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 1-4 of 4 results.

A210208 Triangle read by rows in which row n lists the divisors of n that are prime powers, A000961.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 1, 11, 1, 2, 3, 4, 1, 13, 1, 2, 7, 1, 3, 5, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 9, 1, 19, 1, 2, 4, 5, 1, 3, 7, 1, 2, 11, 1, 23, 1, 2, 3, 4, 8, 1, 5, 25, 1, 2, 13, 1, 3, 9, 27, 1, 2, 4, 7
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 18 2012

Keywords

Comments

{T(n,k): k = 1..A073093(n)} subset of {A027750(n,k): k = 1..A000005(n)} for all n.

Examples

			Table begins:
  1;
  1, 2;
  1, 3;
  1, 2, 4;
  1, 5;
  1, 2, 3;
  1, 7;
  1, 2, 4, 8;
  1, 3, 9;
  1, 2, 5;
  1, 11;
  1, 2, 3, 4; - _Geoffrey Critzer_, Feb 08 2015

Links

Crossrefs

Cf. A073093 (row lengths), A023888 (row sums), A034699 (row maxima), A183091 (row products).
Cf. A000005, A010055, A027750, A141809.

Programs

  • Haskell
    a210208 n k = a210208_tabf !! (n-1) !! (n-1)
a210208_row n = a210208_tabf !! (n-1)
a210208_tabf = map (filter ((== 1) . a010055)) a027750_tabf
  • Mathematica
    Table[Prepend[Select[Divisors[n], PrimeNu[#] == 1 &], 1], {n, 1, 10}]//Grid (* Geoffrey Critzer, Feb 08 2015 *)
  • PARI
    row(n) = select(x -> omega(x) < 2, divisors(n)); \\ Amiram Eldar, May 02 2025

Formula

A034699(n) = T(n,A073093(n)) = maximum of n-th row.

A183092 a(n) is the product of divisors d of n such that d is not equal to m^k where m = noncomposite number, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 72, 1, 14, 15, 1, 1, 108, 1, 200, 21, 22, 1, 1728, 1, 26, 1, 392, 1, 27000, 1, 1, 33, 34, 35, 46656, 1, 38, 39, 8000, 1, 74088, 1, 968, 675, 46, 1, 82944, 1, 500, 51, 1352, 1, 5832, 55, 21952, 57, 58, 1, 388800000, 1, 62, 1323, 1, 65, 287496, 1, 2312, 69, 343000, 1, 80621568, 1, 74, 1125, 2888, 77
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2010

Keywords

Comments

For n = 12, the set of such divisors is {6, 12}; a(12) = 6*12 = 72.
a(n) is also the product of divisors d of n such that d is not equal to p^k where p = prime, k >=1. For n = 12, the set of such divisors is {1, 6, 12}; a(12) = 1*6*12 = 72.

Links

Crossrefs

Cf. A007955, A183091.

Programs

  • Maple
    A183092 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) minus {1} do if nops( numtheory[factorset](d)) > 1 then a := a*d; end if; end do: a ; end proc: # R. J. Mathar, Apr 14 2011
  • PARI
    A183092(n) = factorback(apply(d -> if(isprimepower(d),1,d), divisors(n))); \\ Antti Karttunen, Aug 06 2018

Formula

a(n) = A007955(n) / A183091(n).
a(1) = 1, a(p) = 1, a(pq) = pq, a(pq...z) = (pq...z)^(2^(k-1)-1), a(p^k) = 1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

Extensions

More terms from Antti Karttunen, Aug 06 2018

A290480 Product of proper unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 12, 1, 14, 15, 1, 1, 18, 1, 20, 21, 22, 1, 24, 1, 26, 1, 28, 1, 27000, 1, 1, 33, 34, 35, 36, 1, 38, 39, 40, 1, 74088, 1, 44, 45, 46, 1, 48, 1, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 216000, 1, 62, 63, 1, 65, 287496, 1, 68, 69, 343000, 1, 72, 1, 74, 75, 76, 77, 474552, 1, 80
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 03 2017

Keywords

Examples

			a(12) = 12 because 12 has 6 divisors {1, 2, 3, 4, 6, 12} among which 3 are proper unitary {1, 3, 4} and 1*3*4 = 12.

Links

Crossrefs

Cf. A001221, A007774 (fixed points), A007955, A007956, A034460, A061537, A062758, A063919, A078599, A087652, A126192, A136655, A183091.

Programs

  • Maple
    with(numtheory):
a:= n-> mul(d, d=select(x-> igcd(x, n/x)=1, divisors(n) minus {n})):
seq(a(n), n=1..80);  # Alois P. Heinz, Aug 03 2017
  • Mathematica
    Table[Product[d, {d, Select[Divisors[n], GCD[#, n/#] == 1 &]}]/n, {n, 80}]
Table[n^(2^(PrimeNu[n] - 1) - 1), {n, 80}]
  • PARI
    A290480(n) = if(1==n,n,n^(2^(omega(n)-1)-1)); \\ Antti Karttunen, Aug 06 2018
  • Python
    from sympy import divisors, gcd, prod
def a(n): return prod(d for d in divisors(n) if gcd(d, n//d) == 1)//n
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 04 2017

Formula

a(n) = A061537(n)/n.
a(n) = n^(2^(omega(n)-1)-1), where omega() is the number of distinct primes dividing n (A001221).
a(n) = 1 if n is a prime power.

A290479 Product of nonprime squarefree divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 6, 1, 14, 15, 1, 1, 6, 1, 10, 21, 22, 1, 6, 1, 26, 1, 14, 1, 27000, 1, 1, 33, 34, 35, 6, 1, 38, 39, 10, 1, 74088, 1, 22, 15, 46, 1, 6, 1, 10, 51, 26, 1, 6, 55, 14, 57, 58, 1, 27000, 1, 62, 21, 1, 65, 287496, 1, 34, 69, 343000, 1, 6, 1, 74, 15, 38, 77, 474552, 1, 10
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 03 2017

Keywords

Examples

			a(30) = 27000 because 30 has 8 divisors {1, 2, 3, 5, 6, 10, 15, 30} among which 5 are nonprime squarefree {1, 6, 10, 15, 30} and 1*6*10*15*30 = 27000.

Links

Crossrefs

Cf. A000005, A000469, A006881 (fixed points), A007947, A007955, A007956, A061537, A062758, A078599, A087652, A126192, A136655, A183091, A284118.

Programs

  • Mathematica
    Table[Product[d, {d, Select[Divisors[n], !PrimeQ[#] && SquareFreeQ[#] &]}], {n, 80}]
Table[Last[Select[Divisors[n], SquareFreeQ]]^(DivisorSigma[0, Last[Select[Divisors[n], SquareFreeQ]]]/2 - 1), {n, 80}]
  • PARI
    A290479(n) = if(1==n, n, my(r=factorback(factorint(n)[, 1])); (r^((numdiv(r)/2)-1))); \\ Antti Karttunen, Aug 06 2018

Formula

a(n) = A078599(n)/A007947(n).
a(n) = rad(n)^(d(rad(n))/2-1), where d() is the number of divisors of n (A000005) and rad() is the squarefree kernel of n (A007947).
a(n) = 1 if n is a prime power.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.