A133900 -id:A133900 - cp's OEIS frontend

A135127 Numbers such that the digital sums in bases 2, 3, 5 and 7 all are equal.

0, 1, 882, 883, 1386, 1387, 2502, 2503, 3453, 7555, 7652, 7665, 7931, 9751, 10101, 12250, 12251, 16893, 17010, 17011, 17515, 17550, 17551, 18285, 20301, 22050, 22051, 24406, 24407, 25053, 27503, 31654, 40930, 40931, 41951, 50878, 50879

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=882, since ds_2(882 )=ds_3(882 )=ds_5(882 )=ds_7(882 )=6, where ds_x=digital sum base x.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308, A212222.

Select[Range[0, 32000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 7]] &] (* G. C. Greubel, Sep 27 2016 *)
Select[Range[0,51000],Length[Union[Total/@IntegerDigits[#,{2,3,5,7}]]] == 1&] (* Harvey P. Dale, Sep 18 2019 *)

Added 0, Stanislav Sykora, May 06 2012

A133907 Least prime number p such that binomial(n+p, p) mod p = 1.

Original entry on oeis.org

2, 3, 5, 2, 2, 7, 11, 2, 2, 3, 3, 2, 2, 17, 17, 2, 2, 3, 3, 2, 2, 23, 29, 2, 2, 5, 3, 2, 2, 31, 37, 2, 2, 37, 37, 2, 2, 3, 41, 2, 2, 43, 47, 2, 2, 3, 3, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 59, 61, 2, 2, 67, 3, 2, 2, 67, 71, 2, 2, 71, 73, 2, 2, 3, 5, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 89, 89, 2, 2, 3, 3, 2, 2, 97

Offset: 1

Hieronymus Fischer, Oct 20 2007

nonn

Also the least prime number p such that p divides floor(n/p) or p > n.
a(n) = 2 if and only if n is in A042948. - Robert Israel, May 11 2017
Conjecture: a(n) is the smallest prime p such that Sum_{k=1..n} k^(p-1) == n (mod p). Thus a(n) >= A317358(n). - Thomas Ordowski, Jul 29 2018

			a(2)=3, since binomial(2+3,3) mod 3 = 10 mod 3 = 1 and 3 is the minimal prime number with this property.
a(7)=11 because of binomial(7+11, 11) = 31824 = 2893*11 + 1, but binomial(7+k, k) mod k <> 1 for all primes < 11.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A042948, A133620, A133621, A133623, A133630, A133635.

Cf. A133872, A133880, A133890, A133900, A133906, A133910, A317358.

f:= proc(n) local m;
  m:= 2:
  while floor(n/m) mod m <> 0 do m:= nextprime(m) od:
  m
end proc:
map(f, [$1..100]); # Robert Israel, May 11 2017

a[n_] := Module[{p}, For[p = 2, True, p = NextPrime[p], If[Mod[Binomial[n+p, p], p] == 1, Return[p]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 05 2023 *)

a(n) = my(p=2); while (binomial(n+p, p) % p != 1, p = nextprime(p+1)); p; \\ Michel Marcus, Dec 17 2022

from sympy import nextprime, ff
def A133907(n):
    p, m = 2, (n+2)*(n+1)>>1
    while m%p != 1:
        q = nextprime(p)
        m = m*ff(n+q,q-p)//ff(q,q-p)
        p = q
    return p # Chai Wah Wu, Feb 22 2023

A133624 Binomial(n+p, n) mod n, where p=4.

Original entry on oeis.org

0, 1, 2, 2, 1, 0, 1, 7, 4, 1, 1, 8, 1, 8, 6, 13, 1, 7, 1, 6, 8, 12, 1, 3, 1, 1, 10, 8, 1, 26, 1, 25, 12, 1, 1, 22, 1, 20, 14, 31, 1, 15, 1, 12, 16, 24, 1, 5, 1, 1, 18, 14, 1, 46, 1, 43, 20, 1, 1, 36, 1, 32, 22, 49, 1, 23, 1, 18, 24, 36, 1, 7, 1, 1, 26, 20, 1, 66, 1, 61, 28, 1, 1, 50, 1, 44, 30

Offset: 1

Hieronymus Fischer, Sep 30 2007

nonn

Let d(m)...d(2)d(1)d(0) be the base-n representation of n+p. The relation a(n)=d(1) holds, if n is a prime index. For this reason there are infinitely many terms which are equal to 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636, A133874, A133884, A133880, A133890, A133900, A133910, A362686, A362687, A362688, A362689.

[Binomial(n+4,4) mod n: n in [1..100]]; // Vincenzo Librandi, Apr 27 2014

Table[Mod[Binomial[n+4,n],n],{n,90}] (* Harvey P. Dale, Apr 26 2014 *)

a(n) = binomial(n+4,4) mod n.
a(n)=1 if n is a prime > 4, since binomial(n+4,n) == (1+floor(4/n))(mod n), provided n is a prime.
From Chai Wah Wu, May 26 2016: (Start)
a(n) = (n^4 + 10*n^3 + 11*n^2 + 2*n + 24)/24 mod n.
For n > 6:
if n mod 24 == 0, then a(n) = n/12 + 1.
if n mod 24 is in {1, 2, 5, 7, 10, 11, 13, 17, 19, 23}, then a(n) = 1.
if n mod 24 is in {3, 9, 15, 18, 21}, then a(n) = n/3 + 1.
if n mod 24 is in {4, 20}, then a(n) = n/4 + 1.
if n mod 24 == 6, then a(n) = 5*n/6 + 1.
if n mod 24 is in {8, 16}, then a(n) = 3*n/4 + 1.
if n mod 24 == 12, then a(n) = 7*n/12 + 1.
if n mod 24 is in {14, 22}, then a(n) = n/2 + 1.
(End)
For n > 54, a(n) = 2*a(n-24) - a(n-48). - Ray Chandler, Apr 23 2023

A133876 n modulo 6 repeated 6 times.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 0, 0, 0

Offset: 0

Hieronymus Fischer, Oct 10 2007

nonn

Periodic with length 6^2=36.

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

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133886, A133880, A133890, A133900, A133910.

Table[PadRight[{},6,Mod[n,6]],{n,20}]//Flatten (* Harvey P. Dale, Nov 15 2023 *)

a(n)=(1+floor(n/6)) mod 6.
a(n)=1+floor(n/6)-6*floor((n+6)/36).
a(n)=(((n+6) mod 36)-(n mod 6))/6.
a(n)=((n+6-(n mod 6))/6) mod 6.
G.f. g(x)=(1-x^6)(1+2x^5+3x^12+4x^18+5x^24)/((1-x)(1-x^36)).
G.f. g(x)=(5x^36-6x^30+1)/((1-x)(1-x^6)(1-x^36)).

A133877 n modulo 7 repeated 7 times.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Hieronymus Fischer, Oct 10 2007

nonn

Periodic with length 7^2=49.

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

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133887, A133880, A133890, A133900, A133910.

a(n)=(1+floor(n/7)) mod 7.
a(n)=1+floor(n/7)-7*floor((n+7)/49).
a(n)=(((n+7) mod 49)-(n mod 7))/7.
a(n)=((n+7-(n mod 7))/7) mod 7.
a(n)=binomial(n+7,n) mod 7 =binomial(n+7,7) mod 7.
G.f. g(x)=(1-x^7)(1+2x^7+3x^14+4x^21+5x^28+6x^35)/((1-x)(1-x^49)).
G.f. g(x)=(6x^49-7x^42+1)/((1-x)(1-x^7)(1-x^49)).

A133887 Binomial(n+7,n) mod 7^2.

Original entry on oeis.org

1, 8, 36, 22, 36, 8, 1, 2, 16, 23, 44, 23, 16, 2, 3, 24, 10, 17, 10, 24, 3, 4, 32, 46, 39, 46, 32, 4, 5, 40, 33, 12, 33, 40, 5, 6, 48, 20, 34, 20, 48, 6, 7, 7, 7, 7, 7, 7, 7, 8, 15, 43, 29, 43, 15, 8, 9, 23, 30, 2, 30, 23, 9, 10, 31, 17, 24, 17, 31, 10, 11, 39, 4, 46, 4, 39, 11, 12, 47

Offset: 0

Hieronymus Fischer, Oct 10 2007

nonn

Periodic with length 7^3=343.

Ray Chandler, Table of n, a(n) for n = 0..1500
Index entries for linear recurrences with constant coefficients, order 337.

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133877, A133880, A133890, A133900, A133910.
For the sequence regarding binomial(n+7, n) mod 7 see A133877.

Table[Mod[Binomial[n+7,n],49],{n,0,80}] (* Harvey P. Dale, Apr 08 2018 *)

a(n)=binomial(n+7,7) mod 7^2.

G.f. g(x)=sum{0<=k<343, a(k)*x^k}/(1-x^343).

A133888 Binomial(n+8,n) mod 8.

Original entry on oeis.org

1, 1, 5, 5, 7, 7, 3, 3, 6, 6, 6, 6, 2, 2, 2, 2, 7, 7, 3, 3, 1, 1, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 1, 1, 3, 3, 7, 7, 2, 2, 2, 2, 6, 6, 6, 6, 3, 3, 7, 7, 5, 5, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 5, 7, 7, 3, 3, 6, 6, 6, 6, 2, 2, 2, 2, 7, 7, 3, 3, 1, 1, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 1, 1, 3, 3, 7, 7, 2

Offset: 0

Hieronymus Fischer, Oct 10 2007

nonn

Periodic with length 8^2=64.

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

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133878, A133880, A133890, A133900, A133910.

Table[Mod[Binomial[n+8,n],8],{n,0,110}] (* Harvey P. Dale, Aug 08 2011 *)

a(n)=binomial(n+8,8) mod 8.

A133889 a(n) = binomial(n+9,n) mod 9.

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 1, 1, 1, 2, 2, 2, 8, 8, 8, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 7, 7, 7, 4, 4, 4, 5, 5, 5, 2, 2, 2, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 1, 1, 1, 7, 7, 7, 8, 8, 8, 5, 5, 5, 8, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 4, 4, 4, 1, 1, 1, 2, 2, 2, 8, 8, 8, 2, 2, 2, 3, 3, 3, 3, 3, 3

Offset: 0

Hieronymus Fischer, Oct 10 2007

nonn

Periodic with length 9^2 = 81.

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

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133879, A133880, A133890, A133900, A133910.

Table[Mod[Binomial[n+9,n],9],{n,0,110}] (* Harvey P. Dale, Jan 14 2012 *)

a(n) = binomial(n+9,9) mod 9.

A135122 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 4 all are equal.

Original entry on oeis.org

1, 21, 261, 273, 17748, 17749, 20820, 20821, 65620, 65621, 70740, 70741, 83268, 83269, 86292, 86293, 1066068, 1066069, 1070420, 1135701, 1135893, 1135953, 5326161, 5330001, 5330241, 5330260, 5330261, 5506389, 5525829, 5526801, 5571909, 5574933, 5592321

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=21, since ds_2(21)=ds_3(21)=ds_10(21)=3, where ds_x=digital sum base x.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Select[Range[5600000],Length[Union[Table[Total[IntegerDigits[#,n]],{n,2,4}]]]==1&] (* Harvey P. Dale, Aug 14 2013 *)

isok(n) = my(sd2=sumdigits(n, 2)); (sd2==sumdigits(n, 3)) && (sd2==sumdigits(n, 4)); \\ Michel Marcus, Aug 08 2018

a(31)-a(33) from Giovanni Resta, Aug 06 2018

A133878 n modulo 8 repeated 8 times.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6

Offset: 0

Hieronymus Fischer, Oct 10 2007

nonn

Periodic with length 8^2=64.

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

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133888, A133880, A133890, A133900, A133910.

Flatten[Join[Table[PadRight[{},8,n],{n,7}],Table[PadRight[{},8,n],{n,0,7}]]] (* Harvey P. Dale, Nov 06 2011 *)

a(n)=(1+floor(n/8)) mod 8.
a(n)=1+floor(n/8)-8*floor((n+8)/64).
a(n)=(((n+8) mod 64)-(n mod 8))/8.
a(n)=((n+8-(n mod 8))/8) mod 8.
G.f. g(x)=(1-x^8)(1+2x^8+3x^16+4x^24+5x^32+6x^40+7x^48)/((1-x)(1-x^64)).
G.f. g(x)=(1-x^8)*sum{0<=k<7, (k+1)*x^(8*k)}/((1-x)(1-x^64)).
G.f. g(x)=(7x^64-8x^56+1)/((1-x)(1-x^8)(1-x^64)).

cp's OEIS Frontend

A135127 Numbers such that the digital sums in bases 2, 3, 5 and 7 all are equal.

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A133907 Least prime number p such that binomial(n+p, p) mod p = 1.

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Maple

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A133624 Binomial(n+p, n) mod n, where p=4.

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Magma

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A133876 n modulo 6 repeated 6 times.

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A133877 n modulo 7 repeated 7 times.

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A133887 Binomial(n+7,n) mod 7^2.

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A133888 Binomial(n+8,n) mod 8.

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A133889 a(n) = binomial(n+9,n) mod 9.

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