Kyle D. Balliet - cp's OEIS frontend

A157676 Numbers n such that n + (product of digits of n) is prime.

1, 21, 23, 27, 29, 61, 67, 81, 83, 101, 103, 107, 109, 161, 163, 169, 233, 239, 253, 259, 283, 289, 293, 299, 307, 329, 341, 343, 347, 349, 361, 401, 409, 431, 437, 439, 441, 443, 449, 471, 473, 477, 493, 499, 503, 509, 529, 563, 569, 601, 607, 611, 613, 617

Offset: 1

Kyle D. Balliet, Mar 04 2009

			a(21) = 21 + (2)(1) = 23 (prime). a(67) = 67 + (6)(7) = 109 (prime). a(169) = 169 + (1)(6)(9) = 223 (prime).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A157677, A092518.

fQ[n_] := PrimeQ[n + Times @@ IntegerDigits@n]; Select[ Range@1000, fQ@# &] (* Robert G. Wilson v, May 04 2009 *)

dprod(n)=n=digits(n);prod(i=1,#n,n[i])
is(n)=isprime(dprod(n)+n) \\ Charles R Greathouse IV, Dec 27 2013

a(n) ~ n log n. - Charles R Greathouse IV, Dec 27 2013

More terms from Robert G. Wilson v, May 04 2009

A157678 Numbers k such that k + floor(average of digits of k) is prime.

Original entry on oeis.org

1, 12, 16, 27, 30, 34, 38, 41, 56, 63, 67, 70, 74, 89, 92, 96, 101, 102, 105, 107, 112, 125, 128, 130, 136, 146, 147, 154, 161, 164, 168, 175, 186, 188, 190, 193, 208, 210, 219, 226, 229, 231, 236, 237, 247, 254, 258, 265, 273, 276, 278, 280, 290, 305, 308, 309

Offset: 1

Kyle D. Balliet, Mar 04 2009

			n = 89 -> 89 + floor((8+9)/2) = 89 + 8 = 97 (prime).
n = 190 -> 190 + floor((1+9+0)/3) = 190 + 3 = 193 (prime).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A172367.

a := proc (n) local nn: nn := convert(n, base, 10): if isprime(n+floor(add(nn[j], j = 1 .. nops(nn))/nops(nn))) = true then n else end if end proc: seq(a(n), n = 1 .. 350); # Emeric Deutsch, Mar 07 2009

is(n)=isprime(sumdigits(n)\#digits(n)+n) \\ Charles R Greathouse IV, Dec 27 2013

Corrected and extended by Emeric Deutsch, Mar 07 2009

A157677 Primes p such that p + (product of digits of p) is also prime.

Original entry on oeis.org

23, 29, 61, 67, 83, 101, 103, 107, 109, 163, 233, 239, 283, 293, 307, 347, 349, 401, 409, 431, 439, 443, 449, 499, 503, 509, 563, 569, 601, 607, 613, 617, 619, 653, 659, 677, 683, 701, 709, 743, 809, 907, 929, 941, 1009, 1013, 1019, 1021, 1031, 1033, 1039

Offset: 1

Kyle D. Balliet, Mar 04 2009

If p contains a zero, then p is trivially a member.

			83 is prime, and 83 + 8*3 = 89 which is also prime. 103 is prime, and 103 + 1*0*3 = 103 is also prime. Thus 89 and 103 are members.

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

Union of A092518 and A056709.

Cf. A225303.

a := proc (n) local nn: nn := convert(ithprime(n), base, 10): if isprime(ithprime(n)+product(nn[j], j = 1 .. nops(nn))) = true then ithprime(n) else end if end proc: seq(a(n), n = 1 .. 180); # Emeric Deutsch, Mar 08 2009

Select[Prime[Range[175]], PrimeQ[# + Times @@ IntegerDigits[#]] &] (* Jayanta Basu, Apr 22 2013 *)

dprod(n)=n=digits(n); prod(i=1,#n,n[i])
is(n)=isprime(n) && isprime(n+dprod(n)) \\ Charles R Greathouse IV, Dec 27 2013

a(n) ~ n log n. - Charles R Greathouse IV, Apr 22 2013

More terms from Emeric Deutsch, Mar 08 2009

A157681 Fibonacci sequence beginning 29, 31.

Original entry on oeis.org

29, 31, 60, 91, 151, 242, 393, 635, 1028, 1663, 2691, 4354, 7045, 11399, 18444, 29843, 48287, 78130, 126417, 204547, 330964, 535511, 866475, 1401986, 2268461, 3670447, 5938908, 9609355, 15548263, 25157618, 40705881, 65863499, 106569380

Offset: 1

Kyle D. Balliet, Mar 04 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 1).

List([1..40], n -> 27*Fibonacci(n)+2*Fibonacci(n+1)); # G. C. Greubel, Nov 17 2018

[27*Fibonacci(n) + 2*Fibonacci(n+1): n in [1..40]]; // G. C. Greubel, Nov 17 2018

LinearRecurrence[{1,1},{29,31},40] (* Harvey P. Dale, Dec 05 2014 *)
Table[27*Fibonacci[n] +2*Fibonacci[n+1], {n, 1, 40}] (* G. C. Greubel, Nov 17 2018 *)

vector(40, n, 27*fibonacci(n) + 2*fibonacci(n+1)) \\ G. C. Greubel, Nov 17 2018

[27*fibonacci(n)+2*fibonacci(n+1) for n in (1..10)] # G. C. Greubel, Nov 17 2018

a(n) = a(n-1) + a(n-2), a(0)=29, a(1)=31.
From G. C. Greubel, Nov 17 2018: (Start)
a(n) = 27*Fibonacci(n) + 2*Fibonacci(n+1).
G.f.: x*(29+2*x)/(1-x-x^2). (End)

A157834 Numbers n such that 3n-2 and 3n+2 are both prime.

Original entry on oeis.org

3, 5, 7, 13, 15, 23, 27, 33, 35, 37, 43, 55, 65, 75, 77, 93, 103, 105, 117, 127, 133, 147, 153, 155, 163, 167, 205, 215, 225, 247, 253, 257, 275, 285, 287, 293, 295, 303, 313, 323, 337, 363, 365, 405, 427, 433, 435, 475, 477, 483, 495, 497, 517

Offset: 1

Kyle D. Balliet, Mar 07 2009

nonn

Barycenter of cousin primes (A029708; see also A029710, A023200, A046132), divided by 3. When p>3 and p+4 both are prime, then p = 1 (mod 6) and p+2 = 3 (mod 6). - M. F. Hasler, Jan 14 2013

			15*3 +/- 2 = 43,47 (both prime).

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

Intersection of A024893 and A153183.

[n: n in [1..1000]|IsPrime(3*n-2)and IsPrime(3*n+2)] // Vincenzo Librandi, Dec 13 2010

select(t -> isprime(3*t+2) and isprime(3*t-2), [seq(t,t=3..1000,2)]); # Robert Israel, May 28 2017

Select[Range[600],AllTrue[3#+{2,-2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 03 2019 *)

Intersection of A024893 and A153183.
a(n) = A029708(n)/3. - Zak Seidov, Aug 07 2009
a(n) = A056956(n)*2+1 = (A029710(n)+2)/3 = (A023200(n+1)+2)/3. - M. F. Hasler, Jan 14 2013

A157932 Numbers k such that (3^(35k) + 5^(21k) + 7^(15*k)) mod 105 is prime.

Original entry on oeis.org

0, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 52, 54, 56, 60, 64, 66, 68, 72, 76, 78, 80, 84, 88, 90, 92, 96, 100, 102, 104, 108, 112, 114, 116, 120, 124, 126, 128, 132, 136, 138, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 174

Offset: 1

Kyle D. Balliet, Mar 09 2009

Let b(k) = (3^(35*k) + 5^(21*k) + 7^(15*k)) mod 105, then sequence {b(k)} is 3, repeat (60, 68, 75, 17, 30, 23, 60, 47, 75, 38, 30, 2), with primes 3, 17, 23, 47, 2. First differences of {a(n)} are 4, 2, 2, 4, 4, 2, 2, 4, .... - Michel Marcus, Aug 15 2013
3^(35*k) + 5^(21*k) + 7^(15*k) = (4^k)*(3^k + 5^k + 7^k) mod 105, then by the division algorithm a simple proof yields that only numbers k of the form 24*m, 24*m+4, 24*m+6, 24*m+8, 24*m+12, 24*m+16, 24*m+18, 24*m+20 will be congruent to a prime modulo 105. Thus the pattern 4, 2, 2, 4, 4, 2, 2, ... will repeat infinitely. - Kyle D. Balliet, Jan 01 2014
Even numbers that can be written as the sum of 3 of their divisors, not necessarily distinct (see A355200). Also, numbers k of the form 12*m, 12*m+4, 12*m+6, 12*m+8. - Bernard Schott, Sep 08 2023

			a(4) = 3^(35*4) + 5^(21*4) + 7^(15*4) mod 105 = 17 (prime).

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

Equals {0} Union (A355200 \ A016945) <=> subsequence of even terms in A355200.

Select[Range[0,180],PrimeQ[Mod[3^(35#)+5^(21#)+7^(15#),105]]&] (* Harvey P. Dale, Oct 10 2017 *)

isok(n) = isprime((3^(35*n)+5^(21*n)+7^(15*n)) % 105); \\ Michel Marcus, Aug 15 2013

a(n)=n\4*12+[-4,0,4,6][n%4+1] \\ Charles R Greathouse IV, Dec 27 2013

is(n)=n%=12;n==0||n==4||n==6||n==8 \\ Charles R Greathouse IV, Dec 27 2013

a(n) = (-6-(-I)^n-I^n+6*n)/2 \\ Colin Barker, Oct 19 2015

concat(0, Vec(2*x^2*(2*x^2-x+2)/((x-1)^2*(x^2+1)) + O(x^100))) \\ Colin Barker, Oct 19 2015

3n - 4 <= a(n) <= 3n - 2. - Charles R Greathouse IV, Dec 27 2013
From Colin Barker, Oct 19 2015: (Start)
a(n) = (-6 - (-i)^n - i^n + 6*n)/2, where i = sqrt(-1).
G.f.: 2*x^2*(2*x^2-x+2) / ((x-1)^2*(x^2+1)). (End)

More terms from Michel Marcus, Aug 15 2013

A157833 Numbers k such that 1 + Sum_{j=0..k} (-1)^j*(k-j)! is prime.

Original entry on oeis.org

2, 3, 5, 7, 15, 19, 41, 59, 61, 105, 661, 2653, 3069, 3943, 4053, 8275

Offset: 1

Kyle D. Balliet, Mar 07 2009

Note that when k is even, 3 divides the sum.

The sums corresponding to terms >= 661 are only probable primes. - Klaus Brockhaus, Mar 13 2009

			a(2) = ((-1)^0)(2-0)! + ((-1)^1)(2-1)! + ((-1)^2)(2-2)! + 1 = 2! - 1! + 0! + 1 = 3 (prime).

a(7)-a(15) from Klaus Brockhaus, Mar 13 2009

a(16) from Michael S. Branicky, Aug 30 2024

A157829 Numbers k such that k! - (k-1)! + (k-2)! + 1 is prime.

Original entry on oeis.org

2, 5, 7, 11, 13, 16, 22, 29, 43, 56, 67, 83, 85, 107, 317, 325, 517, 1660, 2010, 2368, 4483, 4697, 14524

Offset: 1

Kyle D. Balliet, Mar 07 2009

			a(11) = 11! - 10! + 9! + 1 = 36650881 (prime).
a(16) = 16! - 15! + 14! + 1 = 19702293811201 (prime).

a := proc (n) if isprime(factorial(n-2)*((n-1)^2+1)+1) = true then n else end if end proc: seq(a(n), n = 2 .. 700); # Emeric Deutsch, Mar 18 2009

is(n)=ispseudoprime(1+(n^2-2*n+2)*(n-2)!) \\ Charles R Greathouse IV, Dec 27 2013

ABC2 $a!-($a-1)!+($a-2)!+1
a: from 2 to 10000
Charles R Greathouse IV, Dec 27 2013

More terms from Emeric Deutsch, Mar 18 2009
a(18)-a(22) from Charles R Greathouse IV, Dec 27 2013
a(23) from Michael S. Branicky, Jan 04 2025

A157202 Numbers k such that 66*k + 5 is prime.

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 9, 12, 13, 14, 16, 18, 19, 23, 26, 27, 36, 37, 39, 41, 42, 43, 44, 46, 51, 56, 57, 58, 64, 68, 71, 74, 76, 77, 78, 81, 82, 83, 88, 89, 91, 93, 98, 102, 103, 104, 106, 111, 114, 117, 118, 123, 127, 133, 134, 141, 142, 149, 152, 153, 154, 156, 158, 159, 161

Offset: 1

Kyle D. Balliet, Feb 25 2009

			23 is a term because 66*23 + 5 = 1523 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1001 [Offset shifted by _Georg Fischer_, Sep 27 2022]

Cf. A142052 (the corresponding primes).

[n: n in [0..500]|IsPrime(66*n+5)]; // Vincenzo Librandi, Dec 08 2010

Select[Range[0, 200], PrimeQ[66 # + 5] &] (* Vincenzo Librandi, Mar 20 2014 *)

is(n)=isprime(66*n+5) \\ Charles R Greathouse IV, Dec 27 2013

Offset changed to 1 by Georg Fischer, Sep 27 2022

A157194 Fibonacci sequence beginning 41, 43.

Original entry on oeis.org

41, 43, 84, 127, 211, 338, 549, 887, 1436, 2323, 3759, 6082, 9841, 15923, 25764, 41687, 67451, 109138, 176589, 285727, 462316, 748043, 1210359, 1958402, 3168761, 5127163, 8295924, 13423087, 21719011, 35142098, 56861109, 92003207, 148864316

Offset: 0

Kyle D. Balliet, Feb 24 2009

a(n) = a(n-1) + a(n-2), a(0)=41, a(1)=43.

cp's OEIS Frontend

User: Kyle D. Balliet

A157676 Numbers n such that n + (product of digits of n) is prime.

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A157678 Numbers k such that k + floor(average of digits of k) is prime.

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A157677 Primes p such that p + (product of digits of p) is also prime.

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A157681 Fibonacci sequence beginning 29, 31.

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A157834 Numbers n such that 3n-2 and 3n+2 are both prime.

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A157932 Numbers k such that (3^(35*k) + 5^(21*k) + 7^(15*k)) mod 105 is prime.

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A157932 Numbers k such that (3^(35k) + 5^(21k) + 7^(15*k)) mod 105 is prime.