A002064 -id:A002064 - cp's OEIS frontend

A367009 Number of divisors of n*2^n + 1.

1, 2, 3, 3, 4, 4, 8, 8, 4, 4, 12, 4, 8, 6, 8, 8, 4, 12, 16, 16, 16, 4, 16, 16, 16, 16, 24, 16, 6, 8, 48, 48, 8, 4, 32, 4, 12, 8, 12, 8, 8, 16, 16, 32, 16, 16, 32, 32, 16, 10, 64, 8, 64, 8, 32, 16, 12, 32, 32, 16, 16, 128, 8, 8, 96, 32, 256, 24, 16, 32, 32, 48

Offset: 0

Sean A. Irvine, Oct 31 2023

nonn

The numbers n*2^n+1 are called Cullen numbers.

Amiram Eldar, Table of n, a(n) for n = 0..865

Cf. A002064, A000005, A366898, A367007, A367008.

Table[DivisorSigma[0, n*2^n + 1], {n, 0, 100}] (* Amiram Eldar, Jan 06 2024 *)

a(n) = numdiv(n*2^n + 1); \\ Amiram Eldar, Jan 06 2024

a(n) = sigma0(n*2^n + 1) = A000005(A002064(n)).

A369324 Array read by ascending antidiagonals: A(n,k) is the number of words of length n on an alphabet [k], avoiding 120 and 210, and sortable by a stack of depth 2, where k >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 9, 4, 1, 0, 1, 16, 25, 16, 5, 1, 0, 1, 32, 65, 56, 25, 6, 1, 0, 1, 64, 161, 176, 105, 36, 7, 1, 0, 1, 128, 385, 512, 385, 176, 49, 8, 1, 0, 1, 256, 897, 1408, 1281, 736, 273, 64, 9, 1, 0, 1, 512, 2049, 3712, 3969, 2752, 1281, 400, 81, 10, 1

Offset: 0

Stefano Spezia, Jan 20 2024

			The array begins:
  0, 1,  1,   1,   1,    1, ...
  0, 1,  2,   3,   4,    5, ...
  0, 1,  4,   9,  16,   25, ...
  0, 1,  8,  25,  56,  105, ...
  0, 1, 16,  65, 176,  385, ...
  0, 1, 32, 161, 512, 1281, ...
  ...

Toufik Mansour, Howard Skogman, and Rebecca Smith, Sorting inversion sequences, arXiv:2401.06662 [math.CO], 2024. See Theorem 3.18 at page 10.

Cf. A000004 (k=0), A000012 (k=1), A000079 (k=2), A002064 (k=3), A340257 (k=4).
Cf. A000290 (n=2), A001477 (n=1), A057427 (n=0), A131423 (n=3), A164039.
Cf. A000035, A369325 (main diagonal), A369326.

A[n_,k_]:=(1-(-1)^k)/2+2^n Sum[Binomial[n+k-3-2i,n-1],{i,0,Floor[(k-2)/2]}]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

A(n,k) = A000035(k) + 2^n*Sum_{i=0..floor((k-2)/2)} binomial(n + k - 3 - 2*i, n - 1).

Sum_{k=0..n} A(n-k,k) = A164039(n-1).

A373099 Last digit of n*2^n + 1.

Original entry on oeis.org

1, 3, 9, 5, 5, 1, 5, 7, 9, 9, 1, 9, 3, 7, 7, 1, 7, 5, 3, 3, 1, 3, 9, 5, 5, 1, 5, 7, 9, 9, 1, 9, 3, 7, 7, 1, 7, 5, 3, 3, 1, 3, 9, 5, 5, 1, 5, 7, 9, 9, 1, 9, 3, 7, 7, 1, 7, 5, 3, 3, 1, 3, 9, 5, 5, 1, 5, 7, 9, 9, 1, 9, 3, 7, 7, 1, 7, 5, 3, 3

Offset: 0

Javier Rodríguez Ríos, May 23 2024

This is a cyclic sequence of 20 numbers, using only 1,3,5,7 and 9 (4 times each).

James Cullen, Question 15897, Educational Times, Vol. 58 (December 1905), p. 534.
Richard K. Guy (2004), Unsolved Problems in Number Theory (3rd ed.), New York: Springer Verlag, pp. section B20, ISBN 0-387-20860-7.

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

Cf. A010879, A002064.

Cf. A373098, A373100.

lastDigit := proc(n)
    return (n * 2^n + 1) mod 10;
end proc:
# Example usage
minN := 1; maxN := 10;
lastDigits := [seq(lastDigit(n), n = minN .. maxN)];
print(lastDigits);

lastDigit[n_] := Mod[n * 2^n + 1, 10]
(* Example usage *)
minN = 1; maxN = 10;
lastDigits = Table[lastDigit[n], {n, minN, maxN}]
Print[lastDigits]

PARI
```
a(n) = lift(Mod(n*2^n + 1, 10))
```

def last_digit(n):
    return (n * 2**n + 1) % 10
# Example usage
min_n, max_n = 1, 10
last_digits = [last_digit(n) for n in range(min_n, max_n + 1)]
print(last_digits)

a(n) = A010879(A002064(n)).
From Chai Wah Wu, Jul 06 2024: (Start)
a(n) = a(n-2) - a(n-4) + a(n-5) + a(n-6) - a(n-7) - a(n-8) + a(n-9) - a(n-11) + a(n-13) for n > 12.
G.f.: (-3*x^12 - 3*x^11 - 2*x^10 - 4*x^9 + x^8 - 5*x^6 + 2*x^5 + 3*x^4 - 2*x^3 - 8*x^2 - 3*x - 1)/((x - 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^6 + x^4 - x^2 + 1)). (End)

A383473 Integers k such that d*2^k + 1 is prime for some divisor of k.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 15, 16, 18, 25, 30, 36, 51, 55, 63, 66, 69, 75, 81, 85, 134, 141, 162, 189, 201, 209, 220, 245, 276, 324, 408, 438, 446, 456, 534, 616, 675, 693, 726, 892, 900, 1305, 1326, 1494, 1824, 2208, 2394, 2766, 2826, 3024, 3168, 3189, 3690, 3703, 3880, 3912, 3927, 4410, 4543, 4713

Offset: 1

Juri-Stepan Gerasimov, Apr 27 2025

nonn

			6 is in the sequence a term because 3*2^6 + 1 = 193 prime for divisor 3 of k = 6.

Supersequence of A005849.

Cf. A002064, A027750, A382811.

[k: k in [1..900] | not #[d: d in Divisors(k) | IsPrime(d*2^k+1)] eq 0];

q[k_] := AnyTrue[Divisors[k], PrimeQ[# * 2^k +1] &]; Select[Range[4000], q] (* Amiram Eldar, Apr 28 2025 *)

isok(k) = fordiv(k, d, if (ispseudoprime(d*2^k+1), return(1))); return(0); \\ Michel Marcus, Apr 28 2025

A111049 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 9, 4, 1, 11, 27, 25, 8, 1, 20, 70, 100, 65, 16, 1, 37, 170, 330, 325, 161, 32, 1, 70, 399, 980, 1295, 966, 385, 64, 1, 135, 917, 2723, 4515, 4501, 2695, 897, 128, 1, 264, 2076, 7224, 14406, 17976, 14364, 7176, 2049, 256

Offset: 0

Philippe Deléham, Oct 07 2005

			Rows begin:
  1;
  1,   1;
  1,   3,    2;
  1,   6,    9,    4;
  1,  11,   27,   25,     8;
  1,  20,   70,  100,    65,    16;
  1,  37,  170,  330,   325,   161,    32;
  1,  70,  399,  980,  1295,   966,   385,   64;
  1, 135,  917, 2723,  4515,  4501,  2695,  897,  128;
  1, 264, 2076, 7224, 14406, 17976, 14364, 7176, 2049, 256;

Cf. A002064, A006127.

With[{m = 9}, CoefficientList[CoefficientList[Series[(1 - 2*x - 2*x*y + x^2 *y + x^2*y^2)/(1 - 3*x - 3*x*y + 2*x^2 + 4*x^2*y + 2*x^2*y^2), {x, 0 , m}, {y, 0, m} ], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)

T(n, k) = if (k<=n, 2^(n-1)*binomial(n-1, k-1)+binomial(n-1, k));
matrix(10, 10, n, k, T(n-1, k-1)) \\ to see the triangle \\ Michel Marcus, Feb 17 2020

T(n, k) = 2^(n-1)binomial(n-1, k-1) + binomial(n-1, k).
Sum_{k=0..n} T(n, k) = 2^(n-1)*(1+2^(n-1)) = A063376(n-1) for n >= 1.
From Peter Bala, Mar 20 2013: (Start)
O.g.f.: (1 - 2*t + x*t*(t-2) + x^2*t^2)/((1 - t*(1+x))*(1 - 2*t*(1+x))) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2 + ....
E.g.f.: (x + 2*exp((1+x)*t) + x*exp(2*t*(1+x)))/(2*(1+x)) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2/2! + ....
Recurrence equation: for n >= 1, T(n+1,k) = 2*T(n,k) + 2*T(n,k-1) - binomial(n,k). (End)
From Philippe Deléham, Oct 18 2013: (Start)
G.f.: (1 - 2*x - 2*x*y + x^2*y + x^2*y^2)/(1 - 3*x - 3*x*y + 2*x^2 + 4*x^2*y + 2*x^2*y^2).
T(n,k) = 3*T(n-1,k) + 3*T(n-1,k-1) - 2*T(n-2,k) - 4*T(n-2,k-1) - 2*T(n-2,k-2), T(0,0) = T(1,1) = T(1,0) = T(2,0) = 1, T(2,1) = 3, T(2,2) = 2, T(n,k) = 0 if k > n or if k < 0. (End)

Wrong a(42) removed by Georg Fischer, Feb 17 2020

A131837 Multiplicative persistence of Cullen numbers.

Original entry on oeis.org

0, 0, 0, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 3, 2, 1, 2, 2, 2, 1, 1, 3, 2, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 20 2007

After the 111th term, all the numbers have some digits equal to zero, thus the persistence is equal to 1.

			Cullen number 65 --> 6*5=30 --> 3*0=0 thus persistence is 2.

Cf. A002064, A031346, A131840.

P:=proc(n) local i,k,w,ok,cont; for i from 0 by 1 to n do w:=1; k:=i*2^i+1; ok:=1; if k<10 then print(0); else cont:=1; while ok=1 do while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w<10 then ok:=0; print(cont); else cont:=cont+1; k:=w; w:=1; fi; od; fi; od; end: P(120);

Table[cn=n*2^n+1;Length[NestWhileList[Times@@IntegerDigits[#]&, cn, #>=10&]], {n, 0, 104}]-1 (* James C. McMahon, Mar 01 2025 *)

a(n) = A031346(A002064(n)). - Michel Marcus, Mar 01 2025

A131840 Additive persistence of Cullen numbers.

Original entry on oeis.org

0, 0, 1, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 2, 2, 3, 2, 3, 3, 3, 3, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 20 2007

			Cullen number 385 --> 3+8+5=16 -->1+6=7 thus persistence is 2

Cf. A002064, A031286, A131837.

with(numtheory): with(combinat): P:=proc(n) local a,t;t:=0; a:=n*2^n+1; while a>9 do t:=t+1; a:=convert(convert(a,base,10),`+`); od; t;
end: seq(P(i),i=1..10^2);

f[n_] := Length@ NestWhileList[Plus @@ IntegerDigits@# &, n*2^n + 1, UnsameQ@## &, All] - 2; Array[f, 88] (* James C. McMahon, Mar 01 2025 *)

a(n) = A031286(A002064(n)). - James C. McMahon, Mar 01 2025

Corrected entries and Maple code by Paolo P. Lava, Dec 19 2017

A137716 Number of digits in the decimal expansion of the n-th Cullen prime.

Original entry on oeis.org

1, 45, 1423, 1749, 1994, 5573, 9726, 9779, 17964, 27347, 79002, 108761, 145072, 407850, 1905090, 2010852

Offset: 1

Ant King, Feb 09 2008

Cullen primes are prime numbers of the form k*2^k+1. This sequence is complete for all values of n up to 3500000.

			As the sixth Cullen prime, 18496*2^18496 + 1 = 1.311...*10^5572, is a 5573-digit number, we have a(6) = 5573.

Ray Ballinger and Mark Rodenkirch, Cullen Primes: Definition and Status.
James Cullen, Question 15897, The Educational Times, Dec. 1905, p. 534.
Allan Cunningham and H. J. Woodall, Factorisation of Q=(2^q ± q) and (q*2^q ± 1), The Messenger of Mathematics, Vol. 47 (1917-18), pp. 1-38.
Wilfrid Keller, New Cullen Primes, Mathematics of Computation, Vol. 64, No. 212 (Ocober 1995), pp. 1733-1741.

Cf. A055642, A005849, A002064.

a(n) = A055642(A050920(n)). [Corrected by Georg Fischer, Nov 18 2023]

a(15)-a(16) from Amiram Eldar, Oct 27 2024

A143038 Triangle read by rows, A000012 * A134309 * A000012, where A134309 = an infinite lower triangular matrix with (1, 1, 2, 4, 8, 16, ...) in the main diagonal and the rest zeros.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 6, 7, 8, 8, 12, 14, 15, 16, 16, 24, 28, 30, 31, 32, 32, 48, 56, 60, 62, 63, 64, 64, 96, 112, 120, 124, 126, 127, 128, 128, 192, 224, 240, 248, 252, 254, 255, 256, 256, 384, 448, 480, 496, 504, 508, 510, 511, 512

Offset: 0

Gary W. Adamson, Jul 18 2008

Row sums = A002064, the Cullen numbers (1, 3, 9, 25, 65, 161, 385, ...).

			First few rows of the triangle:
   1;
   1,  2;
   2,  3,  4;
   4,  6 , 7,  8;
   8, 12, 14, 15, 16;
  16, 24, 28, 30, 21, 32;
  32, 48, 56, 60, 62, 63, 64;
  ...

Cf. A134309, A002064.

Cf. A002262, A003056, A023532.

a(n) = 2^i - 2^(i-1-j)*[jA003056(n), j = A002262(n). - Yuchun Ji, May 15 2020

A146179 Digit sums of Cullen numbers.

Original entry on oeis.org

1, 3, 9, 7, 11, 8, 16, 24, 15, 19, 8, 20, 22, 27, 30, 22, 32, 23, 37, 39, 27, 25, 47, 53, 34, 42, 42, 55, 53, 56, 31, 63, 57, 58, 50, 50, 55, 57, 54, 52, 47, 71, 79, 87, 69, 82, 62, 65, 76, 90, 66, 94, 77, 77, 82, 75, 72, 79, 92, 89, 88, 69, 87, 91, 71, 92, 103, 117, 93, 103

Offset: 1

Parthasarathy Nambi, Oct 27 2008

			The sum of all the digits of the Cullen number 49153 is 22.

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

Cf. A002064.

A002064 := proc(n) n*2^n+1; end: A007953 := proc(n) if n < 10 then n; else add ( d, d=convert(n,base,10) ) ; fi; end: A146179 := proc(n) A007953(A002064(n)) ; end: seq(A146179(n),n=0..80) ; # R. J. Mathar, Jul 08 2009

Table[Total[IntegerDigits[2^n n+1]],{n,0,70}] (* Harvey P. Dale, Jun 10 2013 *)

More terms from R. J. Mathar, Jul 08 2009

cp's OEIS Frontend

A367009 Number of divisors of n*2^n + 1.

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A369324 Array read by ascending antidiagonals: A(n,k) is the number of words of length n on an alphabet [k], avoiding 120 and 210, and sortable by a stack of depth 2, where k >= 0.

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A373099 Last digit of n*2^n + 1.

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A383473 Integers k such that d*2^k + 1 is prime for some divisor of k.

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A111049 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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A131837 Multiplicative persistence of Cullen numbers.

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A131840 Additive persistence of Cullen numbers.

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Maple

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Extensions

A137716 Number of digits in the decimal expansion of the n-th Cullen prime.