A147580
Numbers with exactly 7 distinct odd prime divisors {3,5,7,11,13,17,19}.
Original entry on oeis.org
4849845, 14549535, 24249225, 33948915, 43648605, 53348295, 63047985, 72747675, 82447365, 92147055, 101846745, 121246125, 130945815, 160044885, 169744575, 189143955, 218243025, 237642405, 247342095, 266741475, 276441165, 305540235, 315239925, 363738375, 373438065
Offset: 1
-
a = {}; Do[If[EulerPhi[4849845 x] == 1658880 x, AppendTo[a, 4849845 x]], {x, 1, 100}]; a
A147575
Numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19}.
Original entry on oeis.org
9699690, 19399380, 29099070, 38798760, 48498450, 58198140, 67897830, 77597520, 87297210, 96996900, 106696590, 116396280, 126095970, 135795660, 145495350, 155195040, 164894730, 174594420, 184294110, 193993800, 203693490, 213393180, 232792560, 242492250, 252191940
Offset: 1
-
a = {}; Do[If[EulerPhi[9699690 x] == 1658880 x, AppendTo[a, 9699690 x]], {x, 1, 100}]; a
A307534
Heinz numbers of strict integer partitions with 3 parts, all of which are odd.
Original entry on oeis.org
110, 170, 230, 310, 374, 410, 470, 506, 590, 670, 682, 730, 782, 830, 902, 935, 970, 1030, 1034, 1054, 1090, 1265, 1270, 1298, 1370, 1394, 1426, 1474, 1490, 1570, 1598, 1606, 1670, 1705, 1790, 1826, 1886, 1910, 1955, 1970, 2006, 2110, 2134, 2162, 2255, 2266
Offset: 1
The sequence of terms together with their prime indices begins:
110: {1,3,5}
170: {1,3,7}
230: {1,3,9}
310: {1,3,11}
374: {1,5,7}
410: {1,3,13}
470: {1,3,15}
506: {1,5,9}
590: {1,3,17}
670: {1,3,19}
682: {1,5,11}
730: {1,3,21}
782: {1,7,9}
830: {1,3,23}
902: {1,5,13}
935: {3,5,7}
970: {1,3,25}
1030: {1,3,27}
1034: {1,5,15}
1054: {1,7,11}
Cf.
A001221,
A001222,
A001399,
A005117,
A007304,
A014612,
A037144,
A051037,
A056239,
A080193,
A080257,
A112798,
A143207,
A304636.
-
Select[Range[1000],SquareFreeQ[#]&&PrimeNu[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]
-
from math import isqrt
from sympy import primepi, primerange, integer_nthroot, nextprime
def A307534(n):
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 int(n+x-sum((primepi(x//(k*m))+1>>1)-(b+1>>1) for a,k in filter(lambda x:x[0]&1,enumerate(primerange(2,integer_nthroot(x,3)[0]+1),1)) for b,m in filter(lambda x:x[0]&1,enumerate(primerange(nextprime(k)+1,isqrt(x//k)+1),a+2))))
return bisection(f,n,n) # Chai Wah Wu, Oct 20 2024
A147572
Numbers with exactly 5 distinct prime divisors {2,3,5,7,11}.
Original entry on oeis.org
2310, 4620, 6930, 9240, 11550, 13860, 16170, 18480, 20790, 23100, 25410, 27720, 32340, 34650, 36960, 41580, 46200, 48510, 50820, 55440, 57750, 62370, 64680, 69300, 73920, 76230, 80850, 83160, 92400, 97020, 101640, 103950, 110880, 113190, 115500, 124740, 127050
Offset: 1
-
a = {}; Do[If[EulerPhi[x]/x == 16/77, AppendTo[a, x]], {x, 1, 100000}]; a
Select[Range[130000],FactorInteger[#][[All,1]]=={2,3,5,7,11}&] (* Harvey P. Dale, Oct 04 2020 *)
-
from sympy import integer_log, prevprime
def A147572(n):
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 g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
def f(x): return n+x-g(x,11)
return 2310*bisection(f,n,n) # Chai Wah Wu, Sep 16 2024
A147577
Numbers with exactly 4 distinct odd prime divisors {3,5,7,11}.
Original entry on oeis.org
1155, 3465, 5775, 8085, 10395, 12705, 17325, 24255, 28875, 31185, 38115, 40425, 51975, 56595, 63525, 72765, 86625, 88935, 93555, 114345, 121275, 139755, 144375, 155925, 169785, 190575, 202125, 218295, 259875, 266805, 280665, 282975, 317625
Offset: 1
-
a = {}; Do[If[EulerPhi[x]/x == 32/77, AppendTo[a, x]], {x, 1, 1000000}]; a
Select[Range[350000],EulerPhi[#]/#==32/77&] (* Harvey P. Dale, Mar 25 2016 *)
-
from sympy import integer_log
def A147577(n):
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 i11 in range(integer_log(x,11)[0]+1):
for i7 in range(integer_log(x11:=x//11**i11,7)[0]+1):
for i5 in range(integer_log(x7:=x11//7**i7,5)[0]+1):
c -= integer_log(x7//5**i5,3)[0]+1
return c
return 1155*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024
A147578
Numbers with exactly 5 distinct odd prime divisors {3,5,7,11,13}.
Original entry on oeis.org
15015, 45045, 75075, 105105, 135135, 165165, 195195, 225225, 315315, 375375, 405405, 495495, 525525, 585585, 675675, 735735, 825825, 945945, 975975, 1126125, 1156155, 1216215, 1366365, 1486485, 1576575, 1756755, 1816815, 1876875, 2027025, 2147145, 2207205, 2477475
Offset: 1
-
a = {}; Do[If[EulerPhi[x]/x == 384/1001, AppendTo[a, x]], {x, 1, 1000000}]; a
A147579
Numbers with exactly 6 distinct odd prime divisors {3,5,7,11,13,17}.
Original entry on oeis.org
255255, 765765, 1276275, 1786785, 2297295, 2807805, 3318315, 3828825, 4339335, 5360355, 6381375, 6891885, 8423415, 8933925, 9954945, 11486475, 12507495, 13018005, 14039025, 16081065, 16591575, 19144125, 19654635, 20675655, 21696675, 23228205, 25270245, 26801775
Offset: 1
-
a = {}; Do[If[EulerPhi[255255 x] == 92160 x, AppendTo[a, 255255 x]], {x, 1, 100}]; a
A147581
Numbers with exactly 8 distinct odd prime divisors {3,5,7,11,13,17,19,23}.
Original entry on oeis.org
111546435, 334639305, 557732175, 780825045, 1003917915, 1227010785, 1450103655, 1673196525, 1896289395, 2119382265, 2342475135, 2565568005, 2788660875, 3011753745, 3681032355, 3904125225, 4350310965, 5019589575, 5465775315, 5688868185, 6135053925, 6358146795
Offset: 1
-
a = {}; Do[If[EulerPhi[111546435 x] == 36495360 x, AppendTo[a, 111546435 x]], {x, 1, 100}]; a
-
from sympy import integer_log
def A147581(n):
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 i23 in range(integer_log(x,23)[0]+1):
for i19 in range(integer_log(x23:=x//23**i23,19)[0]+1):
for i17 in range(integer_log(x19:=x23//19**i19,17)[0]+1):
for i13 in range(integer_log(x17:=x19//17**i17,13)[0]+1):
for i11 in range(integer_log(x13:=x17//13**i13,11)[0]+1):
for i7 in range(integer_log(x11:=x13//11**i11,7)[0]+1):
for i5 in range(integer_log(x7:=x11//7**i7,5)[0]+1):
c -= integer_log(x7//5**i5,3)[0]+1
return c
return 111546435*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024
A147573
Numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13}.
Original entry on oeis.org
30030, 60060, 90090, 120120, 150150, 180180, 210210, 240240, 270270, 300300, 330330, 360360, 390390, 420420, 450450, 480480, 540540, 600600, 630630, 660660, 720720, 750750, 780780, 810810, 840840, 900900, 960960, 990990, 1051050, 1081080, 1171170, 1201200, 1261260
Offset: 1
-
a = {}; Do[If[EulerPhi[x]/x == 192/1001, AppendTo[a, x]], {x, 1, 100000}]; a
-
is(n)=if(n%30030, return(0)); my(g=30030); while(g>1, n/=g; g=gcd(n,30030)); n==1 \\ Charles R Greathouse IV, Sep 14 2015
A147574
Numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17}.
Original entry on oeis.org
510510, 1021020, 1531530, 2042040, 2552550, 3063060, 3573570, 4084080, 4594590, 5105100, 5615610, 6126120, 6636630, 7147140, 7657650, 8168160, 8678670, 9189180, 10210200, 10720710, 11231220, 12252240, 12762750, 13273260, 13783770
Offset: 1
-
a = {}; Do[If[EulerPhi[x 510510] == 92160 x, AppendTo[a, 510510 x]], {x, 1, 100}]; a
sdpdQ[n_]:=Module[{f=FactorInteger[n][[All,1]]},Length[f]==7&&Max[f]==17]; Select[Range[510510,138*10^5,510510],sdpdQ] (* Harvey P. Dale, Aug 03 2019 *)
Comments