A253213 -id:A253213 - cp's OEIS frontend

A253208 a(n) = 4^n + 3.

4, 7, 19, 67, 259, 1027, 4099, 16387, 65539, 262147, 1048579, 4194307, 16777219, 67108867, 268435459, 1073741827, 4294967299, 17179869187, 68719476739, 274877906947, 1099511627779, 4398046511107, 17592186044419, 70368744177667, 281474976710659

Offset: 0

Vincenzo Librandi, Dec 29 2014

Subsequence of A226807.

Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A141725, A226807.

Cf. Numbers of the form k^n+k-1: A000057 (k=2), A168607 (k=3), this sequence (k=4), A242329 (k=5), A253209 (k=6), A253210 (k=7), A253211 (k=8), A253212 (k=9), A253213 (k=10).

Magma
```
[4^n+3: n in [0..30]];
```

Table[4^n + 3, {n, 0, 30}] (* or *) CoefficientList[Series[(4 - 13 x) / ((1 - x) (1 - 4 x)), {x, 0, 40}], x]

a(n)=4^n+3 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (4 - 13*x)/((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
From Elmo R. Oliveira, Nov 14 2023: (Start)
a(n) = 4*a(n-1) - 9 with a(0) = 4.
E.g.f.: exp(4*x) + 3*exp(x). (End)

A348487 Positive numbers whose square starts and ends with exactly one 1.

Original entry on oeis.org

1, 11, 39, 41, 101, 111, 119, 121, 129, 131, 139, 141, 319, 321, 329, 331, 349, 351, 359, 361, 369, 371, 379, 381, 389, 391, 399, 401, 409, 411, 419, 421, 429, 431, 439, 441, 1001, 1009, 1011, 1019, 1021, 1029, 1031, 1039, 1041, 1099, 1101, 1109, 1111, 1119, 1121, 1129, 1131, 1139

Offset: 1

Bernard Schott, Oct 21 2021

When a square ends with 1, this square ends with exactly one 1.

Sequences A000533 and A253213 show that there are an infinity of terms. The square of their terms, for n >= 3, starts and ends with exactly one 1. Also, the numbers 119, 1119, 11119, ..., ((10^k + 71) / 9)^2, (k >= 3) are terms. The squares ((10^k + 71) / 9)^2, have the last digit 1 and because 12*10^(2*k - 3) < ((10^k + 71) / 9)^2 <13*10^(2*k - 3), for k >= 3, the squares ((10^k + 71) / 9)^2, k >= 4, start with 12. - Marius A. Burtea, Oct 21 2021

			39 is a term since 39^2 = 1521.
109 is not a term since 109^2 = 11881.
119 is a term since 119^2 = 14161.

Cf. A045855, A090771, A253213, A273372 (squares ending with 1), A017281, A017377.
Cf. A000533, A253213 for n >= 2 (subsequences).
Subsequence of A305719.

[1] cat [n:n in [2..1200]|Intseq(n*n)[1] eq 1 and Intseq(n*n)[#Intseq(n*n)] eq 1 and Intseq(n*n)[-1+#Intseq(n*n)] ne 1]; // Marius A. Burtea, Oct 21 2021

Join[{1}, Select[Range[11, 1200], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 1 && d[[2]] != 1 &]] (* Amiram Eldar, Oct 21 2021 *)

isok(k) = my(d=digits(sqr(k))); (d[1]==1) && (d[#d]==1) && if (#d>2, (d[2]!=1) && (d[#d-1]!=1), 1); \\ Michel Marcus, Oct 21 2021

from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("1")) == len(s.lstrip("1")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [1, 9]))
  return [k for k in r if ok(k)]
print(aupto(1140)) # Michael S. Branicky, Oct 21 2021

A320771 Primes p for which p-1 and p+1 are Niven numbers.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 41, 71, 101, 109, 113, 151, 191, 199, 223, 229, 307, 401, 409, 449, 593, 701, 881, 911, 1009, 1013, 1091, 1129, 1231, 1301, 1303, 1451, 1559, 1811, 1999, 2029, 2089, 2213, 2281, 2311, 2351, 2399, 2531, 2609, 2711, 2753, 3037, 3079, 3109, 3221, 3251, 3329

Offset: 1

Marius A. Burtea, Oct 21 2018

All of the prime numbers in sequences A253213, A199684, A199687 are part of the sequence.

			For p = 11, p-1 = 10 and p + 1 = 12. 10 is divisible by 1 = 1 + 0, 12 is divisible by 3 = 1 + 2. Thus, p = 11 is a term.
For p = 229, p-1 = 228 and p + 1 = 230. 228 is divisible by 12 = 2 + 2 + 8, and 230 is divisible by 5 = 2 + 3 + 0. Thus, p = 229 is a term.

Marius A. Burtea, Table of n, a(n) for n = 1..11188

Cf. A000040, A005349, A007953, A253213, A177506, A199684, A199687.

Filtered([2..2400],p->IsPrime(p) and (p-1) mod List(List([1..p-1],ListOfDigits),Sum)[p-1]=0 and (p+1) mod List(List([1..p+1],ListOfDigits),Sum)[p+1]=0); # Muniru A Asiru, Oct 29 2018

[p: p in PrimesUpTo(2000) | IsIntegral((p-1)/&+Intseq(p-1)) and IsIntegral((p+1)/&+Intseq(p+1))]; // Marius A. Burtea, Jan 06 2019

nivenQ[n_] := Divisible[n, Total[IntegerDigits[n]]]; Select[Range[10000], PrimeQ[#] && nivenQ[#-1] && nivenQ[#+1] &] (* Amiram Eldar, Oct 31 2018 *)
nnQ[p_]:=Divisible[p,Total[IntegerDigits[p]]]; Select[Prime[Range[500]],AllTrue[#+{1,-1},nnQ]&] (* Harvey P. Dale, Jul 19 2023 *)

isniven(n) = frac(n/sumdigits(n)) == 0;
isok(p) = isprime(p) && isniven(p-1) && isniven(p+1); \\ Michel Marcus, Oct 22 2018

cp's OEIS Frontend

A253208 a(n) = 4^n + 3.

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A348487 Positive numbers whose square starts and ends with exactly one 1.

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Author

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Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Python

A320771 Primes p for which p-1 and p+1 are Niven numbers.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI