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
A358865
a(n) is the smallest n-gonal pyramidal number with exactly n prime factors (counted with multiplicity).
Original entry on oeis.org
20, 140, 405, 2856, 25296, 111720, 25984, 5474000, 237600, 223826688, 3852800, 268565760, 1834725376, 175861400000, 335674368, 2863363937280, 4383831556096, 206015846400, 3400704000, 938209120583680, 2981338216980480, 21463949229465600, 45410367307776, 72056803765911552
Offset: 3
a(3) = 20, because 20 is a tetrahedral (or triangular pyramidal) number with 3 prime factors (counted with multiplicity) {2, 2, 5} 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(bigomega(t) == n, return(t))); \\ Daniel Suteu, Dec 05 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
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
A359014
a(n) is the index of the smallest n-gonal number with exactly n prime factors (counted with multiplicity).
Original entry on oeis.org
7, 4, 11, 50, 60, 22, 315, 264, 1295, 256, 315, 4480, 4727, 2634, 25123, 8192, 15903, 18432, 314315, 368640, 1859975, 95326, 2068659, 3145728, 2181039, 1028412, 23612379, 83886080, 18512791, 72421650, 536870912, 251658240
Offset: 3
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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(k))); \\ Daniel Suteu, Dec 12 2022
A359854
a(n) is the least n-gonal number that is the product of n distinct primes, or 0 if there are none.
Original entry on oeis.org
6, 66, 0, 11310, 303810, 28962934, 557221665, 15529888374, 1219300152070, 23900058257790, 1231931106828345, 500402553453949510, 14990069451769732194, 610385355391371697410
Offset: 2
a(3) = 66 because 66 = 11*12/2 is the 11th triangular number and is the product of 3 distinct primes 2*3*11.
a(4) = 0 because a 4-gonal number is a square, and thus not the product of distinct primes.
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f:= proc(s) local n,p,F;
for n from 1 do
p:= (s-2)*n*(n-1)/2 + n;
F:= ifactors(p)[2];
if nops(F) = s and andmap(t -> t[2]=1, F) then return p fi
od
end proc:
f(2):= 0:
map(f, [$2..11]);
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f[s_] := f[s] = Module[{n, p, F}, For[n = 1, True, n++, p = (s - 2)*n*(n-1)/2 + n; F = FactorInteger[p]; If[Length[F] == s && AllTrue[F, #[[2]] == 1&], Return[ p]]]];
f[4] = 0;
Table[Print[n, " ", f[n]]; f[n], {n, 2, 11}] (* Jean-François Alcover, Jan 24 2023, after Maple program *)
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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); if(ceil(A/t) <= B\t, list=concat(list, f(t, q+1, j-1))))); list); vecsort(Vec(f(1, 2, n)));
a(n, k=n) = if(n < 2, return()); if(n==2, return(6)); if(n==4, return(0)); 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
Showing 1-7 of 7 results.
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