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A265827 Carmichael numbers n such that n-1 is not a practical number. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 6 00:51 EST 2021. Contains 349558 sequences. (Running on oeis4.)