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.

A175081 Values taken by the sum of perfect divisors of n (A175067) sorted into ascending order.

Original entry on oeis.org

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

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Author

Jaroslav Krizek, Jan 24 2010

Keywords

Comments

Perfect divisor of n is divisor d such that d^k = n for some k >= 1. See A089723 (number of perfect divisors of n) and A175067 (sum of perfect divisors of n).

A175082 Possible values for sum of perfect divisors of n.

Original entry on oeis.org

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

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Author

Jaroslav Krizek, Jan 24 2010

Keywords

Comments

Perfect divisor of n is divisor d such that d^k = n for some k >= 1. See A175067 (sum of perfect divisors of n) and A175081 (values taken by the sum of perfect divisors of n (A175067) sorted into ascending order).
Complement of A001597(n+1) for n >= 1 (perfect powers >= 4). a(n) = A007916(n-1) for n >= 2. [From Jaroslav Krizek, Jan 30 2010]

A175070 a(n) is the sum of perfect divisors of n - n, where a perfect divisor of n is a divisor d such that d^k = n for some k >= 1.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10
Offset: 1

Views

Author

Jaroslav Krizek, Jan 23 2010

Keywords

Comments

a(1) = 0, for n >=2: a(n) = sum of perfect divisors of n less than n.
a(n) > 0 for perfect powers n = A001597(m) for m > 2.

Links

Crossrefs

Cf. A175067.

Programs

  • Maple
    a:= n-> add(`if`(n=d^ilog[d](n), d, 0), d=numtheory[divisors](n) minus {n}):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 12 2024
  • PARI
    A175070(n) = if(!ispower(n),0,sumdiv(n,d,if((d>1)&&(dAntti Karttunen, Jun 12 2018
  • PARI
    first(n) = {my(res = vector(n)); for(i = 2, sqrtint(n), for(j = 2, logint(n, i), res[i^j] += i)); res} \\ David A. Corneth, Jun 12 2018

Formula

a(n) = A175067(n) - n.

A175083 Number of numbers whose sum of perfect divisors is equal to n.

Original entry on oeis.org

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

Views

Author

Jaroslav Krizek, Jan 24 2010

Keywords

Comments

Perfect divisor of m is divisor d such that d^k = m for some k >= 1. See A175067 (sum of perfect divisors of n) and A175081 (values taken by the sum of perfect divisors of n (A175067) sorted into ascending order).

Links

Crossrefs

Cf. A175067, A175081.

Programs

  • PARI
    up_to = 65537;
A175067(n) = (n+if(!ispower(n),0,sumdiv(n,d,if((d>1)&&(dA175083list(up_to) = { my(range = Map(), v = vector(up_to), x); for(n=1,up_to,x=A175067(n); mapput(range,x,1+if(!mapisdefined(range,x), 0, mapget(range,x)))); for(n=1,up_to,v[n]=if(!mapisdefined(range,n), 0, mapget(range,n))); (v); };
v175083 = A175083list(up_to);
A175083(n) = v175083[n]; \\ Antti Karttunen, Sep 25 2018

Extensions

More terms from Antti Karttunen, Sep 25 2018

A178645 a(n) = sum of divisors d of n such that d^k is not equal to n for any k >= 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 6, 1, 5, 1, 8, 1, 16, 1, 10, 9, 9, 1, 21, 1, 22, 11, 14, 1, 36, 1, 16, 10, 28, 1, 42, 1, 29, 15, 20, 13, 49, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 1, 43, 21, 46, 1, 66, 17, 64, 23, 32, 1, 108, 1, 34, 41, 49, 19, 78, 1, 58, 27, 74, 1, 123, 1, 40, 49, 64, 19, 90, 1, 106, 28, 44, 1, 140, 23, 46, 33, 92, 1, 144, 21, 76, 35, 50, 25, 156, 1, 73, 57, 107
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2010

Keywords

Examples

			For n = 16, set of such divisors is {1, 8}; a(16) = 1+8=9.
For n = 90, which is not a perfect power (A001597), the only divisor d for which d^k = 90 is 90 itself, with k=1, thus a(90) = A001065(90) = A000203(90) - 90 = 144. - _Antti Karttunen_, Jun 12 2018

Links

Crossrefs

Cf. A000203, A001065, A175067, A175070.

Programs

Formula

a(n) = A000203(n) - A175067(n).
a(1) = 0, a(p) = 1, a(pq) = p+q+1, a(pq...z) = [(p+1)*(q+1)*…*(z+1)] - (pq…z), for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

Extensions

Term a(90) corrected from 204 to 144 by Antti Karttunen, Jun 12 2018
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