A086780 -id:A086780 - cp's OEIS frontend

A033846 Numbers whose prime factors are 2 and 5.

10, 20, 40, 50, 80, 100, 160, 200, 250, 320, 400, 500, 640, 800, 1000, 1250, 1280, 1600, 2000, 2500, 2560, 3200, 4000, 5000, 5120, 6250, 6400, 8000, 10000, 10240, 12500, 12800, 16000, 20000, 20480, 25000, 25600, 31250, 32000, 40000, 40960

Offset: 1

Jeff Burch

Numbers k such that Sum_{d prime divisor of k} 1/d = 7/10. - Benoit Cloitre, Apr 13 2002
Numbers k such that phi(k) = (2/5)*k. - Benoit Cloitre, Apr 19 2002
Numbers k such that Sum_{d|k} A008683(d)*A000700(d) = 7. - Carl Najafi, Oct 20 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A033845, A033847, A033848, A033849, A033850, A033851, A003592.
Cf. A086780, A143201.
Cf. A000700, A008683.

import Data.Set (singleton, deleteFindMin, insert)
a033846 n = a033846_list !! (n-1)
a033846_list = f (singleton (2*5)) where
   f s = m : f (insert (2*m) $ insert (5*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

[n:n in [1..100000]| Set(PrimeDivisors(n)) eq {2,5}]; // Marius A. Burtea, May 10 2019

A033846 := proc(n)
if (numtheory[factorset](n) = {2,5}) then
   RETURN(n)
fi: end:  seq(A033846(n),n=1..50000); # Jani Melik, Feb 24 2011

Take[Union[Times@@@Select[Flatten[Table[Tuples[{2,5},n],{n,2,15}],1], Length[Union[#]]>1&]],45] (* Harvey P. Dale, Dec 15 2011 *)

isA033846(n)=factor(n)[,1]==[2,5]~ \\ Charles R Greathouse IV, Feb 24 2011

a(n) = 10*A003592(n).
A143201(a(n)) = 4. - Reinhard Zumkeller, Sep 13 2011
Sum_{n>=1} 1/a(n) = 1/4. - Amiram Eldar, Dec 22 2020

Offset fixed by Reinhard Zumkeller, Sep 13 2011

A147571 Numbers with exactly 4 distinct prime divisors {2,3,5,7}.

Original entry on oeis.org

210, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2520, 2940, 3150, 3360, 3780, 4200, 4410, 5040, 5250, 5670, 5880, 6300, 6720, 7350, 7560, 8400, 8820, 9450, 10080, 10290, 10500, 11340, 11760, 12600, 13230, 13440, 14700, 15120, 15750, 16800

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Successive numbers k such that EulerPhi(x)/x = m:
( Family of sequences for successive n primes )
m=1/2 numbers with exactly 1 distinct prime divisor {2} see A000079
m=1/3 numbers with exactly 2 distinct prime divisors {2,3} see A033845
m=4/15 numbers with exactly 3 distinct prime divisors {2,3,5} see A143207
m=8/35 numbers with exactly 4 distinct prime divisors {2,3,5,7} see A147571
m=16/77 numbers with exactly 5 distinct prime divisors {2,3,5,7,11} see A147572
m=192/1001 numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13} see A147573
m=3072/17017 numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17} see A147574
m=55296/323323 numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19} see A147575

David A. Corneth, Table of n, a(n) for n = 1..10688

Cf. A002473, A086780, A143207, A147572, A147573, A147574, A147575, A147576, A147577, A147578, A147579, A147580.

[n: n in [1..2*10^4] | PrimeDivisors(n) eq [2,3,5,7]]; // Vincenzo Librandi, Sep 15 2015

a = {}; Do[If[EulerPhi[x]/x == 8/35, AppendTo[a, x]], {x, 1, 100000}]; a
Select[Range[20000],PrimeNu[#]==4&&Max[FactorInteger[#][[;;,1]]]<11&] (* Harvey P. Dale, Nov 05 2024 *)

is(n)=n%210==0 && n==2^valuation(n,2) * 3^valuation(n,3) * 5^valuation(n,5) * 7^valuation(n,7) \\ Charles R Greathouse IV, Jun 19 2016

a(n) = 210 * A002473(n). - David A. Corneth, May 14 2019

Sum_{n>=1} 1/a(n) = 1/48. - Amiram Eldar, Nov 12 2020

A051664 a(n) is the number of nonzero coefficients in the n-th cyclotomic polynomial.

Original entry on oeis.org

2, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 7, 2, 17, 3, 19, 5, 9, 11, 23, 3, 5, 13, 3, 7, 29, 7, 31, 2, 15, 17, 17, 3, 37, 19, 17, 5, 41, 9, 43, 11, 7, 23, 47, 3, 7, 5, 23, 13, 53, 3, 17, 7, 25, 29, 59, 7, 61, 31, 9, 2, 31, 15, 67, 17, 31, 17, 71, 3, 73, 37, 7, 19, 31, 17, 79, 5, 3

Offset: 1

Jud McCranie

nonn

a(n)=p(n) if n=p(n); a(n) is not always A006530(n). - Labos Elemer, May 03 2002

This sequence is the Mobius transform of A087073. Let m be the squarefree part of n, then a(n) = a(m). When n = pq, the product of two distinct odd primes, then there is a formula for a(pq). Let x = 1/p (mod q) and y = 1/q (mod p). Then a(pq) = 2xy-1. There are also formulas for the number of positive and negative terms. See papers by Carlitz or Lam and Leung. - T. D. Noe, Aug 08 2003

			9th cyclotomic polynomial is x^6+x^3+1 which has 3 terms, so a(9)=3.

T. D. Noe, Table of n, a(n) for n = 1..1000
L. Carlitz, Number of terms in the cyclotomic polynomial F(pq,x), Amer. Math. Monthly, Vol. 73, No. 9, 1966, pp. 979-981.
T. Y. Lam and K. H. Leung, On the Cyclotomic Polynomial Phi(pq,x), Amer. Math. Monthly, Vol. 103, No. 7, 1996, pp. 562-564.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A086765 (number of positive terms in n-th cyclotomic polynomial), A086780 (number of negative terms in n-th cyclotomic polynomial), A086798 (number of zero terms in n-th cyclotomic polynomial), A087073.

A051664 := proc(n)
        numtheory[cyclotomic](n,x) ;
        nops([coeffs(%)]) ;
end proc: # R. J. Mathar, Sep 15 2012

Table[Count[CoefficientList[Cyclotomic[n, x], x], _?(#!=0&)], {n, 0, 100}]
Table[Length[Cyclotomic[n, x]], {n, 1, 100}] (* Artur Jasinski, Jan 15 2007 *)

a(n)=sum(k=0,eulerphi(n),if(polcoeff(polcyclo(n),k),1,0))

a(n) = #select(x->x!=0, Vec(polcyclo(n))); \\ Michel Marcus, Mar 05 2017

a(n) = phi(n) + 1 - A086798(n). - T. D. Noe, Aug 08 2003

More terms from Labos Elemer, May 03 2002

A288162 Numbers whose prime factors are 2 and 13.

Original entry on oeis.org

26, 52, 104, 208, 338, 416, 676, 832, 1352, 1664, 2704, 3328, 4394, 5408, 6656, 8788, 10816, 13312, 17576, 21632, 26624, 35152, 43264, 53248, 57122, 70304, 86528, 106496, 114244, 140608, 173056, 212992, 228488, 281216, 346112, 425984, 456976, 562432, 692224, 742586, 851968, 913952

Offset: 1

Bernard Schott, Jun 06 2017

nonn

Numbers k such that phi(k)/k = 6/13.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A033845, A033846, A033847, A033848, A033849, A033850, A033851, A086780, A107326, A143207.

[n:n in [1..100000] | Set(PrimeDivisors(n)) eq {2,13}];  // Marius A. Burtea, May 10 2019

Select[Range[920000],FactorInteger[#][[All,1]]=={2,13}&] (* Harvey P. Dale, Jun 18 2021 *)

is(n) = factor(n)[, 1]~==[2, 13] \\ Felix Fröhlich, Jun 06 2017

list(lim)=my(v=List(),t); for(n=1,logint(lim\2,13), t=13^n; while((t<<=1)<=lim, listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jun 11 2017

a(n) = 26 * A107326(n). - David A. Corneth, Jun 06 2017

Sum_{n>=1} 1/a(n) = 1/12. - Amiram Eldar, Dec 22 2020

A086798 Number of coefficients equal to zero in n-th cyclotomic polynomial.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 2, 7, 0, 4, 0, 4, 4, 0, 0, 6, 16, 0, 16, 6, 0, 2, 0, 15, 6, 0, 8, 10, 0, 0, 8, 12, 0, 4, 0, 10, 18, 0, 0, 14, 36, 16, 10, 12, 0, 16, 24, 18, 12, 0, 0, 10, 0, 0, 28, 31, 18, 6, 0, 16, 14, 8, 0, 22, 0, 0, 34, 18, 30, 8, 0, 28, 52, 0, 0, 16, 24, 0, 18, 30, 0

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 05 2003

nonn

See A051664.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from T. D. Noe)

Cf. A086765, A086780.

Cf. A051664 (number of nonzero terms in n-th cyclotomic polynomial).

Table[Count[CoefficientList[Cyclotomic[n, x], x], _?(#==0&)], {n, 0, 100}]

a(n)=sum(k=0,eulerphi(n),if(polcoeff(polcyclo(n),k),0,1))

A086798(n) = (1 + eulerphi(n) - length(select(x->x!=0, Vec(polcyclo(n))))); \\ Antti Karttunen, Sep 21 2018

From Benoit Cloitre, Aug 06 2003: (Start)
a(4n+2) = a(2n+1); a(4n) = a(2n) + phi(2n).
When p is an odd prime and m integer >= 1: a(p^m) = a(2*p^m) = p^m - p^(m-1) - p + 1. In particular a(p) = a(2p) = 0. (End)
a(n) = 1 + phi(n) - A051664(n). - T. D. Noe, Aug 08 2003

More terms from Benoit Cloitre and T. D. Noe, Aug 06 2003

A085900 Numbers k such that k-th cyclotomic polynomial has exactly 3 negative coefficients.

Original entry on oeis.org

14, 15, 28, 30, 45, 56, 60, 75, 90, 98, 112, 120, 135, 150, 180, 196, 224, 225, 240, 270, 300, 360, 375, 392, 405, 448, 450, 480, 540, 600, 675, 686, 720, 750, 784, 810, 896, 900, 960, 1080, 1125, 1200, 1215, 1350, 1372, 1440, 1500, 1568, 1620, 1792, 1800, 1875

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 16 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..96

Cf. A085459, A086780.

Select[Range@ 2000, Count[CoefficientList[Cyclotomic[#, x], x], ?(# < 0 &)] == 3 &] (* _Michael De Vlieger, Oct 26 2017 *)

More terms from Paolo P. Lava, Oct 26 2017

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A033846 Numbers whose prime factors are 2 and 5.

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A147571 Numbers with exactly 4 distinct prime divisors {2,3,5,7}.

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A051664 a(n) is the number of nonzero coefficients in the n-th cyclotomic polynomial.

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A288162 Numbers whose prime factors are 2 and 13.

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A086798 Number of coefficients equal to zero in n-th cyclotomic polynomial.

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A085900 Numbers k such that k-th cyclotomic polynomial has exactly 3 negative coefficients.

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