A056729 -id:A056729 - cp's OEIS frontend

A006972 Lucas-Carmichael numbers: squarefree composite numbers k such that p | k => p+1 | k+1.

399, 935, 2015, 2915, 4991, 5719, 7055, 8855, 12719, 18095, 20705, 20999, 22847, 29315, 31535, 46079, 51359, 60059, 63503, 67199, 73535, 76751, 80189, 81719, 88559, 90287, 104663, 117215, 120581, 147455, 152279, 155819, 162687, 191807, 194327, 196559, 214199

Offset: 1

Richard Pinch and Jeffrey Shallit

nonn

Wright proves that this sequence is infinite (Main Theorem 2). - Charles R Greathouse IV, Nov 03 2015
Conjecture: if k = p*q*r, p = a*d - 1, q = b*d - 1, r = c*d - 1 are distinct odd primes, with d = gcd(p + 1, q + 1, r + 1) and a*b*c*d divides k + 1, then k is a Lucas-Carmichael number. - Davide Rotondo, Dec 23 2020
A composite k is a Lucas-Carmichael number if and only if k | A322702(k+1). - Thomas Ordowski, May 06 2021

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 399, p. 89, Ellipses, Paris 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 550 terms from Paolo P. Lava)
Ed Copeland and Brady Haran, Something special about 399, Numberphile video (2015).
Sridhar Tamilvanan and Subramani Muthukrishnan, On Lucas-Carmichael Integer, arXiv:2311.08012 [math.NT], 2023.
Daniel Suteu, Table of n, a(n) for terms a(n) < 10^15.
Samuel S. Wagstaff, Jr., Ramanujan's Taxicab Number and its Ilk, Purdue Univ. (2024). See p. 2.
Wikipedia, Lucas-Carmichael number
Thomas Wright, There are infinitely many elliptic Carmichael numbers
Thomas Wright, There are infinitely many elliptic Carmichael numbers, arXiv:1609.00231 [math.NT], 2016.
Index entries for sequences related to Carmichael numbers.

Intersection of A024556 and A056729.
Cf. A216925, A216926, A216927, A217002, A217003, A217091 (terms having 3, 4, 5, 6, 7 and 8 factors).
Cf. A216929, A322702.

with(numtheory):
a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, 3, a(n-1)) while isprime(k) or not issqrfree(k)
       or add(irem(k+1,i+1), i=factorset(k))>0 do od; k
    end:
seq(a(n), n=1..15);  # Alois P. Heinz, Apr 05 2018

Select[ Range[ 2, 10^6 ], !PrimeQ[ # ] && Union[ Transpose[ FactorInteger[ # ] ][ [ 2 ] ] ] == {1} && Union[ Mod[ # + 1, Transpose[ FactorInteger[ # ] ][ [ 1 ] ] + 1 ] ] == {0} & ]

is(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 \\ Charles R Greathouse IV, Sep 23 2012

lucas_carmichael(A, B, k) = A=max(A, vecprod(primes(k+1))\2); (f(m, l, lo, k) = my(list=List()); my(hi=min(sqrtint(B+1)-1, 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); f(1, 1, 3, k);
upto(n) = my(list=List()); for(k=3, oo, if(vecprod(primes(k+1))\2 > n, break); list=concat(list, lucas_carmichael(1, n, k))); vecsort(Vec(list)); \\ Daniel Suteu, Dec 01 2023

A080062 Composite numbers n such that for all primes p dividing n, p-1 divides n-1.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 45, 49, 64, 81, 121, 125, 128, 169, 225, 243, 256, 289, 325, 343, 361, 405, 512, 529, 561, 625, 637, 729, 841, 891, 961, 1024, 1105, 1125, 1225, 1331, 1369, 1377, 1681, 1729, 1849, 2025, 2048, 2187, 2197, 2209, 2401, 2465, 2809, 2821

Offset: 1

Robert G. Wilson v, Jan 23 2003

nonn

The subsequence of squarefree terms gives the Carmichael numbers (A002997); cf. Korselt's criterion. - Joerg Arndt, May 17 2016

Donovan Johnson, Table of n, a(n) for n = 1..10000
Wikipedia, Korselt's criterion

Cf. A056729, A079543.

Cf. A002997 (Carmichael numbers).

Select[ Range[2, 10^4], !PrimeQ[ # ] && Union[ Mod[ # - 1, Transpose[ FactorInteger[ # ]][[1]] - 1]] == {0} &]

is080062(n)=if(isprime(n),return(0)); my(f=factor(n)[, 1]); for(j=1, #f, if((n-1)%(f[j]-1), return(0))); 1; \\ Joerg Arndt, May 17 2016

A304291 Composite numbers k such that for all primes p dividing k, p-1 divides k-1 and p+1 divides k+1.

Original entry on oeis.org

8, 27, 32, 125, 128, 243, 343, 512, 1331, 2048, 2187, 2197, 3125, 4913, 6859, 8192, 12167, 16807, 19683, 24389, 29791, 32768, 50653, 68921, 74431, 78125, 79507, 103823, 131072, 148877, 161051, 177147, 205379, 226981, 300763, 357911, 371293, 389017, 493039, 524288

Offset: 1

Paolo P. Lava, May 17 2018

nonn

Intersection of A080062 and A056729.
Mainly odd powers of a prime: A056824 is a subset of this sequence.
If the additional limitations p-2|n-2 and p+2|n+2 should be added, only 243, 19683, 78125, 1594323 would be terms of the sequence for n <= 10^7.
Terms that are not perfect powers are 31*7^4, 31^3*7^4, 71*11^6, .... - Altug Alkan, May 17 2018
It appears that this is the intersection of A002808 and A171561. - Michel Marcus, May 19 2018
From Robert Israel, May 25 2018: (Start)
  If i is odd and 4|j, then 31^i*7^j is a member.
  If i is odd and 6|j, then 71^i*11^j is a member.
  If i is odd and 12|j, then 17^i*5^j is a member.
  If i is odd and 36|j, then 53^i*5^j is a member.
  If i == 9 (mod 18) and 6|j, then 13^i*37^j is a member.
  If i == 9 (mod 18) and 12|j, then 29^i*53^j is a member.
  If i == 18 (mod 36), j == 3 (mod 6) and k == 2 (mod 4), then 5^i*17^j*53^k is a member.
(End)
Composite numbers k such that for all primes p dividing k, p+1 divides k-1 and p-1 divides k+1 are the union of 2^2j and 3^2j, with j>0. - Paolo P. Lava, May 16 2019

			Prime factors of 74431 are 7 and 31 and (74431-1)/(7-1) = 12405, (74431-1)/(31-1) = 2481, (74431+1)/(7+1) = 9304, (74431+1)/(31+1) = 2326.

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 192 terms from Robert Israel)

Cf. A056729, A056824, A080062.

sol:=[]; m:=1; p:=[]; for u in [1..600000] do if not IsPrime(u) then p:=PrimeDivisors(u);  s:=0; for i in [1..#p] do if IsIntegral((u-1)/(p[i]-1)) and  IsIntegral((u+1)/(p[i]+1)) then  s:=s+1; end if; if s eq #p then sol[m]:=u; m:=m+1; end if; end for; end if; end for; sol; // Marius A. Burtea, May 16 2019

with(numtheory): P:=proc(q) local a,b,k,n,ok;
for n from 2 to q do if not isprime(n) then a:=factorset(n); ok:=1;
for k from 1 to nops(a) do if frac((n-1)/(a[k]-1))>0 or frac((n+1)/(a[k]+1))>0 then ok:=0; break; fi; od;
if ok=1 then print(n); fi; fi; od; end: P(10^6);

Select[Range[4, 2^19], Function[k, And[CompositeQ@ k, AllTrue[FactorInteger[k][[All, 1]], And[Mod[k - 1, # - 1] == 0, Mod[k + 1, # + 1] == 0] &]]]] (* Michael De Vlieger, May 22 2018 *)

lista(nn) = {forcomposite(c=1, nn, my(f = factor(c)); ok = 1; for (k=1, #f~, my(p = f[k,1]); if (((c-1) % (p-1)) || ((c+1) % (p+1)), ok = 0; break);); if (ok, print1(c, ", ")););} \\ Michel Marcus, May 19 2018

A079543 Numbers k such that k has at least two distinct prime factors and if a prime p divides k then (p-1) | (k-1) and (p+1) | (k+1).

Original entry on oeis.org

74431, 71528191, 125780831, 178708831, 4150390625, 68738591551, 171739186591, 429079903231, 634061169071

Offset: 1

Don Reble, Jan 22 2003

nonn

			a(1) = 74431 = 7^4 * 31 because 6 and 30 divide 74430 and 8 and 32 divide 74432.

Donovan Johnson, 1384 terms > a(9)

Intersection of A056729 and A080062, minus A001597.

Do[ f = Transpose[ FactorInteger[n]][[1]]; If[ Length[f] > 1 && Union[ Mod[n - 1, f - 1]] == {0} && Union[ Mod[n + 1, f + 1]] == {0}, Print[n]], {n, 6, 10^10}]

a(6)-a(7) from Donovan Johnson, Apr 09 2010

a(8)-a(9) from Donovan Johnson, Jan 21 2013

A227348 Nonsquarefree integers m such that, for prime p, if p^k | m then 1+p^k | 1+m.

Original entry on oeis.org

26999, 122499, 193599, 599975, 2206775, 2620175, 3501575, 4798079, 8278599, 11631059, 14242175, 16956575, 17578799, 19048799, 49061375, 55504175, 57354725, 70963775, 75271559, 107499699, 114930639, 153536525, 165887189, 202729175, 241430399, 248688719, 257552735, 258969887, 275089919

Offset: 1

Emmanuel Vantieghem, Jul 08 2013

nonn

The sequence is part of A056729.

All members are odd.

			26999 = 49*19*29 is in the list because 27000 is divisible by 8,50,20 and 30;
193599 = 9*49*439 is in the list because 193600 is divisible by 4,10,8, 50,440.

Cf. A056729.

PPDivs[m_Integer]:=Module[{f=FactorInteger[m]},Flatten[Table[First[f[[i]]]^Range[Last[f[[i]]]],{i,1,Length[f]}]]]; Select[Select[ Range[1000000], !SquareFreeQ[#]&], Union[ Mod[#+1, 1+PPDivs[#] ] ]== {0} &]

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A006972 Lucas-Carmichael numbers: squarefree composite numbers k such that p | k => p+1 | k+1.

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A080062 Composite numbers n such that for all primes p dividing n, p-1 divides n-1.

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A304291 Composite numbers k such that for all primes p dividing k, p-1 divides k-1 and p+1 divides k+1.

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A079543 Numbers k such that k has at least two distinct prime factors and if a prime p divides k then (p-1) | (k-1) and (p+1) | (k+1).

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A227348 Nonsquarefree integers m such that, for prime p, if p^k | m then 1+p^k | 1+m.

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