A046316
Numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.
Original entry on oeis.org
27, 45, 63, 75, 99, 105, 117, 125, 147, 153, 165, 171, 175, 195, 207, 231, 245, 255, 261, 273, 275, 279, 285, 325, 333, 343, 345, 357, 363, 369, 385, 387, 399, 423, 425, 429, 435, 455, 465, 475, 477, 483, 507, 531, 539, 549, 555, 561, 575, 595, 603, 605
Offset: 1
A369979 sorted into ascending order.
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a046316 n = a046316_list !! (n-1)
a046316_list = filter ((== 3) . a001222) [1, 3 ..]
-- Reinhard Zumkeller, May 05 2015
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list(lim)=my(v=List(),pq); forprime(p=3,lim\9, forprime(q=3,min(lim\3\p,p), pq=p*q; forprime(r=3,lim\pq, listput(v, pq*r)))); Set(v) \\ Charles R Greathouse IV, Aug 23 2017
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from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A046316(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))-b+1 for a,k in enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024
A373848
Numbers k such that k is not divisible by p^p for any prime p, and for which 1 < A373842(k) <= k, where A373842 is the arithmetic derivative of the primorial base log-function.
Original entry on oeis.org
5, 9, 15, 25, 30, 42, 45, 63, 75, 105, 110, 125, 126, 147, 150, 165, 175, 198, 210, 225, 231, 245, 275, 294, 315, 330, 343, 363, 375, 385, 441, 462, 495, 525, 539, 605, 625, 650, 686, 693, 726, 735, 750, 770, 825, 847, 875, 882, 990, 1029, 1050, 1089, 1125, 1155, 1170, 1190, 1210, 1225, 1250, 1331, 1375, 1386, 1430
Offset: 1
Cf.
A001222,
A003415,
A024451,
A087112,
A276085,
A276086,
A359550,
A369979,
A370129,
A370138,
A373842,
A373844,
A373845.
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\\ Uses the code from A373842, or its precomputed data:
A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]A373848(n) = if(!A359550(n), 0, my(u=A373842(n)); ((1
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A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
\\ The following routine checks that n is not a prime larger than five, is in A048103, and in case n is odd, rules out cases that certainly cannot give A373842(n) <= n:
prefilter_for_A373848(n) = if(n < 3 || (isprime(n) && n > 5), 0, my(f=factor(n), k=#f~, lpf=f[1,1], p=f[k,1], m=f[k,2]); for(i=1, k, if(f[i, 2]>=f[i, 1], return(0))); if(2==lpf, return(1)); while(p>lpf, p = precprime(p-1); m *= p; if(m>n, return(0))); (1));
isA373848(n) = if(!prefilter_for_A373848(n), 0, my(x=A276085(n)); if(x>A002620(n), 0, (!isprime(x) && A003415(x)<=n)));
A370138
Arithmetic derivatives of the sums of three primorials > 1.
Original entry on oeis.org
5, 7, 9, 21, 19, 21, 41, 33, 61, 123, 109, 111, 191, 165, 211, 459, 213, 361, 705, 951, 1361, 1319, 3537, 1173, 2195, 2479, 1481, 2111, 3295, 3421, 2313, 5415, 5885, 5891, 11091, 15019, 16371, 35067, 15033, 25061, 33373, 15123, 26057, 31309, 42955, 16691, 48573, 36329, 45845, 62385, 31167, 72201, 62123, 80969, 141399, 151113
Offset: 1
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up_to = 15180;
A002110(n) = prod(i=1,n,prime(i));
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A370137list(up_to) = { my(v = vector(up_to), i=0); for(x=1,oo, for(y=1,x, for(z=1,y, i++; if(i > up_to, return(v)); v[i] = A002110(x)+A002110(y)+A002110(z)))); (v); };
v370137 = A370137list(up_to);
A370137(n) = v370137[n];
A370138(n) = A003415(A370137(n));
A370137
Sums of three primorials > 1.
Original entry on oeis.org
6, 10, 14, 18, 34, 38, 42, 62, 66, 90, 214, 218, 222, 242, 246, 270, 422, 426, 450, 630, 2314, 2318, 2322, 2342, 2346, 2370, 2522, 2526, 2550, 2730, 4622, 4626, 4650, 4830, 6930, 30034, 30038, 30042, 30062, 30066, 30090, 30242, 30246, 30270, 30450, 32342, 32346, 32370, 32550, 34650, 60062, 60066, 60090, 60270, 62370, 90090
Offset: 1
6 = A002110(1) + A002110(1) + A002110(1) = 2+2+2.
10 = A002110(2) + A002110(1) + A002110(1) = 6+2+2.
14 = A002110(2) + A002110(2) + A002110(1) = 6+6+2.
18 = A002110(2) + A002110(2) + A002110(2) = 6+6+6.
38 = A002110(3) + A002110(2) + A002110(1) = 30+6+2.
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nn = 6; MapIndexed[Set[P[First[#2] - 1], #1] &, FoldList[Times, 1, Prime@ Range[nn]]]; Table[P[i] + P[j] + P[k], {i, nn}, {j, i}, {k, j}] // Flatten (* Michael De Vlieger, Mar 09 2024 *)
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up_to = 15180;
A002110(n) = prod(i=1,n,prime(i));
A370137list(up_to) = { my(v = vector(up_to), i=0); for(x=1,oo, for(y=1,x, for(z=1,y, i++; if(i > up_to, return(v)); v[i] = A002110(x)+A002110(y)+A002110(z)))); (v); };
v370137 = A370137list(up_to);
A370137(n) = v370137[n];
A373844
Triangle read by rows: T(n,k) = A276086(1 + A002110(n) + A002110(k)), 1 <= k <= n, where A276086 is the primorial base exp-function.
Original entry on oeis.org
18, 30, 50, 42, 70, 98, 66, 110, 154, 242, 78, 130, 182, 286, 338, 102, 170, 238, 374, 442, 578, 114, 190, 266, 418, 494, 646, 722, 138, 230, 322, 506, 598, 782, 874, 1058, 174, 290, 406, 638, 754, 986, 1102, 1334, 1682, 186, 310, 434, 682, 806, 1054, 1178, 1426, 1798, 1922, 222, 370, 518, 814, 962, 1258, 1406, 1702, 2146, 2294, 2738
Offset: 1
Triangle begins as:
18,
30, 50,
42, 70, 98,
66, 110, 154, 242,
78, 130, 182, 286, 338,
102, 170, 238, 374, 442, 578,
114, 190, 266, 418, 494, 646, 722,
138, 230, 322, 506, 598, 782, 874, 1058,
174, 290, 406, 638, 754, 986, 1102, 1334, 1682,
186, 310, 434, 682, 806, 1054, 1178, 1426, 1798, 1922,
222, 370, 518, 814, 962, 1258, 1406, 1702, 2146, 2294, 2738,
etc.
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A002110(n) = prod(i=1,n,prime(i));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A373844(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2, x=A002110(1+n - binomial(c + 1, 2))); A276086(1+(A002110(1+c)+x)); };
Showing 1-5 of 5 results.
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