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A381015
a(n) = n + (number of trailing 0's of n).
0
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 71, 72, 73, 74, 75, 76, 77
OFFSET
1,2
COMMENTS
Constant congruence speed of (10^n + 1)^n, i.e., a(n) = A373387((10^n + 1)^n).
Since 10^n + 1 is never a perfect power by Catalan's conjecture (Mihăilescu's theorem), it follows that if 10 does not divide n, then (10^n + 1)^n is exactly an n-th perfect power with a constant congruence speed of a(n) = n.
Moreover, for any positive integer n, the congruence speed of (10^n + 1)^n equals 2*a(n) at height 1 and then becomes stable at height 2.
LINKS
FORMULA
a(n) = n + A122840(n).
a(n) = A373387(A121520(n)).
EXAMPLE
a(10) = 11 since A373387((10^10 + 1)^10) = 11.
MATHEMATICA
a[n_]:=n+IntegerExponent[n, 10]; Array[a, 77] (* Stefano Spezia, Feb 13 2025 *)
PROG
(PARI) a(n) = n + valuation(n, 10); \\ Michel Marcus, Feb 13 2025
CROSSREFS
KEYWORD
nonn,base,easy,new
AUTHOR
Marco Ripà, Feb 11 2025
STATUS
approved