A287840 -id:A287840 - cp's OEIS frontend

A339245 Partrich numbers: positive integers whose square part and squarefree part are divisible by 2 and an odd prime.

216, 360, 504, 600, 792, 864, 936, 1000, 1080, 1176, 1224, 1368, 1400, 1440, 1512, 1656, 1944, 1960, 2016, 2088, 2200, 2232, 2376, 2400, 2520, 2600, 2664, 2744, 2808, 2904, 2952, 3000, 3096, 3168, 3240, 3384, 3400, 3456, 3672, 3744, 3800, 3816, 3960, 4000, 4056, 4104, 4200

Offset: 1

Peter Munn, Nov 28 2020

Not named after anyone, partrich numbers have the square part of their odd part, the square part of their even part (A234957), the squarefree part of their odd part and the squarefree part of their even part (A056832) all greater than 1.
Numbers whose odd part and even part are nonsquare and nonsquarefree.
All terms are divisible by 8. If m is present, 2m is absent and 4m is present.
Closed under multiplication by any square and under application of A059896: for n, k >= 1, A059896(a(n), k) is in the sequence.
From Peter Munn, Apr 07 2021: (Start)
The first deficient partrich number is 39304 = 2^3 * 17^3. (ascertained by Amiram Eldar)
The first 7 terms generate Carmichael numbers using the method of Erdős described in A287840.
(End)

			A positive integer is present if and only if it factorizes as 2 times an odd squarefree number > 1, an even square that is a power of 4 and an odd square > 1. This factorization of the initial terms is shown below.
   n  a(n)
   1   216 = 2 *  3 *  4 *  9,
   2   360 = 2 *  5 *  4 *  9,
   3   504 = 2 *  7 *  4 *  9,
   4   600 = 2 *  3 *  4 * 25,
   5   792 = 2 * 11 *  4 *  9,
   6   864 = 2 *  3 * 16 *  9,
   7   936 = 2 * 13 *  4 *  9,
   8  1000 = 2 *  5 *  4 * 25,
   9  1080 = 2 * 15 *  4 *  9,
  10  1176 = 2 *  3 *  4 * 49,
  ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Even Part, Odd Part, Square Part, Squarefree Part.

Subsequence of A008586, A028983, A036554, A036785, A038838, A190892.
Subsequences: A017139, A017643.
Cf. A056832, A059896, A234957, A287840.

q[n_] := Module[{ie = IntegerExponent[n, 2], odd}, ie > 2 && OddQ[ie] && !SquareFreeQ[(odd = n/2^ie)] && !IntegerQ @ Sqrt[odd]]; Select[Range[4200], q] (* Amiram Eldar, Dec 04 2020 *)

A008586 INTERSECT A028983 INTERSECT A036554 INTERSECT A038838.

Asymptotic density is 1/12 - 2/(3 * Pi^2) = 0.01578587757... . (Formula due to Amiram Eldar.)

A292352 Numbers that generate Lucas-Carmichael numbers using an adjusted version of Erdős's method.

Original entry on oeis.org

24, 36, 40, 48, 60, 72, 80, 84, 96, 108, 120, 144, 168, 180, 192, 200, 216, 240, 252, 270, 300, 324, 336, 360, 384, 400, 420, 432, 440, 468, 480, 504, 528, 540, 576, 588, 600, 624, 648, 660, 672, 714, 720, 744, 756, 768, 792, 810, 840, 864, 900, 912, 936, 960

Offset: 1

Amiram Eldar, Sep 14 2017

nonn

Erdős showed in 1956 how to construct Carmichael numbers from a given number n (see A287840). With appropriate sign changes the method can be used to generate Lucas-Carmichael numbers. Given a number n, let P be the set of primes p such that (p+1)|n but p is not a factor of n. Let c be a product of a subset of P with at least 3 elements. If c == -1 (mod n) then c is a Lucas-Carmichael number.

Numbers with only one generated Lucas-Carmichael number: 24, 36, 40, 48, 60, 80, 84, 96, 108, 200, 252, 270, 300, 324, 336, 400, 440, 468, ...

			The set of primes for n = 24 is P={2, 3, 5, 7, 11, 23}. One subset, {5, 7, 11, 23} have c == -1 (mod n): c = 5*7*11*23 = 8855. 24 is the least number that generates Lucas-Carmichael numbers thus a(1)=24.

Cf. A006972, A287840.

a = {}; Do[p = Select[Divisors[n] - 1, PrimeQ]; pr = Times @@ p; pr = pr/GCD[n, pr]; ps = Divisors[pr]; c = 0; Do[p1 = FactorInteger[ps[[j]]][[;; , 1]]; If[Length[p1] < 3, Continue[]]; c1 = Times @@ p1; If[Mod[c1, n] == 1, c++], {j, 1, Length[ps]}]; If[c > 0, AppendTo[a, n]], {n, 1, 1000}]; a

A287861 Numbers n with a record number of Carmichael numbers that can be generated from them using Erdős's method.

Original entry on oeis.org

36, 120, 180, 240, 360, 540, 720, 1080, 1200, 1260, 1680, 2160, 2520, 3780, 5040, 7560, 10080, 15120, 25200

Offset: 1

Amiram Eldar, Sep 01 2017

Erdős showed in 1956 how to construct Carmichael numbers from a given number n (typically with many divisors). Given a number n, let P be the set of primes p such that (p-1)|n but p is not a factor of n. Let c be a product of a subset of P with at least 3 elements. If c == 1 (mod n) then c is a Carmichael number.

The corresponding number of generated Carmichael numbers are 2, 3, 4, 8, 11, 16, 26, 30, 36, 57, 79, 204, 466, 610, 7253, 9778, 58058, 1244090, 5963529.

			The set of primes for n = 36 is P={5, 7, 13, 19, 37}. Two subsets, {7, 13, 19} and {7, 13, 19, 37} have c == 1 (mod n): c = 7*13*19 = 1729 and c = 7*13*19*37 = 63973. 36 is the first number that generates Carmichael numbers thus a(1)=36.

Paul Erdős, On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen 4 (1956), pp. 201-206.
Andrew Granville, Primality testing and Carmichael numbers, Notices of the American Mathematical Society, Vol. 39 No. 6 (1992), pp. 696-700.
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.

Cf. A002997, A287840.

a = {}; cmax = 0; Do[p = Select[Divisors[n] + 1, PrimeQ]; pr = Times @@ p; pr = pr/GCD[n, pr]; ps = Divisors[pr]; c = 0; Do[p1 = FactorInteger[ps[[j]]][[;; , 1]]; If[Length[p1] < 3, Continue[]]; c1 = Times @@ p1; If[Mod[c1, n] == 1, c++], {j, 1, Length[ps]}];
If[c > cmax, cmax = c; AppendTo[a, n]], {n, 1, 1000}]; a

A287862 Numbers n with a record size of the largest Carmichael number that can be generated from them using Erdős's method.

Original entry on oeis.org

36, 60, 108, 112, 120, 180, 216, 360, 540, 840, 1200, 1620, 2016, 2160, 2520, 3360, 3780, 4800, 5040, 6480, 7560, 8400, 10080, 12600, 15120, 25200

Offset: 1

Amiram Eldar, Sep 01 2017

Erdős showed in 1956 how to construct Carmichael numbers from a given number n (typically with many divisors). Given a number n, let P be the set of primes p such that (p-1)|n but p is not a factor of n. Let c be a product of a subset of P with at least 3 elements. If c == 1 (mod n) then c is a Carmichael number.

The corresponding largest Carmichael numbers are 63973, 172081, 188461, 278545, 852841, 31146661, 509033161, 416937760921, ...

			The set of primes for n = 36 is P={5, 7, 13, 19, 37}. Two subsets, {7, 13, 19} and {7, 13, 19, 37} have c == 1 (mod n): c = 7*13*19 = 1729 and c = 7*13*19*37 = 63973. 36 is the first number that generates Carmichael numbers thus a(1)=36.

Paul Erdős, On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen 4 (1956), pp. 201-206.
Andrew Granville, Primality testing and Carmichael numbers, Notices of the American Mathematical Society, Vol. 39 No. 6 (1992), pp. 696-700.
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.

Cf. A002997, A287840.

a = {}; cmax = 0; Do[p = Select[Divisors[n] + 1, PrimeQ]; pr = Times @@ p; pr = pr/GCD[n, pr]; ps = Divisors[pr]; c = 0; Do[p1 = FactorInteger[ps[[j]]][[;; , 1]]; If[Length[p1] < 3, Continue[]]; c1 = Times @@ p1; If[Mod[c1, n] == 1, c = Max[c, c1]], {j, 1, Length[ps]}];
If[c > cmax, cmax = c; AppendTo[a, n]], {n, 1, 1000}]; a

A292353 Numbers n with a record number of Lucas-Carmichael numbers that can be generated from them using an adjusted version of Erdős's method.

Original entry on oeis.org

24, 72, 216, 240, 360, 720, 1440, 2160, 2520, 4320, 5040, 7560, 10080, 15120, 20160

Offset: 1

Amiram Eldar, Sep 14 2017

Erdős showed in 1956 how to construct Carmichael numbers from a given number n (see A287840). With appropriate sign changes the method can be used to generate Lucas-Carmichael numbers. Given a number n, let P be the set of primes p such that (p+1)|n but p is not a factor of n. Let c be a product of a subset of P with at least 3 elements. If c == -1 (mod n) then c is a Lucas-Carmichael number.

The corresponding number of generated Lucas-Carmichael numbers are 1, 3, 5, 9, 21, 169, 681, 900, 1842, 7250, 29132, 77482, 932187, 4970111, 7456418.

			The set of primes for n = 24 is P={2, 3, 5, 7, 11, 23}. One subset, {5, 7, 11, 23} have c == -1 (mod n): c = 5*7*11*23 = 8855. 24 is the least number that generates Lucas-Carmichael numbers thus a(1)=24.

Cf. A006972, A287861, A292352.

a = {}; cmax = 0; Do[p = Select[Divisors[n] - 1, PrimeQ]; pr = Times @@ p; pr = pr/GCD[n, pr]; ps = Divisors[pr]; c = 0; Do[p1 = FactorInteger[ps[[j]]][[;; , 1]]; If[Length[p1] < 3, Continue[]]; c1 = Times @@ p1; If[Mod[c1, n] == 1, c++], {j, 1, Length[ps]}]; If[c > cmax, cmax = c; AppendTo[a, n]], {n, 1, 1000}]; a

A292354 Numbers n with a record size of the largest Lucas-Carmichael number that can be generated from them using an adjusted version of Erdős's method.

Original entry on oeis.org

24, 48, 60, 144, 168, 240, 360, 720, 1440, 2520, 4320, 5040, 7560, 10080, 15120, 20160

Offset: 1

Amiram Eldar, Sep 14 2017

Erdős showed in 1956 how to construct Carmichael numbers from a given number n (see A287840). With appropriate sign changes the method can be used to generate Lucas-Carmichael numbers. Given a number n, let P be the set of primes p such that (p+1)|n but p is not a factor of n. Let c be a product of a subset of P with at least 3 elements. If c == -1 (mod n) then c is a Lucas-Carmichael number.

The corresponding largest Lucas-Carmichael numbers are 8855, 18095, 357599, 1010735, 406335215, 1087044101759, 4467427448759, ...

			The set of primes for n = 24 is P={2, 3, 5, 7, 11, 23}. One subset, {5, 7, 11, 23} have c == -1 (mod n): c = 5*7*11*23 = 8855. 24 is the least number that generates Lucas-Carmichael numbers thus a(1)=24.

Cf. A006972, A287862, A292352.

a = {}; cmax = 0; Do[p = Select[Divisors[n] - 1, PrimeQ]; pr = Times @@ p; pr = pr/GCD[n, pr]; ps = Divisors[pr]; c = 0; Do[p1 = FactorInteger[ps[[j]]][[;; , 1]]; If[Length[p1] < 3, Continue[]]; c1 = Times @@ p1; If[Mod[c1, n] == 1, c = Max[c, c1]], {j, 1, Length[ps]}]; If[c > cmax, cmax = c; AppendTo[a, n]], {n, 1, 1000}]; a

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A339245 Partrich numbers: positive integers whose square part and squarefree part are divisible by 2 and an odd prime.

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A292352 Numbers that generate Lucas-Carmichael numbers using an adjusted version of Erdős's method.

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A287861 Numbers n with a record number of Carmichael numbers that can be generated from them using Erdős's method.

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A287862 Numbers n with a record size of the largest Carmichael number that can be generated from them using Erdős's method.

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A292353 Numbers n with a record number of Lucas-Carmichael numbers that can be generated from them using an adjusted version of Erdős's method.

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A292354 Numbers n with a record size of the largest Lucas-Carmichael number that can be generated from them using an adjusted version of Erdős's method.

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