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
A358928
a(n) is the smallest centered triangular number with exactly n distinct prime factors.
Original entry on oeis.org
1, 4, 10, 460, 9010, 772210, 20120860, 1553569960, 85507715710, 14932196985010, 1033664429333260, 197628216951078460, 21266854897681220860, 7423007155473283614010, 3108276166302017120182510, 851452464506763307285599610, 32749388246772812069108696710
Offset: 0
a(4) = 9010, because 9010 is a centered triangular number with 4 distinct prime factors {2, 5, 17, 53} 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[PrimeNu[ck = c[k]] != n, k++]; ck]; Array[a, 9, 0] (* Amiram Eldar, Dec 09 2022 *)
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a(n) = for(k=0, oo, my(t=3*k*(k+1)/2 + 1); if(omega(t) == n, return(t))); \\ Daniel Suteu, Dec 10 2022
A359234
a(n) is the smallest centered square number with exactly n distinct prime factors.
Original entry on oeis.org
1, 5, 85, 1105, 99905, 2339285, 294346585, 29215971265, 4274253515545, 135890190846085, 14289540733429585, 10285257499051999685, 659442750659021626765, 386961420250791449193065, 10019680253112694448155885, 7190322949201929673798425205, 944550762877225960238953138865
Offset: 0
a(4) = 99905, because 99905 is a centered square number with 4 distinct prime factors {5, 13, 29, 53} and this is the smallest such number.
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a(n) = for(k=0, oo, my(t=2*k*k + 2*k + 1); if(omega(t) == n, return(t))); \\ Daniel Suteu, Dec 29 2022
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omega_centered_square_numbers(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), if(q%4==1, my(v=m*q, r=nextprime(q+1)); while(v <= B, if(j==1, if(v>=A, if (issquare((8*(v-1))/4 + 1) && ((sqrtint((8*(v-1))/4 + 1)-1)%2 == 0), 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) = if(n==0, return(1)); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_centered_square_numbers(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Dec 29 2022
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