A358862
a(n) is the smallest n-gonal number with exactly n distinct prime factors.
Original entry on oeis.org
66, 44100, 11310, 103740, 3333330, 185040240, 15529888374, 626141842326, 21647593547580, 351877410344460, 82634328555218440, 2383985537862979050, 239213805711830629680
Offset: 3
a(3) = 66, because 66 is a triangular number with 3 distinct prime factors {2, 3, 11} and this is the smallest such number.
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Table[SelectFirst[PolygonalNumber[n,Range[400000]],PrimeNu[#]==n&],{n,3,10}] (* The program generates the first 8 terms of the sequence. *) (* Harvey P. Dale, Sep 09 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(omega(t) == n, return(t))); \\ Daniel Suteu, Dec 04 2022
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omega_polygonals(A, B, n, k) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), my(v=m*q, r=nextprime(q+1)); while(v <= B, if(j==1, if(v>=A && ispolygonal(v,k), listput(list, v)), if(v*r <= B, list=concat(list, f(v, r, j-1)))); v *= q)); list); vecsort(Vec(f(1, 2, n)));
a(n, k=n) = if(n < 3, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Dec 04 2022
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.
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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
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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
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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
A358864
a(n) is the smallest n-gonal pyramidal number with exactly n distinct prime factors.
Original entry on oeis.org
84, 1785, 299880, 1020510, 8897460, 102612510, 33367223274, 249417828660, 9177835927260, 10064864238489060, 5558913993302670, 15633689593760207970, 3792821921183752657200
Offset: 3
a(3) = 84, because 84 is a tetrahedral (or triangular pyramidal) number with 3 distinct prime factors {2, 3, 7} and this is the smallest such number.
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a(n) = if(n<3, return()); for(k=1, oo, my(t=(k*(k+1)*((n-2)*k + (5-n)))\6); if(omega(t) == n, return(t))); \\ Daniel Suteu, Dec 05 2022
A358894
a(n) is the smallest centered n-gonal number with exactly n distinct prime factors.
Original entry on oeis.org
460, 99905, 463326, 808208947, 23089262218, 12442607161209225, 53780356630, 700326051644920151, 46634399568693102, 45573558879962759570353
Offset: 3
a(3) = 460, because 460 is a centered triangular number with 3 distinct prime factors {2, 5, 23} and this is the smallest such number.
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a(n) = if(n<3, return()); for(k=1, oo, my(t=((n*k*(k+1))/2+1)); if(omega(t) == n, return(t))); \\ Daniel Suteu, Dec 05 2022
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.
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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 *)
A358926
a(n) is the smallest centered n-gonal number with exactly n prime factors (counted with multiplicity).
Original entry on oeis.org
316, 1625, 456, 3964051, 21568, 6561, 346528, 3588955448828761, 1684992, 210804461608463437, 36865024, 835904150390625, 2052407296
Offset: 3
a(3) = 316, because 316 is a centered triangular number with 3 prime factors (counted with multiplicity) {2, 2, 79} and this is the smallest such number.
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c[n_, k_] := n*k*(k + 1)/2 + 1; a[n_] := Module[{k = 1, ck}, While[PrimeOmega[ck = c[n, k]] != n, k++]; ck]; Array[a, 7, 3] (* Amiram Eldar, Dec 09 2022 *)
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a(n) = if(n<3, return()); for(k=1, oo, my(t=((n*k*(k+1))/2+1)); if(bigomega(t) == n, return(t))); \\ Daniel Suteu, Dec 09 2022
A359016
a(n) is the index of the smallest n-gonal pyramidal number with exactly n prime factors (counted with multiplicity).
Original entry on oeis.org
4, 7, 9, 16, 31, 48, 28, 160, 54, 512, 128, 512, 946, 4224, 512, 10240, 11566, 4095, 1024, 65535, 94794, 180224, 22796, 262143, 1048575, 7077888, 1339848
Offset: 3
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a(n) = if(n<3, return()); for(k=1, oo, my(t=(k*(k+1)*((n-2)*k + (5-n)))\6); if(bigomega(t) == n, return(k))); \\ Daniel Suteu, Dec 12 2022
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