A076550
Indices of triangular numbers listed in A075088.
Original entry on oeis.org
1, 2, 3, 7, 8, 15, 32, 64, 63, 224, 255, 1280, 512, 3968, 6399, 10240, 4095, 14336, 32768, 65535, 229375, 180224, 483327, 262143, 2097151, 1048575, 14680064, 17432576, 33554432, 67108864, 16777215
Offset: 0
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f[n_] := Plus @@ Last /@ FactorInteger@n; t = Table[0, {100}]; k = 1; While[k < 10^9/4, a = f[k] + f[k + 1] -1; If[ t[[a + 1]] == 0, t[[a + 1]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Jun 28 2010 *)
A358863
a(n) is the smallest n-gonal number with exactly n prime factors (counted with multiplicity).
Original entry on oeis.org
4, 28, 16, 176, 4950, 8910, 1408, 346500, 277992, 7542080, 326656, 544320, 120400000, 145213440, 48549888, 4733575168, 536813568, 2149576704, 3057500160, 938539560960, 1358951178240, 36324805836800, 99956555776, 49212503949312, 118747221196800, 59461613912064, 13749193801728
Offset: 2
a(3) = 28, because 28 is a triangular number with 3 prime factors (counted with multiplicity) {2, 2, 7} and this is the smallest such number.
-
g:= proc(s) local n, p, F;
for n from 1 to 10^7 do
p:= (s-2)*n*(n-1)/2 + n;
if numtheory:-bigomega(p) = s then return p fi;
od
end proc:
map(g, [$2..30]); # Robert Israel, Jan 15 2023
-
sng[n_]:=Module[{k=1},While[PrimeOmega[PolygonalNumber[n,k]]!=n,k++];PolygonalNumber[ n,k]]; Array[sng,21,2] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Feb 19 2023 *)
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a(n) = if(n<3, return()); for(k=1, oo, my(t=(k*(n*k - n - 2*k + 4))\2); if(bigomega(t) == n, return(t))); \\ Daniel Suteu, Dec 04 2022
-
bigomega_polygonals(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p,ceil(A/m)), B\m, my(t=m*q); if(ispolygonal(t,k), listput(list, t))), forprime(q = p, sqrtnint(B\m, n), my(t=m*q); if(ceil(A/t) <= B\t, list=concat(list, f(t, q, n-1))))); list); vecsort(Vec(f(1, 2, n)));
a(n, k=n) = if(k < 3, return()); my(x=2^n, y=2*x); while(1, my(v=bigomega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Dec 04 2022
A127637
Smallest squarefree triangular number with exactly n prime factors.
Original entry on oeis.org
1, 3, 6, 66, 210, 3570, 207690, 930930, 56812470, 1803571770, 32395433070, 265257422430, 91348974206490, 24630635909489610, 438603767516904990, 14193386885746698630, 2378522762792139793830, 351206814022419685159830, 28791787439593010836313310
Offset: 0
a(12) = 91348974206490 = 2*3*5*7*11*13*17*19*29*37*67*131 = A000217(13516580).
-
squarefree_omega_polygonals(A, B, n, k) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); my(s=sqrtnint(B\m, j)); if(j==1, forprime(q=max(p, ceil(A/m)), s, if(ispolygonal(m*q, k), listput(list, m*q))), forprime(q=p, s, my(t=m*q); list=concat(list, f(t, q+1, j-1)))); list); vecsort(Vec(f(1, 2, n)));
a(n, k=3) = if(n==0, return(1)); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=squarefree_omega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Jan 18 2023
A358927
a(n) is the smallest tetrahedral number with exactly n prime factors (counted with multiplicity), or -1 if no such number exists.
Original entry on oeis.org
1, -1, 4, 20, 56, 120, 560, 4960, 19600, 41664, 341376, 695520, 7207200, 22238720, 178433024, 1429559296, 179481600, 11453245440, 11444858880, 393079864320, 3928874471424, 5864598896640, 46910348656640, 975649558118400, 3002365391929344, 7805131503206400
Offset: 0
a(4) = 56, because 56 is a tetrahedral number with 4 prime factors (counted with multiplicity) {2, 2, 2, 7} and this is the smallest such number.
-
t[k_] := k*(k + 1)*(k + 2)/6; a[n_] := Module[{k = 1, tk}, While[PrimeOmega[tk = t[k]] != n, k++]; tk]; a[1] = -1; Array[a, 26, 0] (* Amiram Eldar, Dec 09 2022 *)
A358929
a(n) is the smallest centered triangular number with exactly n prime factors (counted with multiplicity).
Original entry on oeis.org
1, 19, 4, 316, 136, 760, 64, 4960, 22144, 103360, 27136, 5492224, 1186816, 41414656, 271212544, 559980544, 1334788096, 12943360, 7032930304, 527049293824, 158186536960, 1096295120896, 7871801589760, 154690378792960, 13071965224960, 56262393856, 964655941943296
Offset: 0
a(4) = 136, because 136 is a centered triangular number with 4 prime factors (counted with multiplicity) {2, 2, 2, 17} and this is the smallest such number.
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c[k_] := (3*k^2 + 3*k + 2)/2; a[n_] := Module[{k = 0, ck}, While[PrimeOmega[ck = c[k]] != n, k++]; ck]; Array[a, 18, 0] (* Amiram Eldar, Dec 09 2022 *)
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a(n) = if(n==0, return(1)); for(k=1, oo, my(t=3*k*(k+1)/2 + 1); if(bigomega(t) == n, return(t))); \\ Daniel Suteu, Dec 10 2022
A101744
Triangular numbers which are 10-almost primes.
Original entry on oeis.org
32640, 73920, 130816, 165600, 204480, 265356, 294528, 401856, 592416, 839160, 947376, 990528, 1279200, 1445850, 1492128, 1606528, 1842240, 1844160, 2031120, 2049300, 2821500, 2956096, 3571128, 3963520, 4148640, 4250070, 4335040
Offset: 1
a(1) = 32640 because that is the smallest triangular number which is also a 10-almost prime; specifically T(255) = 255*(255+1)/2 = 32640 = 2^7 * 3 * 5 * 17.
- Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 59, 1987.
- Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 33-38, 1996.
- Dudeney, H. E. Amusements in Mathematics. New York: Dover, pp. 67 and 167, 1970.
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BigOmega[n_Integer]:=Plus@@Last[Transpose[FactorInteger[n]]]; Select[Table[n*(n+1)/2, {n, 2, 5000}], BigOmega[ # ]==10&] (* Ray Chandler, Dec 14 2004 *)
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list(lim)=my(v=List(),cur,last=3,n=256,t); while((t=n*(n-1)/2) <= lim, cur=bigomega(n); if(cur+old==11, listput(v,t)); old=cur; n++); Vec(v) \\ Charles R Greathouse IV, Feb 05 2017
A209049
Smallest pentagonal number with n prime factors (counted with multiplicity).
Original entry on oeis.org
1, 5, 22, 12, 210, 176, 1520, 1080, 8400, 55200, 273280, 43776, 2624832, 7173360, 71660160, 100659200, 34255872, 178962432, 3623854080, 17895751680, 5413478400, 43690752000, 927712542720, 733008101376, 1789570252800, 35917382287360, 50649120571392
Offset: 0
-
k = 1; t = Table[0, {50}]; While[k < 500000001, a = PrimeOmega[k] + PrimeOmega[3 k - 1] - 1; If[ t[[a + 1]] == 0, t[[a + 1]] = k; Print[{k, a}]]; k++]; # (3 # - 1)/2 & /@ t
Join[{1},Table[SelectFirst[{#,PrimeOmega[#]}&/@PolygonalNumber[5,Range[ 110000]],#[[2]]==n&],{n,20}][[All,1]]] (* The program generates the first 21 terms of the sequence. *) (* Harvey P. Dale, Dec 27 2022 *)
A375657
The smallest triangular number that begins a run of at least n consecutive triangular numbers with the same number of prime factors (counted with multiplicity).
Original entry on oeis.org
1, 6, 6, 6, 724206, 32365035, 32365035, 9127288495, 497232340606, 54524401634046, 192541553136345, 3119282531272578, 1584264619108935753, 34399764958387086910, 34399764958387086910
Offset: 1
a(4) = 6 because 6 is the smallest triangular number that begins a run of 4 consecutive triangular numbers (6, 10, 15, 21) with the same number of prime factors (counted with multiplicity), i.e. 2.
a(5) = 724206 because 724206 is the smallest triangular number that begins a run of 5 consecutive triangular numbers (724206, 725410, 726615, 727821, 729028) with the same number of prime factors (counted with multiplicity), i.e. 5.
A363847
Numbers k such that Omega(m*(m+1)) < Omega(k*(k+1)) for all m < k, where Omega(k) is the number of prime divisors of k counted with multiplicity (A001222).
Original entry on oeis.org
1, 2, 3, 7, 8, 15, 32, 63, 224, 255, 512, 3968, 4095, 14336, 32768, 65535, 180224, 262143, 1048575, 14680064, 16777215, 134217728, 268435455, 1073741823, 8589934592, 12884901887, 34359738368, 68719476735, 1099511627775, 4398046511103, 17592186044415, 35184372088832
Offset: 1
-
seq[kmax_] := Module[{o1 = 0, o2, om = 0, s = {}}, Do[o2 = PrimeOmega[k]; o = o1 + o2; If[o > om, om = o; AppendTo[s, k - 1]]; o1 = o2, {k, 2, kmax}]; s]; seq[10^5]
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lista(kmax) = {my(o1 = 0, o2, om = 0); for(k = 2, kmax, o2 = bigomega(k); o = o1 + o2; if(o > om, om = o; print1(k-1, ", ")); o1 = o2); }
Showing 1-9 of 9 results.
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