A106262 -id:A106262 - cp's OEIS frontend

A106264 Diagonal sums of number triangle A106262.

1, 0, 1, 2, 2, 4, 2, 4, 6, 7, 7, 8, 14, 10, 13, 12, 15, 19, 22, 22, 23, 25, 25, 44, 44, 40, 28, 50, 44, 54, 52, 55, 50, 66, 53, 72, 83, 80, 58, 73, 82, 110, 114, 123, 127, 113, 91, 112, 158, 137, 117, 122, 152, 135, 166, 160, 211, 206, 171, 219, 240, 201, 188, 194, 236

Offset: 0

Paul Barry, Apr 28 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A106262.

[(&+[2^(n-2*k) mod (k+2): k in [0..Floor(n/2)]]): n in [0..80]]; // G. C. Greubel, Jan 10 2023

Table[Sum[Mod[2^(n-2*k), k+2], {k,0,Floor[n/2]}], {n,0,80}] (* G. C. Greubel, Jan 10 2023 *)

def A106264(n): return sum( (2^(n-2*k)%(k+2)) for k in range(n//2+1) )
[A106264(n) for n in range(81)] # G. C. Greubel, Jan 10 2023

a(n) = Sum_{k=0..floor(n/2)} ( 2^(n-2*k) mod (k+2) ).

A106263 Row sums of number triangle A106262.

Original entry on oeis.org

1, 1, 3, 4, 5, 8, 12, 11, 16, 16, 28, 27, 36, 34, 45, 53, 60, 60, 66, 72, 86, 104, 111, 95, 119, 109, 161, 182, 185, 161, 175, 169, 199, 210, 249, 258, 310, 248, 340, 332, 356, 297, 370, 352, 424, 428, 465, 502, 500, 450, 547, 590, 613, 588, 630, 569, 595, 630

Offset: 0

Paul Barry, Apr 28 2005

a(n)=sum{k=0..n, (2^(n-k) mod k+2)}

A062173 a(n) = 2^(n-1) mod n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 4, 2, 1, 8, 1, 2, 4, 0, 1, 14, 1, 8, 4, 2, 1, 8, 16, 2, 13, 8, 1, 2, 1, 0, 4, 2, 9, 32, 1, 2, 4, 8, 1, 32, 1, 8, 31, 2, 1, 32, 15, 12, 4, 8, 1, 14, 49, 16, 4, 2, 1, 8, 1, 2, 4, 0, 16, 32, 1, 8, 4, 22, 1, 32, 1, 2, 34, 8, 9, 32, 1, 48, 40, 2, 1, 32, 16, 2, 4, 40, 1, 32, 64, 8, 4, 2, 54, 32, 1, 58, 58, 88, 1, 32, 1, 24, 46

Offset: 1

Henry Bottomley, Jun 12 2001

nonn

If p is an odd prime then a(p)=1. However, a(n) is also 1 for pseudoprimes to base 2 such as 341.

			a(5) = 2^(5-1) mod 5 = 16 mod 5 = 1.

Antti Karttunen, Table of n, a(n) for n = 1..101101 (first 1000 terms from Harry J. Smith)
Index entries for sequences related to pseudoprimes

Cf. A000079, A001567, A015910, A015919, A062172.
Cf. A082495, A106262, A257531, A305890.
Cf. A176997 (after the initial term, gives the positions of ones).

import Math.NumberTheory.Moduli (powerMod)
a062173 n = powerMod 2 (n - 1) n  -- Reinhard Zumkeller, Oct 17 2015

[Modexp(2,n-1,n): n in [1..110]]; // G. C. Greubel, Jan 11 2023

Array[Mod[2^(# - 1), #] &, 105] (* Michael De Vlieger, Jul 01 2018 *)
Array[PowerMod[2,#-1,#]&,120] (* Harvey P. Dale, May 17 2023 *)

A062173(n) = if(1==n, 0, lift(Mod(2, n)^(n-1))); \\ Antti Karttunen, Jul 01 2018

[power_mod(2,n-1,n) for n in range(1,110)] # G. C. Greubel, Jan 11 2023

a(n) = A106262(2*n-3, n-2). - G. C. Greubel, Jan 11 2023

More terms from Antti Karttunen, Jul 01 2018

A112983 a(n) = 2^(n+1) mod n.

Original entry on oeis.org

0, 0, 1, 0, 4, 2, 4, 0, 7, 8, 4, 8, 4, 8, 1, 0, 4, 2, 4, 12, 16, 8, 4, 8, 14, 8, 25, 4, 4, 8, 4, 0, 16, 8, 1, 20, 4, 8, 16, 32, 4, 2, 4, 32, 34, 8, 4, 32, 11, 48, 16, 32, 4, 2, 31, 8, 16, 8, 4, 32, 4, 8, 16, 0, 64, 62, 4, 32, 16, 18, 4, 56, 4, 8, 61, 32, 36, 50, 4

Offset: 1

Paul Barry, Oct 08 2005

			a(3) = 2^4 mod 3 = 16 mod 3 = 1.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A106262, A212844.

[Modexp(2,n+1,n): n in [1..120]]; // G. C. Greubel, Jan 11 2023

Table[PowerMod[2, n + 1, n], {n, 100}] (* T. D. Noe, Aug 13 2012 *)

print([2**(n+1) % n for n in range(1,77)])
# Alex Ratushnyak, Aug 12 2012

[power_mod(2,n+1,n) for n in range(1,120)] # G. C. Greubel, Jan 11 2023

a(n) = A106262(2*n-1, n-2). - G. C. Greubel, Jan 11 2023

Name, data and offset corrected by Alex Ratushnyak, Aug 12 2012

A213859 a(n) = 2^n mod (n+2).

Original entry on oeis.org

1, 2, 0, 3, 4, 4, 0, 2, 6, 6, 4, 7, 8, 2, 0, 9, 16, 10, 4, 2, 12, 12, 16, 8, 14, 20, 4, 15, 16, 16, 0, 2, 18, 22, 16, 19, 20, 2, 24, 21, 16, 22, 4, 38, 24, 24, 16, 32, 6, 2, 4, 27, 34, 52, 8, 2, 30, 30, 4, 31, 32, 2, 0, 8, 16, 34, 4, 2, 46, 36, 16, 37, 38, 17

Offset: 0

Alex Ratushnyak, Jun 22 2012

Conjectures:
Indices of zeros: 2^(x+2)-2, x >= 0.
There are infinitely many n's such that a(n)=n.
Every integer k >= 0 appears in a(n) at least once.
Every k >= 0 appears in a(n) infinitely many times.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A015910, A066606, A062173, A106262.

[Modexp(2,n,n+2): n in [0..120]]; // G. C. Greubel, Jan 11 2023

Table[PowerMod[2, n, n+2], {n, 0, 100}] (* T. D. Noe, Jun 26 2012 *)

print([2**n % (n+2) for n in range(222)])

[power_mod(2,n,n+2) for n in range(121)] # G. C. Greubel, Jan 11 2023

a(n) = 2^n mod (n+2).

a(n) = A106262(2*n, n). - G. C. Greubel, Jan 11 2023

cp's OEIS Frontend

A106264 Diagonal sums of number triangle A106262.

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A106263 Row sums of number triangle A106262.

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A062173 a(n) = 2^(n-1) mod n.

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A112983 a(n) = 2^(n+1) mod n.

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A213859 a(n) = 2^n mod (n+2).

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