A173172 Numbers n such that sopf(n) divides sopf(n+1) and sopf(n+1) divides sopf(n+2).

24, 225504, 944108, 10869375, 11506989, 12792675, 20962395, 25457760, 79509528, 89002914, 89460294, 146767704, 161064864, 180173147, 219487320, 235762488, 252508509, 419785344, 434887029, 453160511, 487179000, 545112792, 813133607

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:

Crossrefs

Sequence in context: A007240 A289029 A287964 * A061526 A159422 A028371

Adjacent sequences: A173169 A173170 A173171 * A173173 A173174 A173175

Keyword

nonn

Author

Michel Lagneau, Feb 11 2010

Extensions

a(12)-a(23) from Donovan Johnson, Feb 13 2010

Status

approved