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
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 *)
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
A321852
a(n) is the smallest m for which binomial(m, 6) has exactly n distinct prime factors.
Original entry on oeis.org
6, 7, 8, 9, 10, 18, 26, 40, 77, 120, 210, 477, 715, 2227, 3290, 9065, 17020, 49915, 139195, 240465, 721929, 1124840, 4455445, 16319578, 26683220, 105655905, 134879176, 677868170, 3290262264
Offset: 0
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a[n_] := Module[{m=6, t=1}, While[PrimeNu[t] != n, m++; t*=m/(m-6)]; m]; Array[a, 20] (* Amiram Eldar, Nov 27 2018 *)
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a(n)={my(m=6, t=1); while(omega(t)<>n, m++; t*=m/(m-6)); m} \\ Andrew Howroyd, Nov 26 2018
A359089
a(n) is the index of the smallest tetrahedral number with exactly n distinct prime factors.
Original entry on oeis.org
1, 2, 3, 7, 18, 34, 90, 259, 988, 2583, 5795, 37960, 101268, 424268, 3344614, 17168723, 74282570, 351200178, 1082950218, 5313193819, 31439710664, 317760710839, 1782400663483
Offset: 0
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r = -1; Reap[Do[If[# > r, r = #; Sow[n]] &[PrimeNu[n (n + 1) (n + 2)/6]], {n, 2^19}] ][[-1, -1]] (* Michael De Vlieger, Dec 19 2022 *)
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t(n) = n*(n+1)*(n+2)/6; \\ A000292
a(n) = my(k=1); while (omega(t(k)) != n, k++); k; \\ Michel Marcus, Dec 19 2022
A359229
a(n) is the smallest square pyramidal number with exactly n distinct prime factors.
Original entry on oeis.org
1, 5, 14, 30, 1785, 6930, 149226, 3573570, 139223370, 3708968340, 62366724420, 2279301054030, 1348519628145690, 27928822496705130, 1558931949520935990, 430616881400429491950, 161887663616926971163440
Offset: 0
a(4) = 1785, because 1785 is a square pyramidal number with 4 distinct prime factors {3, 5, 7, 17} and this is the smallest such number.
Showing 1-6 of 6 results.
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