A265827 Carmichael numbers n such that n-1 is not a practical number.

63973, 126217, 748657, 4463641, 6054985, 9890881, 10837321, 19384289, 26921089, 31146661, 34901461, 41471521, 53711113, 76595761, 140241361, 144218341, 150846961, 151530401, 169057801, 171454321, 171679561, 172430401, 228842209, 277241401, 280761481

Offset

1,1

Comments

Most Carmichael numbers (odd) have practical number companions (even) that are 1 less. This sequence identifies the exceptions. However Carmichael numbers appear to have no practical number companions that are 1 more.

Links

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

Example

a(1)=63973=7*13*19*37. It is a Carmichael number because its prime factors less 1 are 6, 12, 18, 36 and this set are all divisors of 63973-1=2^2*3^2*1777 that is not a practical number. It is the first such occurrence.

Mathematica

CarmichaelQ[n_] := If[Mod[n, CarmichaelLambda[n]]==1&&!PrimeQ[n], True, False]; PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; Select[2Range[10^7]+1, CarmichaelQ[#]&&!PracticalQ[#-1] &]

Prog

(PARI) is_c(n) = {my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1}
is_p(n) = bittest(n, 0) && return(n==1); my(P=1); n && !for(i=2, #n=factor(n)~, n[1, i]>1+(P*=sigma(n[1, i-1]^n[2, i-1])) && return);
forcomposite(n=561, 1e10, if(is_c(n) && !is_p(n-1), print1(n, ", "))) \\ Altug Alkan, Dec 16 2015

Crossrefs

Cf. A002997, A261614.

Sequence in context: A236873 A236608 A214758 * A212882 A290793 A182518

Adjacent sequences: A265824 A265825 A265826 * A265828 A265829 A265830

Keyword

nonn

Author

Frank M Jackson, Dec 16 2015

Extensions

More terms from Altug Alkan, Dec 16 2015

Status

approved