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
A359235
a(n) is the smallest centered square number with exactly n prime factors (counted with multiplicity).
Original entry on oeis.org
1, 5, 25, 925, 1625, 47125, 2115625, 4330625, 83760625, 1049140625, 6098828125, 224991015625, 3735483578125, 329495166015625, 8193863401953125, 7604781494140625, 216431299462890625, 148146624615478515625, 25926420587158203125, 11071085186929931640625
Offset: 0
a(4) = 1625, because 1625 is a centered square number with 4 prime factors (counted with multiplicity) {5, 5, 5, 13} and this is the smallest such number.
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cs:= n -> 2*n*(n+1)+1:
V:= Vector(12): count:= 0:
for n from 1 while count < 12 do
v:= cs(n);
w:= numtheory:-bigomega(v);
if V[w] = 0 then V[w]:= v; count:= count+1 fi
od:
convert(V,list); # Robert Israel, Dec 22 2022
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bigomega_centered_square_numbers(A, B, n) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, if(q%4==1, my(t=m*q); if(issquare(2*t-1), listput(list, t)))), forprime(q=p, sqrtnint(B\m, n), if(q%4==1, 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) = if(n==0, return(1)); my(x=2^n, y=2*x); while(1, my(v=bigomega_centered_square_numbers(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Dec 29 2022
Showing 1-3 of 3 results.
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