A128252 T(n,m) is the smallest number that starts a sequence of n+1 consecutive integers whose Euler totient Functions are multiples of m.

0, 0, 3, 0, 3, 13, 0, 3, 26, 12, 0, 3, 35, 15, 61, 0, 3, 35, 32, 99, 13, 0, 3, 151, 32, 121, 26, 86, 0, 3, 151, 32, 121, 35, 637, 15, 0, 3, 151, 72, 3688, 35, 841, 15, 37, 0, 3, 727, 108, 5608, 151, 2694, 87, 216, 61, 0, 3, 1453, 108, 5697, 151, 66668, 87, 216, 99, 267

Offset

1,3

Comments

T(n,m) is the smallest 'a' such that all A000010(a+i), 0<=i<=n, are multiples of m. T(7,3)=151 because phi(151)=2*3*5, phi(152)=2^3*3^2, phi(153)=2^5*3 up to phi(158)=2*3*13 are all multiples of 3 and the numbers up to 150 do not start such a run of 8 elements. Table is read along antidiagonals.

References

Ho-Joo Lee and Gerald Myerson, Consecutive Integers whose totients are multiples of n, Solution to, American Mathematical Monthly 110:2 (2003), pp. 158-159.

Links

Table of n, a(n) for n=1..66.

Example

n\m.1.2....3...4.....5....6.......7...8.....9....10
--------------------------------------------------
1|..0.3...13..12....61...13......86..15....37....61.
2|..0.3...26..15....99...26.....637..15...216....99.
3|..0.3...35..32...121...35.....841..87...216...121.
4|..0.3...35..32...121...35....2694..87..1082...121.
5|..0.3..151..32..3688..151...66668.230..2916..3688.
6|..0.3..151..72..5608..151..168252.285..2916..5608.
7|..0.3..151.108..5697..151..168252.285..2916..5697.
8|..0.3..727.108.31800..727.1201204.403.37366.31800.
9|..0.3.1453.108.31800.1453.1201204.798.48505.31800

Maple

T := proc(n, m) local a, i, fail ; a :=0 ; while true do fail := false ; for i from 0 to n do if numtheory[phi](a+i) mod m <> 0 then fail := true ; break ; fi ; od ; if fail = false then RETURN(a) ; else a := a+1 ; fi ; od ; end: for d from 2 to 12 do for n from d-1 to 1 by -1 do printf("%d, ", T(n, d-n)) ; od ; od;

Mathematica

t[n_, m_] := Module[{a, i, fail}, a = 0; While[True, fail = False; For[i = 0, i <= n, i++, If[Mod[EulerPhi[a+i], m] != 0, fail = True; Break[]]]; If[fail == False, Return[a], a++]]]; Table[t[n-m+1, m], {n, 1, 11}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jan 10 2014, translated from Maple *)

Crossrefs

Sequence in context: A176005 A211963 A275080 * A230675 A327673 A279657

Adjacent sequences: A128249 A128250 A128251 * A128253 A128254 A128255

Keyword

nonn,tabl

Author

R. J. Mathar, May 03 2007

Status

approved