A133910 -id:A133910 - cp's OEIS frontend

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

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.

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)).

A133879 n modulo 9 repeated 9 times.

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Oct 10 2007

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. A133889, A133880, A133890, A133900, A133910.

Rest[Flatten[Table[Table[Table[n,{9}],{n,0,8}],{3}],1]]//Flatten (* Harvey P. Dale, Apr 15 2018 *)

a(n)=(1+floor(n/9)) mod 9.
a(n)=1+floor(n/9)-9*floor((n+9)/81).
a(n)=(((n+9) mod 81)-(n mod 9))/9.
a(n)=((n+9-(n mod 9))/9) mod 9.
G.f. g(x)=(1-x^9)(1+2x^9+3x^18+4x^27+5x^36+6x^45+7x^56+8x^63)/((1-x)(1-x^81)).
G.f. g(x)=(1-x^9)*sum{0<=k<8, (k+1)*x^(9*k)}/((1-x)(1-x^81)).
G.f. g(x)=(8x^81-9x^72+1)/((1-x)(1-x^9)(1-x^81)).

A133905 Least composite number m such that binomial(n+m,m) mod m = 1.

Original entry on oeis.org

4, 9, 25, 10, 26, 9, 9, 9, 6, 4, 4, 34, 34, 85, 289, 4, 4, 57, 87, 8, 8, 25, 25, 25, 134, 4, 4, 15, 15, 111, 111, 4, 4, 8, 8, 10, 10, 121, 121, 82, 86, 4, 4, 49, 49, 49, 49, 4, 4, 265, 68, 10, 10, 8, 8, 6, 9, 4, 4, 194, 194, 469, 249, 4, 4, 44, 44, 146, 146, 16, 16, 6, 6, 4, 4, 162

Offset: 1

Hieronymus Fischer, Oct 20 2007

nonn

			a(1)=4, since binomial(1+4,4) mod 4 = 5 mod 4 = 1 and 4 is the minimal composite number with this property.
a(5)=26 because of binomial(5+26,26)=169911=6535*26+1, but binomial(5+k,k) mod k<>1 for all composite numbers <26.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

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

lcn[n_]:=Module[{m=4},While[PrimeQ[m]||Mod[Binomial[n+m,m],m]!=1,m++];m]; Array[lcn,80] (* Harvey P. Dale, May 13 2022 *)

a(n) = { my(m = 4, ok = 0); until (ok, if (! isprime(m) && (binomial(n+m, m) % m == 1), ok = 1, m++);); return (m);} \\ Michel Marcus, Jul 15 2013

A134335 Numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer, but not a prime.

Original entry on oeis.org

15, 35, 39, 42, 50, 51, 55, 65, 77, 78, 87, 91, 92, 95, 110, 111, 114, 115, 119, 123, 140, 141, 143, 155, 159, 161, 164, 170, 183, 185, 186, 187, 189, 201, 203, 204, 209, 215, 219, 221, 222, 225, 230, 235, 236, 242, 247, 258, 259, 264, 267, 284, 285, 287, 290

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

			a(1) = 15, since 15 = 3*5 and (3+5)/2 = 4 is not prime.
a(5) = 50, since 50 = 2*5*5 and (2+5+5)/3 = 4 is not prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A100118, A046363, A133620, A133621.

Cf. A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334.

fp[{a_,b_}]:=a*b;s={};Do[If[q=Total[fp/@FactorInteger[n]]/Total[Last/@FactorInteger[n]];IntegerQ[q]&&!PrimeQ[q],AppendTo[s,n]],{n,2,290}];s (* James C. McMahon, Apr 05 2025 *)

Definition clarified by the author, May 06 2013

A133881 Even numbers k such that binomial(k+p,k) mod k = 1, where p=10.

Original entry on oeis.org

4, 68, 164, 260, 292, 356, 388, 452, 484, 516, 548, 676, 708, 772, 836, 932, 964, 1028, 1060, 1124, 1156, 1252, 1348, 1412, 1444, 1508, 1572, 1604, 1636, 1732, 1796, 1828, 1892, 1924, 2084, 2116, 2244, 2276, 2308, 2372, 2404, 2468, 2564, 2596, 2692, 2756

Offset: 1

Hieronymus Fischer, Oct 16 2007

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A133620, A133621-A133625, A133630, A038509, A133634-A133636.

Cf. A133880, A133890, A133900, A133910.

Select[2*Range[1500],Mod[Binomial[10+#,#],#]==1&] (* Harvey P. Dale, Jun 05 2023 *)

A133894 Numbers m such that binomial(m+4,m) mod 4 = 0.

Original entry on oeis.org

12, 13, 14, 15, 28, 29, 30, 31, 44, 45, 46, 47, 60, 61, 62, 63, 76, 77, 78, 79, 92, 93, 94, 95, 108, 109, 110, 111, 124, 125, 126, 127, 140, 141, 142, 143, 156, 157, 158, 159, 172, 173, 174, 175, 188, 189, 190, 191, 204, 205, 206, 207, 220, 221, 222, 223, 236, 237

Offset: 1

Hieronymus Fischer, Oct 20 2007

Also numbers m such that floor(1+(m/4)) mod 4 = 0.
Partial sums of the sequence 12,1,1,1,13,1,1,1,13, ... which has period 4.
Numbers congruent to {12,13,14,15} mod 16. Numbers n such that n xor 12 = n - 12. [Brad Clardy, May 06 2013]

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

Cf. A000040, A133620, A133621, A133623, A133630, A133635, A133874, A133884, A133890, A133900, A133910.

a(n)=(n+3)\4*16-[1,4,3,2][n%4+1] \\ Charles R Greathouse IV, May 06 2013

a(n) = 4*n + 12 - 3*(n mod 4).
G.f.: 12/(1-x)+x(1+x+x^2+13x^3)/((1-x^4)(1-x)) = (12+x+x^2+x^3+x^4)/((1-x^4)(1-x)) = (12-11x-x^5)/((1-x^4)(1-x)^2).
a(n) = 4*n+3*((1-i)*i^n+(1+i)*(-i)^n+(-1)^n+5)/2, where i=sqrt(-1). - Bruno Berselli, Apr 08 2011

A133908 Least odd prime number m such that binomial(n+m,m) mod m = 1.

Original entry on oeis.org

3, 3, 5, 5, 7, 7, 11, 11, 3, 3, 3, 13, 17, 17, 17, 17, 19, 3, 3, 3, 23, 23, 29, 29, 5, 5, 3, 3, 3, 31, 37, 37, 37, 37, 37, 3, 3, 3, 41, 41, 43, 43, 47, 47, 3, 3, 3, 53, 7, 5, 5, 5, 5, 3, 3, 3, 59, 59, 61, 61, 67, 67, 3, 3, 3, 67, 71, 71, 71, 71, 73, 3, 3, 3, 5, 5, 5, 5, 5, 83, 3, 3, 3, 89, 89

Offset: 1

Hieronymus Fischer, Oct 20 2007

nonn

Also the least odd prime number m such that m divides floor(n/m) or m>n.

			a(3)=5, since binomial(3+5,5) mod 5 = 56 mod 5 = 1 and 5 is the minimal odd prime number with this property.
a(8)=11 because of binomial(8+11,11)=75582=6871*11+1, but binomial(8+k,k) mod k<>1 for all odd primes <11.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

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

With[{oprs=Rest[Prime[Range[100]]]},Flatten[Table[Select[oprs,Mod[ Binomial[ n+#,#],#]==1&,1],{n,90}]]] (* Harvey P. Dale, Jun 27 2012 *)

cp's OEIS Frontend

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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A133878 n modulo 8 repeated 8 times.

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A133879 n modulo 9 repeated 9 times.

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A133905 Least composite number m such that binomial(n+m,m) mod m = 1.

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PARI

A134335 Numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer, but not a prime.

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Extensions

A133881 Even numbers k such that binomial(k+p,k) mod k = 1, where p=10.

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A133894 Numbers m such that binomial(m+4,m) mod 4 = 0.