A006972 -id:A006972 - cp's OEIS frontend

A056729 If p | n, then p+1 | n+1 for composite n.

8, 27, 32, 63, 125, 128, 243, 275, 343, 399, 512, 567, 575, 935, 1127, 1331, 1539, 2015, 2048, 2187, 2197, 2303, 2783, 2915, 3087, 3125, 4563, 4913, 4991, 5103, 5719, 5831, 6399, 6859, 6875, 6929, 7055, 7139, 7625, 8192, 8855, 12167, 12719, 14027

Offset: 1

Robert G. Wilson v, Aug 31 2000

nonn

The Lucas-Carmichael numbers (A006972) are a subset.
Contains p^(2k+1) for any prime p, since (x+1) | (x^n + 1) when n is odd.
The only even numbers in this sequence are the composite odd powers of 2. [Emmanuel Vantieghem, Jul 08 2013]
If you try to extend this idea to the divisors, the only integer which is satisfied is 1.
Extension to prime power divisors is possible. [Emmanuel Vantieghem, Jul 08 2013]

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A006972.

fQ[n_] := !PrimeQ[n] && Union[ Mod[ n + 1, Transpose[ FactorInteger[n]][[1]] + 1]] == {0}; Select[ Range[20000], fQ[#] &]

is(n)=my(f=factor(n)[,1]);for(i=1,#f,if((n+1)%(f[i]+1), return(0))); !isprime(n) \\ Charles R Greathouse IV, Jan 15 2015

A216928 Least Lucas-Carmichael number with n prime factors.

Original entry on oeis.org

399, 8855, 588455, 139501439, 3512071871, 199195047359, 14563696180319, 989565001538399, 20576473996736735, 4049149795181043839, 409810997884396741919, 46852073639840281125599, 6414735508880546179805759, 466807799396932243821123839, 41222773167337486494297521279

Offset: 3

Tim Johannes Ohrtmann, Sep 20 2012

nonn

Is this sequence infinite? - Charles R Greathouse IV, Sep 23 2012

a(15) <= 6414735508880546179805759. a(16) <= 466807799396932243821123839. - Donovan Johnson, Sep 26 2012

Daniel Suteu, Table of n, a(n) for n = 3..35
Ed Copeland and Brady Haran, Something special about 399, Numberphile video (2015).

Cf. A006972 (Lucas-Carmichael numbers).

lucas_carmichael(A, B, k) = A=max(A, vecprod(primes(k+1))\2); (f(m, l, lo, k) = my(list=List()); my(hi=sqrtnint(B\m, k)); if(lo > hi, return(list)); if(k==1, lo=max(lo, ceil(A/m)); my(t=lift(-1/Mod(m,l))); while(t < lo, t += l); forstep(p=t, hi, l, if(isprime(p), my(n=m*p); if((n+1)%(p+1) == 0, listput(list, n)))), forprime(p=lo, hi, if(gcd(m, p+1) == 1, list=concat(list, f(m*p, lcm(l, p+1), p+1, k-1))))); list); vecsort(Vec(f(1, 1, 3, k)));
a(n) = if(n < 3, return()); my(x=vecprod(primes(n+1))\2,y=2*x); while(1, my(v=lucas_carmichael(x,y,n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 24 2023

a(7)-a(12) from Donovan Johnson, Sep 22 2012
a(13)-a(14) from Donovan Johnson, Sep 26 2012
a(15)-a(16) confirmed and a(17) added by Daniel Suteu, Aug 29 2022

A202157 a(n) = smallest k having at least two prime divisors d such that (d + n) | ( k + n).

Original entry on oeis.org

63, 18, 45, 50, 75, 66, 63, 102, 75, 50, 165, 198, 147, 258, 165, 110, 663, 182, 399, 442, 147, 242, 705, 678, 455, 786, 483, 182, 1015, 950, 1023, 988, 363, 506, 637, 1446, 1083, 322, 885, 590, 1155, 1443, 1935, 2118, 627, 770, 3243, 2502, 1407, 2706, 845

Offset: 1

Michel Lagneau, Dec 13 2011

nonn

The sequence of numbers k composite and squarefree, prime p | k ==> p+n | k+n is given by A029591 (least quasi-Carmichael number of order -n).
If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.
In this sequence, the majority of terms are not squarefree: 63, 18, 45, ...

			a(8) = 102 because the prime divisors of 102 are 2, 3 and 17;
(2 + 8) | (102 + 8) = 110 = 10*11;
(3 + 8) | 110 = 11*10.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 399, p. 89, Ellipses, Paris 2008.

Amiram Eldar, Table of n, a(n) for n = 1..2500

Cf. A006972, A029591, A202158, A202159.

with(numtheory):for n from 1 to 52 do:i:=0:for k from 1 to 5000 while(i=0) do:x:=factorset(k):n1:=nops(x):y:=k+n: j:=0:for m from 1 to n1 do:if  n1>=2 and irem(y,x[m]+n)=0 then j:=j+1:else fi:od:if j>=2 then i:=1:printf(`%d, `,k):else fi:od:od:

numd[n_, k_] := Module[{p=FactorInteger[k][[;;,1]], c=0}, Do[If[Divisible[n+k, n+p[[i]]], c++], {i,1,Length[p]}]; c]; a[n_]:=Module[{k=1}, While[numd[n, k] <= 1, k++]; k]; Array[a, 50] (* Amiram Eldar, Sep 09 2019 *)

a(n) >= n^2 + 4n + 6. [Charles R Greathouse IV, Dec 13 2011]

A202158 a(n) = smallest k having at least three prime divisors d such that (d + n) | (k + n).

Original entry on oeis.org

399, 598, 165, 1886, 715, 2370, 273, 532, 231, 935, 3445, 828, 1547, 2821, 1105, 3710, 12903, 4182, 6669, 4732, 2475, 4466, 2737, 2706, 1595, 5658, 10413, 3542, 7315, 24225, 23769, 22578, 3927, 12818, 1885, 64119, 11063, 20482, 10881, 4370, 52275, 7878, 14645

Offset: 1

Michel Lagneau, Dec 13 2011

nonn

The sequence of numbers k composite and squarefree, prime p | k ==> p+n | k+n is given by A029591 (least quasi-Carmichael number of order -n).
If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.
In this sequence, the majority of terms are not squarefree.

			a(3) = 165 because the prime divisors of 165 are 3, 5, 11 =>
(3 + 3) | (165 + 3) = 168 = 6*28;
(5 + 3) | 168 = 8*21;
(11 + 3) | 168 = 14*12.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 399, p. 89, Ellipses, Paris 2008.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A006972, A029591, A202157, A202159.

with(numtheory):for n from 1 to 45 do:i:=0:for k from 1 to 100000 while(i=0) do:x:=factorset(k):n1:=nops(x):y:=k+n: j:=0:for m from 1 to n1 do:if  n1>=2 and irem(y,x[m]+n)=0 then j:=j+1:else fi:od:if j>2 then i:=1:printf(`%d, `,k):else fi:od:od:

numd[n_, k_] := Module[{p=FactorInteger[k][[;;,1]], c=0}, Do[If[Divisible[n+k, n+p[[i]]], c++], {i,1,Length[p]}]; c]; a[n_]:=Module[{k=1}, While[numd[n, k] <= 2, k++]; k]; Array[a, 40] (* Amiram Eldar, Sep 09 2019 *)

A202159 a(n) = smallest k having at least four prime divisors d such that (d + n) | (k + n).

Original entry on oeis.org

8855, 11590, 27885, 122360, 16555, 10290, 6545, 61642, 71799, 65195, 14245, 142788, 63635, 580930, 39585, 21098, 69003, 258482, 59885, 378952, 8715, 266090, 133285, 690501, 27335, 704790, 1017423, 299222, 187891, 771650, 293405, 1638598, 282315, 553610, 227205

Offset: 1

Michel Lagneau, Dec 13 2011

nonn

The sequence of numbers k composite and squarefree, prime p | k ==> p+n | k+n is given by A029591 (least quasi-Carmichael number of order -n).
If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.
In this sequence, the majority of terms are not squarefree.

			a(3) = 27885 because the prime divisors of 27885 are 3, 5, 11, 13  =>
(3 + 3)| (27885 + 3) = 27888 = 6*4648;
(5 + 3) | 27888 = 8*3486;
(11 + 3) | 27888 = 14*1992;
(13 + 3) | 27888 = 16*1743.

Amiram Eldar, Table of n, a(n) for n = 1..450

Cf. A006972, A029591, A202157, A202158.

with(numtheory):for n from 1 to 33 do:i:=0:for k from 1 to 5000000  while(i=0) do:x:=factorset(k):n1:=nops(x):y:=k+n: j:=0:for m from 1 to n1 do:if  n1>=2 and irem(y,x[m]+n)=0 then j:=j+1:else fi:od:if j>3 then i:=1:printf(`%d, `,k):else fi:od:od:

numd[n_, k_] := Module[{p=FactorInteger[k][[;;,1]], c=0}, Do[If[Divisible[n+k, n+p[[i]]], c++], {i,1,Length[p]}]; c]; a[n_]:=Module[{k=1}, While[numd[n, k] <= 3, k++]; k]; Array[a, 35] (* Amiram Eldar, Sep 09 2019 *)

A292573 Least Lucas-Carmichael number whose Dedekind psi value is an n-th power.

Original entry on oeis.org

399, 935, 8855, 935, 191807, 31535, 2376401921999, 287432495, 3512071871, 443155187061431999, 269174015, 7187117143679, 13153784862132539999, 16794805460009764607, 12239445970372607, 308403604421150344874999, 1940371455953001599, 22735742424704664191, 193426182224210221055, 2674275760170030054482343935

Offset: 1

Amiram Eldar, Sep 19 2017

nonn

There are no other terms below 10^12.

			psi(8855) = 24^3 while the psi values of smaller Lucas-Carmichael numbers are not cubes, therefore a(3) = 8855.

Max Alekseyev, Table of n, a(n) for n = 1..45

Cf. A001615, A006972, A290805, A292571, A292572.

a(7),a(10),a(12)-a(26) from Max Alekseyev, May 01 2024

A253597 Least Lucas-Carmichael number divisible by the n-th prime.

Original entry on oeis.org

399, 935, 399, 935, 2015, 935, 399, 4991, 51359, 2015, 1584599, 20705, 5719, 18095, 2915, 46079, 162687, 22847, 46079, 16719263, 12719, 7055, 80189, 104663, 20705, 482143, 196559, 60059, 90287, 162687, 3441239, 13971671

Offset: 2

Tim Johannes Ohrtmann, Jan 05 2015

nonn

Has any odd prime number at least one Lucas-Carmichael multiple?

			a(2) = 399 because this is the least Lucas-Carmichael number which is divisible by 3 (the second prime number).

Tim Johannes Ohrtmann and Charles R Greathouse IV, Table of n, a(n) for n = 2..1000 (terms up to 202 from Ohrtmann)

Cf. A006972, A065091, A253598.

LucasCarmichaelQ[n_] := Block[{fi = FactorInteger@ n}, !PrimeQ@ n && Times @@ (Last@# & /@ fi) == 1 && Plus @@ Mod[n + 1, 1 + First@# & /@ fi] == 0]; f[n_] := Block[{k = p = Prime@ n}, While[ !LucasCarmichaelQ@ k, k += p]; k]; Array[f, 35, 2] (* Robert G. Wilson v, Feb 11 2015 *)

is_A006972(n)=my(f=factor(n)); for(i=1, #f[, 1], if((n+1)%(f[i, 1]+1) || f[i, 2]>1, return(0))); #f[, 1]>1
a(n) = pn = prime(n); ln = 1; until (is_A006972(ln) && (ln % pn == 0), ln++); ln;

is_A006972(n)=my(f=factor(n)); for(i=1, #f~, if((n+1)%(f[i, 1]+1) || f[i, 2]>1, return(0))); #f~>1
a(n)=my(p=prime(n), c=p^2+p, t=p); while(!is_A006972(t+=c),); t \\ Charles R Greathouse IV, Feb 03 2015

a(n) >> n^2 log^2 n. - Charles R Greathouse IV, Feb 03 2015

A253598 a(n) = least Lucas-Carmichael number which is divisible by b(n), where {b(n)} (A255602) is the list of all numbers which could be a divisor of a Lucas-Carmichael number.

Original entry on oeis.org

399, 399, 935, 399, 935, 2015, 935, 399, 399, 4991, 51359, 2015, 8855, 1584599, 9486399, 20705, 5719, 18095, 2915, 935, 399, 46079, 162687, 2015, 22847, 46079, 16719263, 8855, 12719, 7055, 935, 80189, 189099039, 104663, 20705, 482143, 196559, 60059, 30073928079, 90287, 8855, 31535

Offset: 1

Tim Johannes Ohrtmann, Jan 05 2015

nonn

a(933) <= 266336887317945807999. - Daniel Suteu, Dec 01 2023

			a(12) = 8855 because this is the least Lucas-Carmichael number which is divisible by A255602(12) = 35.

Daniel Suteu, Table of n, a(n) for n = 1..932 (first 95 terms from Tim Johannes Ohrtmann, terms 96..154 from Robert G. Wilson v)

Cf. A006972, A253597, A255602.

LucasCarmichaelQ[n_] := Block[{fi = FactorInteger@ n}, ! PrimeQ@ n && Times @@ (Last@# & /@ fi) == 1 && Plus @@ Mod[n + 1, 1 + First@# & /@ fi] == 0]; LucasCarmichaelQ[1] = False; fQ[n_] := Block[{fi = FactorInteger@ n}, ffi = First@# & /@ fi; Times @@ (Last@# & /@ fi) == 1 && Min@ Flatten@ Table[Mod[1 + ffi, i], {i, ffi}] > 0]; fQ[1] = True; fQ[2] = False; lcdv = Select[ Range@ 3204, fQ]; f[n_] := Block[{k = lcdv[[n]]}, d = 2k; While[ !LucasCarmichaelQ@ k, k += d]; k]; Array[f, 95] (* Robert G. Wilson v, Feb 11 2015 *)

a(96) from Charles R Greathouse IV, Feb 12 2015

A255602 Numbers k which are odd and squarefree and have the property that k is either a prime number or for every prime p dividing k, p+1 is not divisible by any of the other prime factors of k.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 35, 37, 39, 41, 43, 47, 53, 55, 57, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 93, 97, 101, 103, 107, 109, 111, 113, 115, 119, 127, 129, 131, 133, 137, 139, 143, 149, 151, 155

Offset: 1

Tim Johannes Ohrtmann and Robert G. Wilson v, Feb 27 2015

nonn

A proper subset of A056911 and a proper subset of A005117. Any divisor of a Lucas-Carmichael number is in this sequence. It is not known whether every number in this sequence divides at least one Lucas-Carmichael number. All prime numbers except 2 are present. Composite numbers in the sequence include 21, 35, 39, 55, 57, 65, 77, 85, 93, 111, 115, 119, 129, 133, 143, 155, 161, 183, 185, 187, ..., .

			15 is not in the sequence since its two prime factors are 3 and 5, and 5+1 is divisible by 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A006972, A056911, A253598.

fQ[n_] := Block[{fi = FactorInteger@ n}, ffi = First@# & /@ fi; Times @@ (Last@# & /@ fi) == 1 && Min@ Flatten@ Table[ Mod[1 + ffi, i], {i, ffi}] > 0]; fQ[1] = True; fQ[2] = False; Select[ Range@ 190, fQ]

isok(n) = {if (! ((n % 2) && issquarefree(n)), return (0)); vpf = factor(n)[, 1]; for (i=1, #vpf, vpx = vpf[i]+1; for (j=1, #vpf, if (! (vpx % vpf[j]), return (0)); ); ); return (1); } \\ Michel Marcus, Mar 02 2015

A292571 Lucas-Carmichael numbers whose Dedekind psi value is a square.

Original entry on oeis.org

935, 31535, 76751, 1707839, 3106799, 11141999, 24685199, 43383167, 83618639, 151524071, 161841239, 189099039, 212133599, 213884999, 219155615, 233743319, 241485839, 271038599, 287432495, 338340239, 353107799, 624840479, 660423455, 945236159, 1171355471

Offset: 1

Amiram Eldar, Sep 19 2017

nonn

			psi(935) = 36^2.

Amiram Eldar, Table of n, a(n) for n = 1..241 (terms below 10^12)

Intersection of A006972 and A291167.

Cf. A001615, A272798, A292572, A292573.

psi[n_] := If[n < 1, 0, n*Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]]; s = Import["b006972.txt","Data"][[All,-1]]; Select[s, IntegerQ@Sqrt[psi@#] &]

cp's OEIS Frontend

A056729 If p | n, then p+1 | n+1 for composite n.

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A216928 Least Lucas-Carmichael number with n prime factors.

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A202157 a(n) = smallest k having at least two prime divisors d such that (d + n) | ( k + n).

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A202158 a(n) = smallest k having at least three prime divisors d such that (d + n) | (k + n).

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A202159 a(n) = smallest k having at least four prime divisors d such that (d + n) | (k + n).

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A292573 Least Lucas-Carmichael number whose Dedekind psi value is an n-th power.

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A253597 Least Lucas-Carmichael number divisible by the n-th prime.

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A253598 a(n) = least Lucas-Carmichael number which is divisible by b(n), where {b(n)} (A255602) is the list of all numbers which could be a divisor of a Lucas-Carmichael number.