A108497 -id:A108497 - cp's OEIS frontend

A108499 Number of values of k (1<=k<=n) where k^(n+1) = k mod n, or equivalently where sum_i{1<=i<=n} k^i = 0 mod n.

1, 2, 2, 3, 2, 6, 2, 5, 4, 6, 2, 9, 2, 6, 4, 9, 2, 14, 2, 15, 8, 6, 2, 15, 6, 6, 10, 9, 2, 18, 2, 17, 4, 6, 4, 21, 2, 6, 8, 25, 2, 42, 2, 9, 8, 6, 2, 27, 8, 22, 4, 15, 2, 38, 12, 15, 8, 6, 2, 45, 2, 6, 16, 33, 4, 18, 2, 15, 4, 18, 2, 35, 2, 6, 12, 9, 4, 42, 2, 45, 28, 6, 2, 63, 4, 6, 4, 15, 2, 42, 4

Offset: 1

Henry Bottomley, Jun 06 2005

nonn

			a(2)=2 since 1^3 = 1 mod 2 and 2^3 = 8 = 0 mod 2 = 2 mod 2.
a(3)=2 since 1^1+1^2+1^3 = 3 = 0 mod 3 and 3^1+3^2+3^3 = 39 = 0 mod 3 but 2^1+2^2+2^3 = 14 = 2 mod 3 != 0 mod 3.

Numbers of zeros in rows of A108497 or A108498.

a(n)=n-A108500(n). a(n)=n iff n is in A014117.

A108495 a(n) = (n^7 - n)/6.

Original entry on oeis.org

0, 0, 21, 364, 2730, 13020, 46655, 137256, 349524, 797160, 1666665, 3247860, 5971966, 10458084, 17568915, 28476560, 44739240, 68389776, 102036669, 148978620, 213333330, 300181420, 415726311, 567470904, 764411900, 1017252600

Offset: 0

Henry Bottomley, Jun 06 2005

Also integer sequences for (n^2-n)/1 (A002378 offset), (n^3-n)/2 (A027480 offset), (n^43-n)/42 (A108496) and (n^1807-n)/1806.

			a(2) = (2^7 - 2)/6 = 126/6 = 21.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Zagier, Problems posed at the St Andrews Colloquium, 1996.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A014117, A108497, A108499.

[(n^7-n)/6: n in [0..40]]; // Vincenzo Librandi, May 02 2011

Table[(n^7-n)/6,{n,0,30}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{0,0,21,364,2730,13020,46655,137256},30] (* Harvey P. Dale, Apr 16 2014 *)

[(n**7-n)//6 for n in range(41)] # David Radcliffe, Jun 06 2025

a(n) = (n-1)*A059721(n) = -A024004(n)*n/6.

G.f.: 7*x^2*(3 + 28*x + 58*x^2 + 28*x^3 + 3*x^4)/(1-x)^8. [Colin Barker, May 08 2012]

A108496 a(n) = (n^43 - n)/42.

Original entry on oeis.org

0, 0, 209430786243, 7815642080822311372, 1842172677508006361457030, 27068294695622864223661876860, 68747114771196346634599779308105, 51995580380757061883555053636996008

Offset: 0

Henry Bottomley, Jun 06 2005

nonn

Also integer sequences for (n^2-n)/1 (A002378 offset), (n^3-n)/2 (A027480 offset), (n^7-n)/6 (A108495) and (n^1807-n)/1806.

			a(2) = (2^43 - 2)/42 = 8796093022206/42 = 209430786243.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Zagier, Problems posed at the St Andrews Colloquium, 1996.

Cf. A014117, A108497, A108499.

[(n^43-n)/42: n in [0..20]]; // Vincenzo Librandi, May 02 2011

Table[(n^43-n)/42,{n,0,10}] (* Harvey P. Dale, Jan 31 2022 *)

a(n) = (n-1)*A108048(n).

A108498 Triangle read by rows: T(n,k) = sum_i{1<=i<=n} k^i mod n, showing 1<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 4, 5, 6, 0, 0, 6, 0, 4, 0, 2, 0, 0, 0, 5, 3, 0, 2, 6, 0, 8, 0, 0, 6, 2, 0, 0, 0, 6, 2, 0, 0, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 0, 6, 12, 6, 2, 0, 0, 0, 6, 12, 6, 2

Offset: 1

Henry Bottomley, Jun 06 2005

			Rows start: 0; 0,0; 0,2,0; 0,2,0,0; 0,2,3,4,0; 0,0,0,0,0,0; 0,2,3,4,5,6,0; etc.
T(7,3) = 3+9+27+81+243+729+2187 mod 7 = 3279 mod 7 = 3.

Cf. A014117, A059721, A108048, A108497, A108499.

T(n, k+n)=T(n, k). T(n, 0)=T(n, 1)=T(n, n)=T(1, k)=T(2, k)=T(6, k)=T(42, k)=T(1806, k)=0. T(p, k)=k for p prime and 1

A108500 Number of values of k (1<=k<=n) where k^(n+1) != k mod n, or equivalently where sum_i{1<=i<=n} k^i != 0 mod n.

Original entry on oeis.org

0, 0, 1, 1, 3, 0, 5, 3, 5, 4, 9, 3, 11, 8, 11, 7, 15, 4, 17, 5, 13, 16, 21, 9, 19, 20, 17, 19, 27, 12, 29, 15, 29, 28, 31, 15, 35, 32, 31, 15, 39, 0, 41, 35, 37, 40, 45, 21, 41, 28, 47, 37, 51, 16, 43, 41, 49, 52, 57, 15, 59, 56, 47, 31, 61, 48, 65, 53, 65, 52, 69, 37, 71, 68, 63, 67

Offset: 1

Henry Bottomley, Jun 06 2005

nonn

			a(2)=0 since 1^3 = 1 mod 2 and 2^3 = 8 = 0 mod 2 = 2 mod 2.
a(3)=1 since 2^1+2^2+2^3 = 14 = 2 mod 3 != 0 mod 3 but 1^1+1^2+1^3 = 3 = 0 mod 3 and 3^1+3^2+3^3 = 39 = 0 mod 3.

Numbers of nonzeros in rows of A108497 or A108498.

a(n)=n-A108499(n). a(n)=0 iff n is in A014117. a(p)=p-2 for p prime.

Showing 1-5 of 5 results.

cp's OEIS Frontend

A108499 Number of values of k (1<=k<=n) where k^(n+1) = k mod n, or equivalently where sum_i{1<=i<=n} k^i = 0 mod n.

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A108495 a(n) = (n^7 - n)/6.

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A108496 a(n) = (n^43 - n)/42.

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A108498 Triangle read by rows: T(n,k) = sum_i{1<=i<=n} k^i mod n, showing 1<=k<=n.

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A108500 Number of values of k (1<=k<=n) where k^(n+1) != k mod n, or equivalently where sum_i{1<=i<=n} k^i != 0 mod n.

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