A059995 -id:A059995 - cp's OEIS frontend

A327440 a(n) = floor(3*n/10).

0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23

Offset: 0

Bruno Berselli, Sep 11 2019

The sequence can be obtained from A008585 by deleting the last digit of each term.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A008585.

Similar sequences with the formula floor(k*n/10): A059995 (k=1); A002266 (k=2); A057354 (k=4); A004526 (k=5); A057355 (k=6); A188511 (k=7); A090223 (k=8).

[div(3*n, 10) for n in 0:80] |> println

Table[Floor[3 n/10], {n, 0, 80}]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3}, 80]

PARI
```
vector(80, n, n--; floor(3*n/10))
```

O.g.f.: x^4*(1 + x^3 + x^6)/((1 + x)*(1 - x)^2*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) = (x^4 + x^7 + x^10)/(1 - x - x^10 + x^11).

a(n) = a(n-1) + a(n-10) - a(n-11) for n > 10.

A329809 Numbers k such that floor(k/10)^(k mod 10) contains the digit (k mod 10).

Original entry on oeis.org

11, 26, 37, 39, 46, 52, 55, 56, 57, 59, 66, 67, 69, 73, 74, 76, 78, 84, 86, 87, 95, 97, 99, 101, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 128, 129, 131, 136, 137, 138, 139, 141, 144, 145, 146, 148, 151, 152, 153, 155, 156, 157, 158, 159, 161, 162, 165, 166, 169, 171, 172, 173, 175

Offset: 1

Eric Angelini and M. F. Hasler, Nov 21 2019

Or: Numbers k such that A059995(k)^A010879(k) contains the last digit of k, A010879(k).

It's easy to see that all numbers ending in {11, 37, 46, 52, 55, 59, 66, 69, 73, 97, 99} are in the sequence: for these, A059995(k)^A010879(k) mod 100 = (1, 87, 96, 25, 25, 25, 56, 96, 43, 69, 89).

M. F. Hasler, in reply to Éric Angelini, 2019 must be read here as 201^9, SeqFan list, Nov. 21, 2019.

Cf. A059995 (floor(n/10): drop final digit), A010879 (n mod 10; final digit of n).

select( t->setsearch(Set(digits((t\10)^(t%10))),t%10),[0..9999])

A342112 Drop the final digit of n^5.

Original entry on oeis.org

0, 0, 3, 24, 102, 312, 777, 1680, 3276, 5904, 10000, 16105, 24883, 37129, 53782, 75937, 104857, 141985, 188956, 247609, 320000, 408410, 515363, 643634, 796262, 976562, 1188137, 1434890, 1721036, 2051114, 2430000, 2862915, 3355443, 3913539, 4543542, 5252187, 6046617

Offset: 0

Stefano Spezia, Feb 28 2021

Why exponent 5? Because it is the smallest nontrivial exponent e such that for an integer k not ending in 0, 1, 5 and 6, k^e has the same unit digit of k in base 10.

Index entries for linear recurrences with constant coefficients, signature (5,-10, 10,-5,1,0,0,0,0,1,-5,10,-10,5,-1).
Index entries for sequences related to final digits of numbers.

Cf. A000584, A010879, A016813, A056865, A059995, A061167, A164938.

Table[(n^5-Last[IntegerDigits[n]])/10,{n,0,36}]

G.f.: x^2*(3 + 9*x + 12*x^2 + 12*x^3 + 12*x^4 + 12*x^5 + 12*x^6 + 12*x^7 + 13*x^8 + 8*x^9 + 15*x^10 - x^11 + x^12)/((1 - x)^6*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9)).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + a(n-10) - 5*a(n-11) + 10*a(n-12) - 10*a(n-13) + 5*a(n-14) - a(n-15) for n > 14.
a(n) = floor(n^5/10).
a(n) = (A000584(n) - A010879(n))/10.
a(n) = A164938(n) + A059995(n).

cp's OEIS Frontend

A327440 a(n) = floor(3*n/10).

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A329809 Numbers k such that floor(k/10)^(k mod 10) contains the digit (k mod 10).

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A342112 Drop the final digit of n^5.

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