A324972 -id:A324972 - cp's OEIS frontend

A090466 Regular figurative or polygonal numbers of order greater than 2.

6, 9, 10, 12, 15, 16, 18, 21, 22, 24, 25, 27, 28, 30, 33, 34, 35, 36, 39, 40, 42, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 63, 64, 65, 66, 69, 70, 72, 75, 76, 78, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 100, 102, 105, 106, 108, 111, 112, 114, 115, 117, 118

Offset: 1

Robert G. Wilson v, Dec 01 2003

The sorted k-gonal numbers of order greater than 2. If one were to include either the rank 2 or the 2-gonal numbers, then every number would appear.
Number of terms less than or equal to 10^k for k = 1,2,3,...: 3, 57, 622, 6357, 63889, 639946, 6402325, 64032121, 640349979, 6403587409, 64036148166, 640362343980, ..., . - Robert G. Wilson v, May 29 2014
The n-th k-gonal number is 1 + k*n(n-1)/2 - (n-1)^2 = A057145(k,n).
For all squares (A001248) of primes p >= 5 at least one a(n) exists with p^2 = a(n) + 1. Thus the subset P_s(3) of rank 3 only is sufficient. Proof: For p >= 5, p^2 == 1 (mod {3,4,6,8,12,24}) and also P_s(3) + 1 = 3*s - 2 == 1 (mod 3). Thus the set {p^2} is a subset of {P_s(3) + 1}; Q.E.D. - Ralf Steiner, Jul 15 2018
For all primes p > 5, at least one polygonal number exists with P_s(k) + 1 = p when k = 3 or 4, dependent on p mod 6. - Ralf Steiner, Jul 16 2018
Numbers m such that r = (2*m/d - 2)/(d - 1) is an integer for some d, where 2 < d < m is a divisor of 2*m. If r is an integer, then m is the d-th (r+2)-gonal number. - Jianing Song, Mar 14 2021

Albert H. Beiler, Recreations In The Theory Of Numbers, The Queen Of Mathematics Entertains, Dover, NY, 1964, pp. 185-199.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 1000 terms are from T. D. Noe)
Felix Ponomarenko and Vadim Ponomarenko, Polygonal Simplex Numbers, San Diego State Univ., 2025.
Eric Weisstein's World of Mathematics, Figurate Number
Index to sequences related to polygonal numbers

Cf. A057145, A001248, A177028 (A342772, A342805), A177201, A316676, A364693 (characteristic function).
Complement is A090467.
Sequence A090428 (excluding 1) is a subsequence of this sequence. - T. D. Noe, Jun 14 2012
Other subsequences: A324972 (squarefree terms), A324973, A342806, A364694.
Cf. also A275340.

isA090466 := proc(n)
    local nsearch,ksearch;
    for nsearch from 3 do
        if A057145(nsearch,3) > n then
            return false;
        end if;
        for ksearch from 3 do
            if A057145(nsearch,ksearch) = n then
                return true;
            elif A057145(nsearch,ksearch) > n then
                break;
            end if;
        end do:
    end do:
end proc:
for n from 1 to 1000 do
    if isA090466(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 28 2016

Take[Union[Flatten[Table[1+k*n (n-1)/2-(n-1)^2,{n,3,100},{k,3,40}]]],67] (* corrected by Ant King, Sep 19 2011 *)
mx = 150; n = k = 3; lst = {}; While[n < Floor[mx/3]+2, a = PolygonalNumber[n, k]; If[a < mx+1, AppendTo[ lst, a], (n++; k = 2)]; k++]; lst = Union@ lst (* Robert G. Wilson v, May 29 2014 and updated Jul 23 2018; PolygonalNumber requires version 10.4 or higher *)

list(lim)=my(v=List()); lim\=1; for(n=3,sqrtint(8*lim+1)\2, for(k=3,2*(lim-2*n+n^2)\n\(n-1), listput(v, 1+k*n*(n-1)/2-(n-1)^2))); Set(v); \\ Charles R Greathouse IV, Jan 19 2017

is(n)=for(s=3,n\3+1,ispolygonal(n,s)&&return(s)); \\ M. F. Hasler, Jan 19 2017

isA090466(m) = my(v=divisors(2*m)); for(i=3, #v, my(d=v[i]); if(d==m, return(0)); if((2*m/d - 2)%(d - 1)==0, return(1))); 0 \\ Jianing Song, Mar 14 2021

Integer k is in this sequence iff A176774(k) < k. - Max Alekseyev, Apr 24 2018

Verified by Don Reble, Mar 12 2006

A324973 Special polygonal numbers.

Original entry on oeis.org

6, 15, 66, 70, 91, 190, 231, 435, 561, 703, 715, 782, 861, 946, 1045, 1105, 1426, 1653, 1729, 1770, 1785, 1794, 1891, 2035, 2278, 2465, 2701, 2821, 2926, 3059, 3290, 3367, 3486, 3655, 4371, 4641, 4830, 5005, 5083, 5151, 5365, 5551, 5565, 5995, 6441, 6545, 6601

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Mar 21 2019

nonn

Squarefree polygonal numbers P(r,p) = (p^2*(r-2)-p*(r-4))/2 whose greatest prime factor is p >= 3, and whose rank (or order) is r >= 3 (see A324974).

The Carmichael numbers A002997 and primary Carmichael numbers A324316 are subsequences. See Kellner and Sondow 2019.

			P(3,5) = 15 is squarefree, and its greatest prime factor is 5, so 15 is a member.
More generally, if p is an odd prime and P(3,p) is squarefree, then P(3,p) is a member, since P(3,p) = (p^2+p)/2 = p*(p+1)/2, so p is its greatest prime factor.
CAUTION: P(6,7) = 91 = 7*13 is a member even though 7 is NOT its greatest prime factor, as P(6,7) = P(3,13) and 13 is its greatest prime factor.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv preprint, arXiv:1902.10672 [math.NT], 2019-2021.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv preprint, arXiv:1902.11283 [math.NT], 2019-2022.
Wikipedia, Polygonal number.

Subsequence of A324972 = intersection of A005117 and A090466.
A002997, A324316, A324319 and A324320 are subsequences.
Cf. A324974, A324975, A324976.

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[#]] &];
Take[Union[Table[Last[t], {t, T}]], 47]

is(k) = if(issquarefree(k) && k>1, my(p=vecmax(factor(k)[, 1]), r); p>2 && (r=2*(k/p-1)/(p-1)) && denominator(r)==1, 0); \\ Jinyuan Wang, Feb 18 2021

Several missing terms inserted by Jinyuan Wang, Feb 18 2021

A324975 Rank of the n-th Carmichael number.

Original entry on oeis.org

6, 10, 12, 8, 8, 10, 6, 6, 8, 18, 52, 12, 12, 18, 98, 164, 22, 6, 50, 8, 96, 34, 52, 46, 52, 6, 6, 156, 20, 46, 36, 32, 16, 8, 304, 36, 20, 36, 10, 316, 76, 468, 8, 30, 24, 1580, 84, 54, 8, 12, 250, 28, 92, 36, 20, 418, 456, 928, 188, 16, 8, 276, 284, 56, 144

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Mar 24 2019

nonn

See A324974 for definition and explanation of rank of a special polygonal number, hence of rank of a Carmichael number A002997 by Kellner and Sondow 2019.

The ranks of the primary Carmichael numbers A324316 form the subsequence A324976.

			If m = A002997(1) = 561 = 3*11*17, then p = 17, so a(1) = 2+2*((561/17)-1)/(17-1) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Wikipedia, Polygonal number

Subsequence of A324974.
A324976 is a subsequence.
Cf. also A002997, A324316, A324972, A324973, A324977.

T = Cases[Range[1, 10000000, 2], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]];
GPF[n_] := Last[Select[Divisors[n], PrimeQ]];
Table[2 + 2*(T[[i]]/GPF[T[[i]]] - 1)/(GPF[T[[i]]] - 1), {i, Length[T]}]

a(n) = 2+2*((m/p)-1)/(p-1), where m = A002997(n) and p is its greatest prime factor. (See Formula in A324974.) Hence a(n) is even, by Carmichael's theorem that p-1 divides (m/p)-1, for any prime factor p of a Carmichael number m.

A324976 Rank of the n-th primary Carmichael number.

Original entry on oeis.org

12, 8, 18, 12, 52, 52, 20, 32, 16, 54, 8, 36, 124, 34, 12, 72, 96, 26, 28, 76, 98, 1804, 108, 124, 18, 72, 172, 120, 10, 104, 32, 244, 130, 376, 18, 92, 780, 36, 172, 92, 284, 24, 198, 12, 244, 64, 234, 340, 100, 284, 24, 124, 44, 518, 364, 16, 82, 148, 8, 206

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Mar 24 2019

See A324974 for definition and explanation of rank of a special polygonal number, hence of rank of a primary Carmichael number A324316 by Kellner and Sondow 2019.

			If m = A324316(1) = 1729 = 7*13*19, then p = 19, so a(1) = 2+2*((1729/19)-1)/(19-1) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Wikipedia, Polygonal number

Subsequence of A324975 (rank of the n-th Carmichael number A002997) and of A324974 (rank of the n-th special polygonal number A324973).

Cf. also A324316, A324972.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestCP[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] == # &];
T = Select[Range[1, 10^7, 2], TestCP[#] &];
GPF[n_] := Last[Select[Divisors[n], PrimeQ]];
Table[2 + 2*(T[[i]]/GPF[T[[i]]] - 1)/(GPF[T[[i]]] - 1), {i, Length[T]}]

a(n) = 2+2*((m/p)-1)/(p-1), where m = A324316(n) and p is its greatest prime factor. Hence a(n) is even; see Formula in A324975.

More terms from Amiram Eldar, Mar 27 2019

A324974 Rank of the n-th special polygonal number A324973(n).

Original entry on oeis.org

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

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Mar 24 2019

nonn

While two polygonal numbers of different ranks can be equal (e.g., P(6,n) = P(3,2n-1)), 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 well-defined.

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.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv preprint, arXiv:1902.10672 [math.NT], 2019-2021.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv preprint, arXiv:1902.11283 [math.NT], 2019-2022.
Wikipedia, Polygonal number.

A324975 and A324976 are subsequences.

Cf. A002997, A324316, A324972, A324973, A324977.

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}]

a(n) = 2 + 2*((m/p)-1)/(p-1), where m = A324973(n) and p is its greatest prime factor. (Proof: solve m = P(r,p) = (p^2*(r-2) - p*(r-4))/2 for r.)

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

cp's OEIS Frontend

A090466 Regular figurative or polygonal numbers of order greater than 2.

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A324973 Special polygonal numbers.

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A324975 Rank of the n-th Carmichael number.

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A324976 Rank of the n-th primary Carmichael number.

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A324974 Rank of the n-th special polygonal number A324973(n).

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