A324976 -id:A324976 - cp's OEIS frontend

A324316 Primary Carmichael numbers.

1729, 2821, 29341, 46657, 252601, 294409, 399001, 488881, 512461, 1152271, 1193221, 1857241, 3828001, 4335241, 5968873, 6189121, 6733693, 6868261, 7519441, 10024561, 10267951, 10606681, 14469841, 14676481, 15247621, 15829633, 17098369, 17236801, 17316001, 19384289, 23382529, 29111881, 31405501, 34657141, 35703361, 37964809

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 21 2019

Squarefree integers m > 1 such that if prime p divides m, then the sum of the base-p digits of m equals p. It follows that m is then a Carmichael number (A002997).
Dickson's conjecture implies that the sequence is infinite, see Kellner 2019.
If m is a term and p is a prime factor of m, then p <= a*sqrt(m) with a = sqrt(66337/132673) = 0.7071..., where the bound is sharp.
The distribution of primary Carmichael numbers is A324317.
See Kellner and Sondow 2019 and Kellner 2019.
Primary Carmichael numbers are special polygonal numbers A324973. The rank of the n-th primary Carmichael number is A324976(n). See Kellner and Sondow 2019. - Jonathan Sondow, Mar 26 2019
The first term is the Hardy-Ramanujan number. - Omar E. Pol, Jan 09 2020

			1729 = 7 * 13 * 19 is squarefree, and 1729 in base 7 is 5020_7 = 5 * 7^3 + 0 * 7^2 + 2 * 7 + 0 with 5+0+2+0 = 7, and 1729 in base 13 is a30_13 with a+3+0 = 10+3+0 = 13, and 1729 in base 19 is 4f0_19 with 4+f+0 = 4+15+0 = 19, so 1729 is a member.

Bernd C. Kellner, Table of n, a(n) for n = 1..10000 (computed by using Pinch's database, see link below)
Bernd C. Kellner, On primary Carmichael numbers, #A38 Integers 22 (2022), 39 p.; arXiv:1902.11283 [math.NT], 2019.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, #A52 Integers 21 (2021), 21 p.; arXiv:1902.10672 [math.NT], 2019.
R. G. E. Pinch, The Carmichael numbers up to 10^18, 2008.
Index entries for sequences related to Carmichael numbers.

Subsequence of A002997, A324315.
Least primary Carmichael number with n prime factors is A306657.
Cf. also A005117, A195441, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405, A324973, A324976, A001235.

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, #] == # &];
Select[Range[1, 10^7, 2], TestCP[#] &]

use ntheory ":all"; my $m; forsquarefree { $m=$; say if @ > 2 && is_carmichael($m) && vecall { $ == vecsum(todigits($m,$)) } @; } 1e7; # _Dana Jacobsen, Mar 28 2019

from sympy import factorint
from sympy.ntheory import digits
def ok(n):
    pf = factorint(n)
    if n < 2 or max(pf.values()) > 1: return False
    return all(sum(digits(n, p)[1:]) == p for p in pf)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 03 2022

a_1 + a_2 + ... + a_k = p if p is prime and m = a_1 * p + a_2 * p^2 + ... + a_k * p^k with 0 <= a_i <= p-1 for i = 1, 2, ..., k (note that a_0 = 0).

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.

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

A324316 Primary Carmichael numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Perl

Python

Formula

A324973 Special polygonal numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A324975 Rank of the n-th Carmichael number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions