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.

A140215 A140213(n)*A140214(n).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 7, 1, 1, 5, 11, 1, 13, 7, 5, 1, 17, 1, 19, 5, 7, 11, 23, 1, 125, 13, 1, 7, 29, 5, 31, 1, 11, 17, 1225, 1, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 1, 343, 125, 17, 13, 53, 1, 3025, 7, 19, 29, 59, 5, 61, 31, 7, 1, 4225, 11, 67, 17, 23, 1225, 71, 1, 73, 37, 125, 19
Offset: 1

Views

Author

R. J. Mathar, Jun 27 2008

Keywords

Programs

A279060 Number of divisors of n of the form 6*k + 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 4, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 1, 2, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2016

Keywords

Comments

Möbius transform is the period-6 sequence {1, 0, 0, 0, 0, 0, ...}.

Examples

			a(14) = 2 because 14 has 4 divisors {1,2,7,14} among which 2 divisors {1,7} are of the form 6*k + 1.

Links

Crossrefs

Cf. A001227, A001817, A001826, A001876, A035218, A140213, A188169, A319995, A320001.
Cf. A001620, A016629, A222457 (psi(1/6)).

Programs

  • Mathematica
    nmax = 120; CoefficientList[Series[Sum[x^k/(1 - x^(6 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 120; CoefficientList[Series[Sum[x^(6 k + 1)/(1 - x^(6 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
Table[Count[Divisors[n],?(Mod[#,6]==1&)],{n,0,120}] (* _Harvey P. Dale, Apr 27 2018 *)
  • PARI
    A279060(n) = if(!n,n,sumdiv(n, d, (1==(d%6)))); \\ Antti Karttunen, Jul 09 2017
  • Python
    from sympy import divisors
def A279060(n): return sum(d%6 == 1 for d in divisors(n)) # David Radcliffe, Jun 19 2025

Formula

G.f.: Sum_{k>=1} x^k/(1 - x^(6*k)).
G.f.: Sum_{k>=0} x^(6*k+1)/(1 - x^(6*k+1)).
From Antti Karttunen, Oct 03 2018: (Start)
a(n) = A320001(n) + [1 == n (mod 6)], where [ ] is the Iverson bracket, giving 1 only when n = 1 mod 6, and 0 otherwise.
a(n) = A035218(n) - A319995(n). (End)
a(n) = (A035218(n) + A035178(n)) / 2. - David Radcliffe, Jun 19 2025
Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,6) - (1 - gamma)/6 = 0.686263..., gamma(1,6) = -(psi(1/6) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A136655 Product of odd divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 27, 5, 11, 3, 13, 7, 225, 1, 17, 27, 19, 5, 441, 11, 23, 3, 125, 13, 729, 7, 29, 225, 31, 1, 1089, 17, 1225, 27, 37, 19, 1521, 5, 41, 441, 43, 11, 91125, 23, 47, 3, 343, 125, 2601, 13, 53, 729, 3025, 7, 3249, 29, 59, 225, 61, 31, 250047, 1, 4225, 1089
Offset: 1

Views

Author

Jonathan Vos Post, Jun 25 2008

Keywords

Comments

Product of rows of triangle A182469. - Reinhard Zumkeller, May 01 2012

Links

Crossrefs

Cf. A000265, A000593, A007955, A007956, A078701.
Cf. A140210, A140211, A140213, A140214, A140215.
Cf. A125911, A126192.
Cf. A001227, A183063.

Programs

  • Haskell
    a136655 = product . a182469_row  -- Reinhard Zumkeller, May 01 2012
  • Maple
    with(numtheory); f:=proc(n) local t1,i,k; t1:=divisors(n); k:=1; for i in t1 do if i mod 2 = 1 then k:=k*i; fi; od; k; end; # N. J. A. Sloane, Jul 14 2008
  • Mathematica
    Array[Times @@ Select[Divisors@ #, OddQ] &, 66] (* Michael De Vlieger, Aug 03 2017 *)
a[n_] := (oddpart = n/2^IntegerExponent[n, 2])^(DivisorSigma[0, oddpart]/2); Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, if (d[k]%2, d[k], 1)); \\ Michel Marcus, Aug 04 2017
  • Python
    from math import isqrt
from sympy import divisor_count
def A136655(n):
    d = divisor_count(m:=n>>(~n&n-1).bit_length())
    return isqrt(m)**d if d&1 else m**(d>>1) # Chai Wah Wu, Jun 27 2025

Formula

a(p) = p if p noncomposite; a(2^n) = 1; a(pq) = p^2 * q^2 when p, q are odd primes.
a(n) = sqrt(n^od(n)/2^ed(n)), where od(n) = number of odd divisors of n = tau(2*n)-tau(n) and ed(n) = number of even divisors of n = 2*tau(n)-tau(2*n). - Vladeta Jovovic, Jun 25 2008
Also a(n) = A007955(A000265(n)). - David Wilson, Jun 26 2008
a(n) = Product_{h == 1 mod 4 and h | n}*Product_{i == 3 mod 4 and i | n}.
a(n) = Product_{j == 1 mod 6 and j | n}*Product_{k == 5 mod 6 and k | n}.
a(n) = A140210(n)*A140211(n). - R. J. Mathar, Jun 27 2008
a(n) = A007955(n) / A125911(n).

Extensions

More terms from N. J. A. Sloane, Jul 14 2008
Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A170824 a(n) = product of distinct primes of form 6k+1 that divide n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 13, 7, 1, 1, 1, 1, 19, 1, 7, 1, 1, 1, 1, 13, 1, 7, 1, 1, 31, 1, 1, 1, 7, 1, 37, 19, 13, 1, 1, 7, 43, 1, 1, 1, 1, 1, 7, 1, 1, 13, 1, 1, 1, 7, 19, 1, 1, 1, 61, 31, 7, 1, 13, 1, 67, 1, 1, 7, 1, 1, 73, 37, 1, 19, 7, 13, 79, 1, 1, 1, 1, 7, 1, 43, 1, 1, 1, 1, 91, 1, 31, 1
Offset: 1

Views

Author

N. J. A. Sloane, Dec 25 2009, following a suggestion from Jonathan Vos Post

Keywords

Links

Crossrefs

Cf. A002476, A170817.
Cf. A140213. [R. J. Mathar, Jan 21 2010]
Differs from A248909 for the first time at n=49, where a(49) = 7, while A248909(49) = 49.

Programs

  • Maple
    A170824 := proc(n) a := 1 ; for p in numtheory[factorset](n) do if p mod 6 = 1 then a := a*p ; end if ; end do ; a ; end proc:
A140213 := proc(n) a := 1 ; for p in numtheory[divisors](n) do if p mod 6 = 1 then a := a*p ; end if ; end do ; a ; end proc:
seq(A170824(n),n=1..120) ; # R. J. Mathar, Jan 21 2010
  • Mathematica
    test[p_] := IntegerQ[(p - 1)/6]; a[n_]:= Module[{aux = FactorInteger[n]}, Product[If[test[aux[[i, 1]]],aux[[i, 1]],1],{i, Length[aux]}]]; Table[a[n], {i, 1, 200}] (* Jose Grau, Feb 16 2010 *)
Table[Times@@Select[Transpose[FactorInteger[n]][[1]],IntegerQ[(#-1)/6]&],{n,100}] (* Harvey P. Dale, Jul 29 2013 *)
  • PARI
    a(n) = my(f=factor(n)); prod(k=1, #f~, if (((p=f[k,1])%6) == 1, p, 1)); \\ Michel Marcus, Jul 10 2017
  • Scheme
    (define (A170824 n) (if (= 1 n) n (* (if (= 1 (modulo (A020639 n) 6)) (A020639 n) 1) (A170824 (A028234 n))))) ;; Antti Karttunen, Jul 09 2017

Formula

a(1) = 1; for n > 1, if A020639(n) = 1 (mod 6), a(n) = A020639(n) * a(A028234(n)), otherwise a(n) = a(A028234(n)). - Antti Karttunen, Jul 09 2017

Extensions

More terms from R. J. Mathar, Jan 21 2010

A140210 a(n) = Product_{h == 1 (mod 4) and h|n} h.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 9, 5, 1, 1, 13, 1, 5, 1, 17, 9, 1, 5, 21, 1, 1, 1, 125, 13, 9, 1, 29, 5, 1, 1, 33, 17, 5, 9, 37, 1, 13, 5, 41, 21, 1, 1, 2025, 1, 1, 1, 49, 125, 17, 13, 53, 9, 5, 1, 57, 29, 1, 5, 61, 1, 189, 1, 4225, 33, 1, 17, 69, 5, 1, 9, 73, 37, 125, 1, 77, 13, 1, 5, 729, 41, 1, 21
Offset: 1

Views

Author

R. J. Mathar, Jun 27 2008

Keywords

Links

Crossrefs

Cf. A007955, A140211, A136655, A140213, A140214, A140215, A136655.

Programs

  • Maple
    with(numtheory):
a:= n-> mul(i, i=select(h-> irem(h, 4)=1, divisors(n))):
seq(a(n), n=1..120);  # Alois P. Heinz, Jul 28 2009
  • Mathematica
    a[n_] := Times @@ Select[Divisors[n], Mod[#, 4] == 1&]; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 24 2015 *)
  • PARI
    a(n) = my(p=1); fordiv(n, d, if ((d % 4)==1, p*=d)); p; \\ Michel Marcus, Jan 07 2021

Extensions

More terms from Alois P. Heinz, Jul 28 2009
Showing 1-5 of 5 results.

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

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