OFFSET
1,1
COMMENTS
The sum of the distinct primes dividing n (without repetition) is sometimes called sopf(n).
(A008472(a(n)+1) mod A008472(a(n)) = 0) and (A008472(a(n)+2) mod A008472(a(n)+1) = 0). [From Reinhard Zumkeller, Mar 12 2010]
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 225504, p. 264, Ellipses, Paris 2008.
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..100
W. Sierpinski, Number Of Divisors And Their Sum
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
EXAMPLE
sopf(24) = 5, sopf(25) = 5, sopf(26) = 15; and 5 divides 5, 5 and divides 15.
sopf(225504) = 34, sopf(225505) = 408, sopf(225506) = 2448; and 34 divides 408, and 408 divides 2448.
MAPLE
with(numtheory): for n from 2 to 100000000 do : t1 := ifactors(n)[2] : t2 := sum(t1[i][1], i=1..nops(t1)) : tt1 :=ifactors(n+1)[2] :tt2 := sum(tt1[i][1], i=1..nops(tt1)): ttt1:=ifactors(n+2)[2] : ttt2 := sum(ttt1[i][1], i=1..nops(ttt1)): a:= tt2/t2 ; aa:=floor(a) ; b := ttt2/tt2:bb:=floor(b): if a=aa and b=bb then print (n): else fi: od:
MATHEMATICA
Position[Partition[Table[Total[FactorInteger[n][[;; , 1]]], {n, 13*10^6}], 3, 1], _?(Mod[#[[2]], #[[1]]]==Mod[#[[3]], #[[2]]]==0&)]//Quiet//Flatten (* The program generates the first six terms of the sequence. *) (* Harvey P. Dale, Nov 20 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 11 2010
EXTENSIONS
a(12)-a(23) from Donovan Johnson, Feb 13 2010
STATUS
approved