Displaying 1-10 of 25 results found.
1, 2, 3, 1, 6, 2, 12, 4, 17, 7, 8, 34, 14, 16, 68, 28, 32, 87, 49, 56, 64, 174, 98, 112, 128, 348, 196, 224, 256, 353, 343, 392, 448, 512, 706, 686, 784, 896, 1024, 1412, 1372, 1568, 1792, 2048, 423, 2401, 2744, 3136, 3584, 4096, 846, 4802, 5488, 6272, 7168, 8192, 1692, 9604, 10976, 12544, 14336, 2961, 13423, 3384, 19208
EXAMPLE
For n = 5, the 5th number of the form 2^i*7^j with i, j >= 0 is 8, and the 6th number of the form 2^i*7^j with i, j >= 0 is 14, so the difference is 14 - 8 = 6.
PROG
(PARI) diff(v)=vector(#v-1, i, v[i+1]-v[i])
list(lim)=my(v=List(), N); for(n=0, log(lim)\log(7), N=7^n; while(N<=lim, listput(v, N); N<<=1)); diff(vecsort(Vec(v))) \\ Charles R Greathouse IV, Jun 28 2011
AUTHOR
Mick Purcell (mickpurcell(AT)gmail.com), Sep 25 2009
7-smooth numbers: positive numbers whose prime divisors are all <= 7.
(Formerly M0477 N0177)
+10
159
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 125, 126, 128, 135, 140, 144, 147, 150, 160, 162, 168, 175, 180, 189, 192
COMMENTS
Also called humble numbers; sometimes also called highly composite numbers, but this usually refers to A002182. Successive numbers k such that phi(210k) = 48k. - Artur Jasinski, Nov 05 2008 Numbers which are products of single-digit numbers. - N. J. A. Sloane, Jul 02 2017 Phi(a(n)) is 7-smooth. In fact, the Euler Phi function applied to p-smooth numbers, for any prime p, is p-smooth. - Richard Locke Peterson, May 09 2020 Also those integers k, such that, for every prime p > 5, p^(12k) - 1 == 0 (mod 5040k). - Federico Provvedi, Jun 06 2022 The nonprimes with this property are all terms except for 2, 3, 5 and 7, i.e.: (1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, ...); the composite terms are all but the first one of this subsequence. ["Trivial" data provided mainly for search purpose.] - M. F. Hasler, Jun 06 2023
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 52.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
Sum_{n>=1} 1/a(n) = Product_{primes p <= 7} p/(p-1) = (2*3*5*7)/(1*2*4*6) = 35/8. - Amiram Eldar, Sep 22 2020
MATHEMATICA
Select[Range[250], Max[Transpose[FactorInteger[ # ]][[1]]]<=7&]
aa = {}; Do[If[EulerPhi[210 n] == 48 n, AppendTo[aa, n]], {n, 1, 1200}]; aa (* Artur Jasinski, Nov 05 2008 *) mxExp = 8; Select[Union[Times @@@ Flatten[Table[Tuples[{2, 3, 5, 7}, n], {n, mxExp}], 1]], # <= 2^mxExp &] (* Harvey P. Dale, Aug 13 2012 *) mx = 200; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}, {k, 0, Log[5, mx/(2^i*3^j)]}, {l, 0, Log[7, mx/(2^i*3^j*5^k)]}] (* Robert G. Wilson v, Aug 17 2012 *)
PROG
(PARI) test(n)=m=n; forprime(p=2, 7, while(m%p==0, m=m/p)); return(m==1)
for(n=1, 200, if(test(n), print1(n", ")))
(PARI) list(lim)=my(v=List(), t); for(a=0, logint(lim\1, 7), for(b=0, logint(lim\7^a, 5), for(c=0, logint(lim\7^a\5^b, 3), t=3^c*5^b*7^a; while(t<=lim, listput(v, t); t<<=1)))); Set(v) \\ Charles R Greathouse IV, Feb 22 2017 (Haskell)
import Data.Set (singleton, deleteFindMin, fromList, union)
a002473 n = a002473_list !! (n-1)
a002473_list = f $ singleton 1 where
f s = x : f (s' `union` fromList (map (* x) [2, 3, 5, 7]))
where (x, s') = deleteFindMin s
(Magma) [n: n in [1..200] | PrimeDivisors(n) subset PrimesUpTo(7)]; // Bruno Berselli, Sep 24 2012 (Python)
import heapq
from itertools import islice
from sympy import primerange
def A002473gen(p=7): # generate all p-smooth terms
v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
while True:
v = heapq.heappop(h)
if v != oldv:
yield v
oldv = v
for p in psmooth_primes:
heapq.heappush(h, v*p)
(Python)
from sympy import integer_log
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x):
c = n+x
for i in range(integer_log(x, 7)[0]+1):
i7 = 7**i
m = x//i7
for j in range(integer_log(m, 5)[0]+1):
j5 = 5**j
r = m//j5
for k in range(integer_log(r, 3)[0]+1):
c -= (r//3**k).bit_length()
return c
EXTENSIONS
Additional comments from Michel Lecomte, Jun 09 2007
Numbers of the form 2^i*5^j with i, j >= 0.
+10
105
1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 128, 160, 200, 250, 256, 320, 400, 500, 512, 625, 640, 800, 1000, 1024, 1250, 1280, 1600, 2000, 2048, 2500, 2560, 3125, 3200, 4000, 4096, 5000, 5120, 6250, 6400, 8000, 8192, 10000, 10240, 12500, 12800
COMMENTS
These are the natural numbers whose reciprocals are terminating decimals. - David Wasserman, Feb 26 2002 Also numbers that are divisible by neither 10k - 7, 10k - 3, 10k - 1 nor 10k + 1, for all k > 0. - Robert G. Wilson v, Oct 26 2010 Since p = 2 and q = 5 are coprime, sum_{n >= 1} 1/a(n) = sum_{i >= 0} sum_{j >= 0} 1/p^i * 1/q^j = sum_{i >= 0} 1/p^i q/(q - 1) = p*q/((p-1)*(q-1)) = 2*5/(1*4) = 2.5. - Franklin T. Adams-Watters, Jul 07 2014 Conjecture: Each positive integer n not among 1, 4 and 12 can be written as a sum of finitely many numbers of the form 2^a*5^b + 1 (a,b >= 0) with no one dividing another. This has been verified for n <= 3700. - Zhi-Wei Sun, Apr 18 2023
REFERENCES
Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 73.
FORMULA
The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(10*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019
MAPLE
isA003592 := proc(n)
if n = 1 then
true;
else
return (numtheory[factorset](n) minus {2, 5} = {} );
end if;
end proc:
option remember;
if n = 1 then
1;
else
for a from procname(n-1)+1 do
if isA003592(a) then
return a;
end if;
end do:
end if;
MATHEMATICA
twoFiveableQ[n_] := PowerMod[10, n, n] == 0; Select[Range@ 10000, twoFiveableQ] (* Robert G. Wilson v, Jan 12 2012 *) twoFiveableQ[n_] := Union[ MemberQ[{1, 3, 7, 9}, # ] & /@ Union@ Mod[ Rest@ Divisors@ n, 10]] == {False}; twoFiveableQ[1] = True; Select[Range@ 10000, twoFiveableQ] (* Robert G. Wilson v, Oct 26 2010 *) maxExpo = 14; Sort@ Flatten@ Table[2^i * 5^j, {i, 0, maxExpo}, {j, 0, Log[5, 2^(maxExpo - i)]}] (* Or *)
PROG
(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim+.5)\log(5), N=5^n; while(N<=lim, listput(v, N); N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011 (Sage)
def isA003592(n) :
return not any(d != 2 and d != 5 for d in prime_divisors(n))
@CachedFunction
if n == 1 : return 1
while not isA003592(k) : k += 1
return k
(Magma) [n: n in [1..10000] | PrimeDivisors(n) subset [2, 5]]; // Bruno Berselli, Sep 24 2012 (Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a003592 n = a003592_list !! (n-1)
a003592_list = f $ singleton 1 where
f s = y : f (insert (2 * y) $ insert (5 * y) s')
where (y, s') = deleteFindMin s
(Python)
from heapq import heappush, heappop
pq = [1]
seen = set(pq)
while True:
value = heappop(pq)
yield value
seen.remove(value)
for x in 2*value, 5*value:
if x not in seen:
heappush(pq, x)
seen.add(x)
A003592_list = [next(sequence) for _ in range(100)]
(GAP) Filtered([1..10000], n->PowerMod(10, n, n)=0); # Muniru A Asiru, Mar 19 2019
Numbers of the form (11^i)*(13^j).
+10
20
1, 11, 13, 121, 143, 169, 1331, 1573, 1859, 2197, 14641, 17303, 20449, 24167, 28561, 161051, 190333, 224939, 265837, 314171, 371293, 1771561, 2093663, 2474329, 2924207, 3455881, 4084223, 4826809, 19487171, 23030293, 27217619
FORMULA
Sum_{n>=1} 1/a(n) = (11*13)/((11-1)*(13-1)) = 143/120. - Amiram Eldar, Sep 23 2020 a(n) ~ exp(sqrt(2*log(11)*log(13)*n)) / sqrt(143). - Vaclav Kotesovec, Sep 23 2020
MATHEMATICA
mx = 3*10^7; Sort@ Flatten@ Table[ 11^i*13^j, {i, 0, Log[11, mx]}, {j, 0, Log[13, mx/11^i]}] (* Robert G. Wilson v, Aug 17 2012 *) fQ[n_]:=PowerMod[143, n, n] == 0; Select[Range[2 10^7], fQ] (* Vincenzo Librandi, Jun 27 2016 *)
PROG
(Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a108090 n = a108090_list !! (n-1)
a108090_list = f $ singleton (1, 0, 0) where
f s = y : f (insert (11 * y, i + 1, j) $ insert (13 * y, i, j + 1) s')
where ((y, i, j), s') = deleteFindMin s
(Magma) [n: n in [1..10^7] | PrimeDivisors(n) subset [11, 13]]; // Vincenzo Librandi, Jun 27 2016 (PARI) list(lim)=my(v=List(), t); for(j=0, logint(lim\=1, 13), t=13^j; while(t<=lim, listput(v, t); t*=11)); Set(v) \\ Charles R Greathouse IV, Aug 29 2016
CROSSREFS
Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364, A107466, A108056.
AUTHOR
Douglas Winston (douglas.winston(AT)srupc.com), Jun 03 2005
Numbers of the form 3^i*7^j with i, j >= 0.
+10
19
1, 3, 7, 9, 21, 27, 49, 63, 81, 147, 189, 243, 343, 441, 567, 729, 1029, 1323, 1701, 2187, 2401, 3087, 3969, 5103, 6561, 7203, 9261, 11907, 15309, 16807, 19683, 21609, 27783, 35721, 45927, 50421, 59049, 64827, 83349, 107163, 117649
FORMULA
The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(21*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019 Sum_{n>=1} 1/a(n) = (3*7)/((3-1)*(7-1)) = 7/4. - Amiram Eldar, Sep 22 2020
MATHEMATICA
f[upto_]:=Sort[Select[Flatten[3^First[#] 7^Last[#] & /@ Tuples[{Range[0, Floor[Log[3, upto]]], Range[0, Floor[Log[7, upto]]]}]], # <= upto &]]; f[120000] (* Harvey P. Dale, Mar 04 2011 *) fQ[n_] := PowerMod[21, n, n] == 0; Select[Range[120000], fQ] (* Bruno Berselli, Sep 24 2012 *)
PROG
(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim)\log(7), N=7^n; while(N<=lim, listput(v, N); N*=3)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011 (Magma) [n: n in [1..120000] | PrimeDivisors(n) subset [3, 7]]; // Bruno Berselli, Sep 24 2012 (Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a003594 n = a003594_list !! (n-1)
a003594_list = f $ singleton 1 where
f s = y : f (insert (3 * y) $ insert (7 * y) s')
where (y, s') = deleteFindMin s
(GAP) Filtered([1..120000], n->PowerMod(21, n, n)=0); # Muniru A Asiru, Mar 19 2019 (Python)
from sympy import integer_log
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return n+x-sum(integer_log(x//7**i, 3)[0]+1 for i in range(integer_log(x, 7)[0]+1))
Periodic sequence {0,1,1,1,1,1,1} or n^6 mod 7.
+10
17
0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1
COMMENTS
This sequence also represents n^12 mod 7; n^18 mod 7; (exponents are = 0 mod 6).
Characteristic sequence for numbers n>=1 to be relatively prime to 7. - Wolfdieter Lang, Oct 29 2008 a(n+4), n>=0, (periodic 1,1,1,0,1,1,1) is also the characteristic sequence for mod m reduced positive odd numbers (i.e., gcd(2*n+1,m)=1, n>=0) for each modulus m from 7* A003591 = [7,14,28,49,56,98,112,196,...]. [ Wolfdieter Lang, Feb 04 2012]
FORMULA
a(n) = 0 if n=0 mod 7; a(n)= 1 else.
G.f. = (x+x^2+x^3+x^4+x^5+x^6)/(1-x^7)= -x*(1+x)*(1+x+x^2)*(x^2-x+1) / ( (x-1)*(1+x+x^2+x^3+x^4+x^5+x^6) ).
Multiplicative with a(p) = (if p=7 then 0 else 1), p prime. - Reinhard Zumkeller, Nov 30 2009 Dirichlet g.f. (1-7^(-s))*zeta(s). - R. J. Mathar, Mar 06 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013
MATHEMATICA
PadRight[{}, 120, {0, 1, 1, 1, 1, 1, 1}] (* Harvey P. Dale, Jul 09 2018 *)
PROG
(Sage) [power_mod(n, 6, 7)for n in range(0, 105)] # Zerinvary Lajos, Nov 06 2009
AUTHOR
Bruce Corrigan (scentman(AT)myfamily.com), Aug 09 2005
Numbers of the form (2^i)*(13^j).
+10
13
1, 2, 4, 8, 13, 16, 26, 32, 52, 64, 104, 128, 169, 208, 256, 338, 416, 512, 676, 832, 1024, 1352, 1664, 2048, 2197, 2704, 3328, 4096, 4394, 5408, 6656, 8192, 8788, 10816, 13312, 16384, 17576, 21632, 26624, 28561, 32768, 35152, 43264, 53248, 57122
FORMULA
Sum_{n>=1} 1/a(n) = (2*13)/((2-1)*(13-1)) = 13/6. - Amiram Eldar, Sep 23 2020 a(n) ~ exp(sqrt(2*log(2)*log(13)*n)) / sqrt(26). - Vaclav Kotesovec, Sep 23 2020
MATHEMATICA
fQ[n_] := PowerMod[26, n, n]==0; Select[Range[60000], fQ] (* Vincenzo Librandi, Feb 04 2012 *) mx = 60000; Sort@ Flatten@ Table[2^i*13^j, {i, 0, Log[2, mx]}, {j, 0, Log[13, mx/2^i]}] (* Robert G. Wilson v, Aug 17 2012 *)
PROG
(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim)\log(13), N=13^n; while(N<=lim, listput(v, N); N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011
AUTHOR
Douglas Winston (douglas.winston(AT)srupc.com), May 21 2005
Numbers of the form (3^i)*(13^j).
+10
12
1, 3, 9, 13, 27, 39, 81, 117, 169, 243, 351, 507, 729, 1053, 1521, 2187, 2197, 3159, 4563, 6561, 6591, 9477, 13689, 19683, 19773, 28431, 28561, 41067, 59049, 59319, 85293, 85683, 123201, 177147, 177957, 255879, 257049, 369603, 371293, 531441
FORMULA
Sum_{n>=1} 1/a(n) = (3*13)/((3-1)*(13-1)) = 13/8. - Amiram Eldar, Sep 23 2020 a(n) ~ exp(sqrt(2*log(3)*log(13)*n)) / sqrt(39). - Vaclav Kotesovec, Sep 23 2020
MATHEMATICA
mx = 540000; Sort@ Flatten@ Table[3^i*13^j, {i, 0, Log[3, mx]}, {j, 0, Log[13, mx/3^i]}] (* Robert G. Wilson v, Aug 17 2012 *) fQ[n_]:=PowerMod[39, n, n] == 0; Select[Range[2 10^7], fQ] (* Vincenzo Librandi, Jun 27 2016 *)
PROG
(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim)\log(13), N=13^n; while(N<=lim, listput(v, N); N*=3)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011 (Magma) [n: n in [1..10^7] | PrimeDivisors(n) subset [3, 13]]; // Vincenzo Librandi, Jun 27 2016
CROSSREFS
Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326.
AUTHOR
Douglas Winston (douglas.winston(AT)srupc.com), May 23 2005
Numbers of the form (5^i)*(13^j).
+10
10
1, 5, 13, 25, 65, 125, 169, 325, 625, 845, 1625, 2197, 3125, 4225, 8125, 10985, 15625, 21125, 28561, 40625, 54925, 78125, 105625, 142805, 203125, 274625, 371293, 390625, 528125, 714025, 1015625, 1373125, 1856465, 1953125, 2640625
FORMULA
Sum_{n>=1} 1/a(n) = (5*13)/((5-1)*(13-1)) = 65/48. - Amiram Eldar, Sep 23 2020 a(n) ~ exp(sqrt(2*log(5)*log(13)*n)) / sqrt(65). - Vaclav Kotesovec, Sep 23 2020
MATHEMATICA
mx = 2700000; Sort@ Flatten@ Table[5^i*13^j, {i, 0, Log[5, mx]}, {j, 0, Log[13, mx/5^i]}] (* Robert G. Wilson v, Aug 17 2012 *)
PROG
(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim)\log(13), N=13^n; while(N<=lim, listput(v, N); N*=5)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011
CROSSREFS
Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364.
AUTHOR
Douglas Winston (douglas.winston(AT)srupc.com), May 27 2005
Numbers of the form (7^i)*(13^j).
+10
9
1, 7, 13, 49, 91, 169, 343, 637, 1183, 2197, 2401, 4459, 8281, 15379, 16807, 28561, 31213, 57967, 107653, 117649, 199927, 218491, 371293, 405769, 753571, 823543, 1399489, 1529437, 2599051, 2840383, 4826809, 5274997, 5764801, 9796423, 10706059, 18193357, 19882681
FORMULA
Sum_{n>=1} 1/a(n) = (7*13)/((7-1)*(13-1)) = 91/72. - Amiram Eldar, Sep 23 2020 a(n) ~ exp(sqrt(2*log(7)*log(13)*n)) / sqrt(91). - Vaclav Kotesovec, Sep 23 2020
MATHEMATICA
n = 10^7; Flatten[Table[7^i*13^j, {i, 0, Log[7, n]}, {j, 0, Log[13, n/7^i]}]] // Sort (* Amiram Eldar, Sep 23 2020 *)
PROG
(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim)\log(13), N=13^n; while(N<=lim, listput(v, N); N*=7)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011 (Python)
from sympy import integer_log
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return n+x-sum(integer_log(x//13**i, 7)[0]+1 for i in range(integer_log(x, 13)[0]+1))
CROSSREFS
Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364, A107466.
AUTHOR
Douglas Winston (douglas.winston(AT)srupc.com), Jun 02 2005
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