 65, 14, 13, 8, 13, 64, 13, 11, 8, 9, 30, 20, 12, 4, 9, 21, 11, 8, 21, 14, 20, 12, 9, 12, 13, 23, 9, 8, 11, 13, 64, 8, 84, 28, 14, 116, 12, 14, 9, 85, 11, 8, 12, 11, 65, 65, 42, 8, 13, 13, 21, 9, 11, 21, 13, 66, 8, 28, 12, 9, 49, 14, 13, 8, 11, 65, 20, 14, 13, 9, 66

Offset: 1

Antti Karttunen, May 09 2017

nonn

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

Cf. A064216, A278223, A286243, A286366, A286371, A286373.

(define (A286372 n) (A286366 (A064216 n)))

a(n) = A286366(A064216(n)).

A305973 Filter sequence for the prime signature of 2n-1.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 2, 4, 2, 2, 4, 2, 3, 5, 2, 2, 4, 4, 2, 4, 2, 2, 6, 2, 3, 4, 2, 4, 4, 2, 2, 6, 4, 2, 4, 2, 2, 6, 4, 2, 7, 2, 4, 4, 2, 4, 4, 4, 2, 6, 2, 2, 8, 2, 2, 4, 2, 4, 6, 4, 3, 4, 5, 2, 4, 2, 4, 9, 2, 2, 4, 4, 4, 6, 2, 2, 6, 4, 2, 4, 4, 2, 8, 2, 3, 6, 2, 6, 4, 2, 2, 4, 4, 4, 9, 2, 2, 8, 2, 2, 4, 4, 4, 6, 4, 2, 4, 4, 4, 4, 4, 2, 10, 2, 2, 8, 2, 4, 4, 2

Offset: 1

Antti Karttunen, Jun 15 2018

nonn

Restricted growth sequence transform of A278223, the least number with the same prime signature as the n-th odd number.

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

Cf. A046523, A101296, A278223, A305901, A305983, A305985.

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
v305973 = rgs_transform(vector(up_to,n,A046523(n+n-1)));
A305973(n) = v305973[n];

A285716 a(n) = A080791(A245611(n)).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 2, 1, 0, 3, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 2, 0, 0, 1, 0, 1, 2, 1, 1, 1, 2, 0, 1, 0, 1, 3, 0, 0, 1, 1, 1, 2, 0, 0, 2, 1, 0, 1, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 0, 1, 1, 1, 3, 0, 0, 2, 0, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 0, 3, 0, 0, 2, 0, 1, 1, 0

Offset: 1

Antti Karttunen, Apr 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

One less than A091304 after the initial term.
Cf. A000120, A080791, A244153, A245611, A278223, A285712, A285714, A285715.
Cf. A006254 (gives the positions of zeros after initial a(1)=0.)

a[n_] := PrimeOmega[2*n - 1] - 1; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)

;; First implementation uses memoization-macro definec:
(definec (A285716 n) (if (<= n 2) 0 (+ (if (= 2 (modulo n 3)) 1 0) (A285716 (A285712 n)))))
(define (A285716 n) (A080791 (A245611 n)))

a(1) = 0, a(2) = 1, for n > 2, a(n) = a(A285712(n)) + [n == 2 mod 3]. (Where [] is Iverson bracket, giving here 1 only if n is of the form 3k+2, and 0 otherwise.)
a(n) = A080791(A245611(n)).
For all n >= 2,  a(n) = A091304(n)-1 = A000120(A244153(n))-1. - Antti Karttunen, May 31 2017

A305901 Filter sequence for all such sequences b, for which b(A006254(k)) = constant for all k >= 3.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 14 2018

nonn

Restricted growth sequence transform of A305900(A064216(n)).
For all i, j:
  a(i) = a(j) => A278223(i) = A278223(j).
  a(i) = a(j) => A253786(i) = A253786(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A006254, A064216, A305900.

Cf. also A305902.

up_to = 1000;
partialsums(f,up_to) = { my(v = vector(up_to), s=0); for(i=1,up_to,s += f(i); v[i] = s); (v); }
v_partsums = partialsums(x -> isprime(x+x-1), up_to);
A305901(n) = if(n<=3,n,if(isprime(n+n-1),4,3+n-v_partsums[n]));

For n <= 3, a(n) = n, and for n >= 4, a(n) = 4 if 2n-1 is a prime (for all n in A006254[3..] = 4, 6, 7, 9, 10, 12, 15, ...), and for all other n (numbers n such that 2n-1 is composite), a(n) = running count from 5 onward.

A286257 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

1, 5, 5, 14, 12, 27, 5, 86, 14, 27, 23, 90, 12, 84, 27, 152, 23, 148, 5, 148, 27, 27, 80, 324, 25, 61, 44, 148, 23, 495, 5, 935, 61, 27, 61, 702, 5, 142, 61, 324, 138, 495, 23, 148, 90, 61, 23, 1426, 14, 265, 27, 90, 467, 324, 27, 430, 27, 61, 80, 2140, 12, 61, 183, 2144, 61, 495, 23, 607, 27, 495, 23, 2998, 23, 142, 90, 90, 142, 625, 5, 1426, 226, 27, 467

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, A278223, A286255, A286256, A286258.

Cf. A005382 (gives the positions of 5's), A067756 (of 12's), A234098 (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
A286257(n) = (1/2)*(2 + ((A046523(n)+A046523((2*n)-1))^2) - A046523(n) - 3*A046523((2*n)-1));
for(n=1, 10000, write("b286257.txt", n, " ", A286257(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 (A286257 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)).

a(n) = (1/2)*(2 + ((A046523(n)+A278223(n))^2) - A046523(n) - 3*A278223(n)).

cp's OEIS Frontend

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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A291769 Restricted growth sequence transform of A292249; filter combining multiplicative order of 2 mod 2n+1 & prime signature of 2n+1 (A002326 & A278223).

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A286250 Filter-sequence: a(n) = A278223(A064216(n)) = A046523((2*A064216(n))-1).

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A278224 a(n) = A046523(A048673(n)).

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A091304 a(n) = Omega(2n-1) (number of prime factors of the n-th odd number, counted with multiplicity).

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A286372 a(n) = A286366(A064216(n)).

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A305973 Filter sequence for the prime signature of 2n-1.

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A285716 a(n) = A080791(A245611(n)).

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