cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A286255 Compound filter: a(n) = P(A046523(n), A046523(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 5, 12, 14, 23, 27, 38, 63, 40, 27, 80, 90, 23, 61, 216, 152, 80, 90, 80, 148, 61, 27, 302, 375, 40, 84, 179, 90, 467, 495, 530, 698, 61, 61, 826, 702, 23, 61, 412, 324, 467, 495, 80, 265, 148, 27, 1178, 1323, 109, 148, 142, 90, 302, 430, 412, 430, 61, 27, 1832, 1890, 23, 142, 2787, 2410, 601, 495, 80, 148, 601, 495, 2630, 2700, 23, 142, 265, 148, 601, 495, 1178
Offset: 1

Views

Author

Antti Karttunen, May 07 2017

Keywords

Links

Crossrefs

Cf. A000027, A046523, A286240, A286256, A286257, A286258.
Cf. A005383 (after its initial term 3, gives the positions of 23's in this sequence).
Cf. A051950 (one of the matches not matched by A046523 alone).

Programs

  • PARI
    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
A286255(n) = (2 + ((A046523(n)+A046523(1+n))^2) - A046523(n) - 3*A046523(1+n))/2;
for(n=1, 10000, write("b286255.txt", n, " ", A286255(n)));
  • Python
    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(n + 1)) # Indranil Ghosh, May 07 2017
  • Scheme
    (define (A286255 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A046523 (+ 1 n))) 2) (- (A046523 n)) (- (* 3 (A046523 (+ 1 n)))) 2)))

Formula

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

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

Views

Author

Antti Karttunen, May 07 2017

Keywords

Links

Crossrefs

Cf. A000027, A046523, A278223, A286255, A286256, A286258.
Cf. A005382 (gives the positions of 5's), A067756 (of 12's), A234098 (of 23's).

Programs

  • PARI
    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)));
  • Python
    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
  • Scheme
    (define (A286257 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A046523 (+ -1 n n))) 2) (- (A046523 n)) (- (* 3 (A046523 (+ -1 n n)))) 2)))

Formula

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)).

A305978 Filter sequence combining prime signatures of n and 2n+1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 5, 6, 7, 8, 2, 9, 10, 4, 4, 11, 5, 12, 5, 12, 4, 13, 2, 14, 15, 4, 16, 17, 2, 18, 19, 20, 4, 8, 4, 21, 19, 8, 4, 22, 2, 23, 5, 12, 17, 8, 5, 24, 25, 12, 4, 26, 2, 27, 8, 27, 8, 13, 5, 28, 5, 29, 12, 30, 4, 23, 31, 12, 4, 23, 5, 32, 19, 4, 12, 33, 8, 18, 5, 34, 35, 36, 2, 28, 13, 4, 13, 37, 2, 38, 8, 17, 8, 39, 4, 40, 41, 12, 12, 42, 5, 23
Offset: 1

Views

Author

Antti Karttunen, Jun 15 2018

Keywords

Comments

Restricted growth sequence transform of A286258.

Links

Crossrefs

Cf. A046523, A101296, A286258, A305810, A305973, A305977.
Cf. A005384 (positions of 2's), A234095 (of 5's).

Programs

  • PARI
    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
Aux305978(n) = [A046523(n),A046523(n+n+1)];
v305978 = rgs_transform(vector(up_to,n,Aux305978(n)));
A305978(n) = v305978[n];

A286256 Compound filter: a(n) = P(A046523(n), A046523(2+n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 12, 5, 40, 5, 84, 12, 86, 14, 142, 5, 148, 23, 216, 27, 367, 5, 265, 23, 148, 27, 412, 12, 430, 59, 142, 44, 832, 5, 1860, 23, 698, 61, 826, 27, 856, 23, 412, 27, 1402, 5, 850, 80, 148, 90, 1384, 12, 1759, 40, 265, 27, 607, 23, 1105, 61, 430, 27, 2086, 5, 2140, 80, 2352, 148, 4342, 27, 850, 23, 832, 27, 5080, 5, 2998, 80, 142, 148, 832, 27, 2956, 138, 1426
Offset: 1

Views

Author

Antti Karttunen, May 07 2017

Keywords

Links

Crossrefs

Cf. A000027, A046523, A286255, A286257, A286258.
Cf. A001359 (gives the positions of 5's), A049002 (of 12's), A115093 (of 23's).

Programs

  • PARI
    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
A286256(n) = (2 + ((A046523(n)+A046523(2+n))^2) - A046523(n) - 3*A046523(2+n))/2;
for(n=1, 10000, write("b286256.txt", n, " ", A286256(n)));
  • Python
    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(n + 2)) # Indranil Ghosh, May 07 2017
  • Scheme
    (define (A286256 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A046523 (+ 2 n))) 2) (- (A046523 n)) (- (* 3 (A046523 (+ 2 n)))) 2)))

Formula

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

A286466 Compound filter: a(n) = P(A112049(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 12, 2, 16, 5, 38, 7, 16, 9, 94, 2, 16, 23, 138, 2, 67, 5, 80, 16, 16, 9, 355, 7, 16, 38, 80, 2, 436, 5, 530, 16, 16, 40, 706, 2, 16, 23, 302, 2, 436, 5, 80, 67, 16, 9, 1228, 7, 67, 23, 80, 2, 277, 23, 302, 16, 16, 14, 2021, 2, 16, 80, 2082, 16, 436, 5, 80, 16, 436, 9, 2704, 2, 16, 80, 80, 16, 436, 5, 1178, 121, 16, 9, 2086, 16, 16, 23, 302, 2, 1771
Offset: 1

Views

Author

Antti Karttunen, May 10 2017

Keywords

Comments

Here the information combined together to a(n) consists of A046523(n), giving essentially the prime signature of n, 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).

Links

Crossrefs

Cf. A000027, A046523, A112049, A286258, A286465.

Programs

  • PARI
    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
A286466(n) = (1/2)*(2 + ((A112049(n)+A046523(n))^2) - A112049(n) - 3*A046523(n));
for(n=1, 10000, write("b286466.txt", n, " ", A286466(n)));
  • Python
    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 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 T(a112049(n), a046523(n)) # Indranil Ghosh, May 11 2017
  • Scheme
    (define (A286466 n) (* (/ 1 2) (+ (expt (+ (A112049 n) (A046523 n)) 2) (- (A112049 n)) (- (* 3 (A046523 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A112049(n)+A046523(n))^2) - A112049(n) - 3*A046523(n)).
Showing 1-5 of 5 results.

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