

A324974


Rank of the nth special polygonal number A324973(n).


3



3, 3, 3, 5, 3, 3, 6, 3, 6, 3, 11, 5, 3, 3, 8, 10, 5, 6, 12, 3, 15, 9, 3, 5, 3, 8, 3, 8, 19, 14, 5, 7, 3, 6, 6, 36, 21, 66, 22, 3, 10, 5, 6, 3, 3, 50, 10, 20, 5, 14, 11, 51, 3, 10, 21, 6, 13, 5, 16, 25, 3, 3, 6, 6, 12, 14, 10, 68, 5, 28, 3, 11, 29, 3, 56, 6, 19
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OFFSET

1,1


COMMENTS

While two polygonal numbers of different ranks can be equal (e.g., P(6,n) = P(3,2n1)), that cannot occur for special polygonal numbers, since for fixed p the value of P(r,p) is strictly increasing with r. Thus the rank of a special polygonal number is welldefined.
The Carmichael numbers A002997 and primary Carmichael numbers A324316 are special polygonal numbers (see Kellner and Sondow 2019). Their ranks form the subsequences A324975 and A324976.


LINKS



FORMULA

a(n) = 2 + 2*((m/p)1)/(p1), where m = A324973(n) and p is its greatest prime factor. (Proof: solve m = P(r,p) = (p^2*(r2)  p*(r4))/2 for r.)


EXAMPLE

If m = A324973(4) = 70 = 2*5*7, then p = 7, so a(4) = 2+2*((70/7)1)/(71) = 5.


MATHEMATICA

GPF[n_] := Last[Select[Divisors[n], PrimeQ]];
T = Select[Flatten[Table[{p, (p^2*(r  2)  p*(r  4))/2}, {p, 3, 150}, {r, 3, 100}], 1], SquareFreeQ[Last[#]] && First[#] == GPF[Last[#]] &];
TT = Take[Union[Table[Last[T[[i]]], {i, Length[T]}]], 47];
Table[2 + 2*(t/GPF[t]  1)/(GPF[t]  1), {t, TT}]


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

Several missing terms inserted by and more terms from Jinyuan Wang, Feb 18 2021


STATUS

approved



