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.

A286585 a(n) = A053735(A048673(n)).

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 2, 4, 3, 3, 3, 5, 1, 5, 2, 5, 2, 4, 2, 4, 2, 4, 3, 6, 5, 6, 3, 6, 4, 7, 3, 6, 3, 3, 3, 5, 3, 5, 5, 5, 4, 3, 4, 5, 4, 6, 1, 7, 5, 6, 4, 7, 2, 8, 4, 7, 4, 5, 3, 8, 4, 4, 4, 7, 4, 6, 2, 4, 5, 6, 3, 6, 4, 6, 5, 6, 4, 6, 4, 6, 7, 5, 3, 4, 5, 7, 6, 6, 5, 5, 4, 7, 3, 8, 1, 8, 5, 6, 3, 7, 6, 7, 2, 8
Offset: 1

Views

Author

Antti Karttunen, May 31 2017

Keywords

Links

Crossrefs

Cf. A000079, A048673, A053735, A286582, A286583, A286584, A286586.

Programs

  • Python
    from sympy.ntheory.factor_ import digits
from sympy import factorint, nextprime
from operator import mul
def a053735(n): return sum(digits(n, 3)[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 a053735(a048673(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 12 2017
  • Scheme
    (define (A286585 n) (A053735 (A048673 n)))

Formula

a(n) = A053735(A048673(n)).
For all n >= 0, a(A000079(n)) = n+1.

A286582 a(n) = A001222(A048673(n)).

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 2, 2, 1, 1, 1, 1, 2, 1, 3, 1, 2, 2, 3, 5, 3, 3, 2, 3, 2, 2, 3, 3, 4, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 4, 1, 4, 3, 3, 2, 1, 2, 5, 2, 3, 3, 2, 1, 2, 1, 1, 2, 2, 4, 3, 2, 4, 3, 4, 2, 1, 3, 1, 3, 4, 2, 2, 4, 5, 7, 3, 3, 1, 2, 3, 4, 1, 1, 3, 5, 2, 1, 2, 1, 2, 5, 4, 6, 2, 3, 1, 2, 3, 2, 4, 3, 1, 1
Offset: 1

Views

Author

Antti Karttunen, May 31 2017

Keywords

Links

Crossrefs

Cf. A001222, A048673, A091304, A278224, A286583, A286584, A286585.

Programs

  • PARI
    A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A048673(n) = (A003961(n)+1)/2;
A286582(n) = bigomega(A048673(n));
  • Python
    from sympy import factorint, nextprime, primefactors, prod
def a001222(n): return 0 if n==1 else a001222(n//primefactors(n)[-1]) + 1
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + prod(nextprime(i)**f[i] for i in f))//2
def a(n): return a001222(a048673(n))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jun 12 2017
  • Scheme
    (define (A286582 n) (A001222 (A048673 n)))

Formula

a(n) = A001222(A048673(n)).

A286584 a(n) = A048673(n) mod 4.

Original entry on oeis.org

1, 2, 3, 1, 0, 0, 2, 2, 1, 3, 3, 3, 1, 1, 2, 1, 2, 2, 0, 0, 0, 0, 3, 0, 1, 2, 3, 2, 0, 1, 3, 2, 1, 1, 3, 1, 1, 3, 3, 3, 2, 3, 0, 3, 0, 0, 3, 3, 1, 2, 0, 1, 2, 0, 2, 1, 2, 3, 3, 2, 2, 0, 2, 1, 0, 2, 0, 2, 1, 0, 1, 2, 0, 2, 3, 0, 0, 0, 2, 0, 1, 1, 1, 0, 3, 3, 2, 0, 1, 3, 2, 3, 1, 0, 1, 0, 3, 2, 3, 1, 0, 3, 2, 2, 1, 1, 3, 3, 1, 1, 3, 2, 0
Offset: 1

Views

Author

Antti Karttunen, May 31 2017

Keywords

Links

Crossrefs

Cf. A010873, A048673, A286582, A286583, A286585.
Cf. A246261 (positions of odd terms), A246263 (of even terms).

Programs

  • PARI
    A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A048673(n) = (A003961(n)+1)/2;
A286584(n) = (A048673(n)%4);
  • Python
    from sympy import factorint, nextprime
from operator import mul
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 a048673(n)%4 # Indranil Ghosh, Jun 12 2017
  • Scheme
    (define (A286584 n) (modulo (A048673 n) 4))

Formula

a(n) = A010873(A048673(n)) = A048673(n) mod 4.

A305902 Restricted growth sequence transform of A305900(A048673(n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 14 2018

Keywords

Comments

For all i, j:
a(i) = a(j) => A278224(i) = A278224(j).
a(i) = a(j) => A286583(i) = A286583(j).
a(i) = a(j) => A292251(i) = A292251(j).

Links

Crossrefs

Cf. A048673, A305900.
Cf. also A305901.

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; };
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A305900(n) = if(n<=5,n,if(isprime(n),5,3+n-primepi(n)));
v305902 = rgs_transform(vector(up_to,n,A305900(A048673(n))));
A305902(n) = v305902[n];

A289623 a(n) = A055396(A048673(n)).

Original entry on oeis.org

0, 1, 2, 3, 1, 1, 1, 1, 6, 5, 4, 9, 2, 7, 1, 13, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 16, 8, 1, 2, 10, 2, 30, 2, 3, 14, 3, 1, 23, 1, 17, 1, 1, 2, 4, 18, 1, 1, 4, 1, 1, 1, 35, 1, 15, 11, 1, 1, 1, 1, 3, 1, 1, 1, 1, 21, 1, 12, 1, 1, 1, 2, 1, 1, 1, 1, 1, 65, 3, 2, 1, 19, 20, 1, 1, 4, 56, 1, 32, 2, 1, 2, 1, 2, 1, 38, 6
Offset: 1

Views

Author

Antti Karttunen, Jul 16 2017

Keywords

Comments

From the scatter plot it can be seen that the terms are grouped into two distinct populations by their magnitude, with significant gap between them.

Links

Crossrefs

Cf. A048673, A055396, A246263 (the positions of ones).
Cf. also A278224, A286582, A286583, A286584, A286586.

Programs

Formula

a(n) = A055396(A048673(n)).
Showing 1-5 of 5 results.

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