Mikk Heidemaa - cp's OEIS frontend

A371631 Primes whose product of nonzero digits divided by the sum of its digits is also prime.

167, 257, 523, 541, 617, 761, 1447, 1607, 1861, 2053, 2251, 2503, 2521, 2851, 4051, 5023, 5281, 5821, 6701, 8161, 8521, 10067, 10607, 10861, 11273, 11471, 12713, 13127, 13217, 13721, 14407, 16007, 17123, 17231, 17321, 18061, 20507, 20521, 21247, 21317, 21713, 22051

Offset: 1

Mikk Heidemaa, May 24 2024

No term N can have a "9" digit. [Proof: The sum of the digits of N is not a multiple of 3, but the numerator would be a multiple of 9, and so the number would be a multiple of 9, so not a prime.]

			167 (prime) is a term because 1*6*7/(1+6+7)=42/14=3 (prime).

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

Subsequence of A038367.

Equals prime terms of A138566.

pQ[n_] := Block[{idp = DeleteCases[IntegerDigits[n], 0]}, PrimeQ[Times @@ idp/Total@ idp]]; Cases[Prime@ Range@ PrimePi[10^5], _?pQ]
Select[Prime[Range[2500]],PrimeQ[Times@@(IntegerDigits[#]/.(0->1))/Total[ IntegerDigits[ #]]]&] (* Harvey P. Dale, Sep 24 2024 *)

A359630 Primes p such that 10^p+3 or 10^p+9 is also prime.

Original entry on oeis.org

2, 3, 5, 11, 17, 101, 107, 26927, 48109

Offset: 1

Mikk Heidemaa, Jan 08 2023

Union of the terms which are prime in A049054 and in A088275.

If it exists, a(10) > 2*10^5 (according to the comment at A088275).

			3 is a term since it is prime and so is 10^3 + 9 = 1009.
11 is a term since it is prime and 10^11 + 3 = 100000000003 is also a prime.

Cf. A049054, A088275.

Block[{p}, ParallelDo[p := Prime @ i; If[(PrimeQ[10^p + 3] || PrimeQ[10^p + 9]), Print @ p], {i, PrimePi @ 48109}, Method -> "FinestGrained"]]
Select[Prime[Range[5000]],AnyTrue[10^#+{3,9},PrimeQ]&] (* Harvey P. Dale, Feb 09 2025 *)

A357915 Concatenation of the decimal digits of {n, 1..n}.

Original entry on oeis.org

11, 212, 3123, 41234, 512345, 6123456, 71234567, 812345678, 9123456789, 1012345678910, 111234567891011, 12123456789101112, 1312345678910111213, 141234567891011121314, 15123456789101112131415, 1612345678910111213141516

Offset: 1

Mikk Heidemaa, Jan 18 2023

Concatenation of the consecutive integers 1..n, with n prepended (n > 0).
The terms a(1), a(7), a(31), and a(337) are primes (of the form n1...n).
a(3643) is a 13469-digit probable prime (the number of digits is also a prime).
These indices 7, 31, 337, 3643 are themselves primes of the form 6m+1.
For the known terms a(n) which are primes and for a(3643), a(n) == 2 (mod 3).
There is no other prime term for n < 15000 (and no prime term with prime index n < 25000).

			a(2) = 212 since it is the concatenation of the consecutive positive integers <= 2, with 2 prepended.

Winston de Greef, Table of n, a(n) for n = 1..367
G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 71234567

Cf. A007908, A078257, A172495.

aUpTo[n_] := Table[ FromDigits @ Flatten @ IntegerDigits @ {i, Range @ i}, {i,n}]; aUpTo[999]

a(n) = my(s=Str(n)); for(k=1, n, s=Str(s, k)); eval(s); \\ Michel Marcus, Jan 20 2023

def a(n): return int(str(n)+"".join(map(str, range(1, n+1))))
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Jan 20 2023

a(n) = concat(n, A007908(n)).

A359406 Integers k such that the concatenation of k consecutive primes starting at 31 is prime.

Original entry on oeis.org

1, 2, 3, 23, 43, 141

Offset: 1

Mikk Heidemaa, Dec 30 2022

The corresponding primes (p) known (31, 3137, 313741, ...) have an even number of digits and p (mod 10) == 1|7. For those at a(1)...a(6), p (mod 3) == p (mod 5) holds.
a(7): 3472 corresponds to a 15968-digit probable prime (certification in progress).
For a(8), k > 15000 (if it exists).
a(8) > 30000. - Tyler Busby, Feb 13 2023

			2 is a term because the consecutive primes 31 and 37 concatenated to 3137 yield another prime.

Cf. A069151, A309191, A030996, A280894.

UpToK[k_] := Block[{a := FromDigits @ Flatten @ IntegerDigits @ Join[{}, Prime @ Range[11, i]]}, Reap[ Do[ If[ PrimeQ[a], Sow[i - 10], Sow[Nothing]], {i, k}]]][[2, 1]]; UpToK[3500] (* or *)
UpToK[k_] := Flatten @ Parallelize @ MapIndexed[ If[ PrimeQ[#1], #2, Nothing] &, DeleteCases[ FromDigits /@ Flatten /@ IntegerDigits @ Prime @ Range[11, Range[k]], 0]]; UpToK[3500]

A343834 Primes with digits in nondecreasing order, only primes, and with sum of digits also a prime.

Original entry on oeis.org

2, 3, 5, 7, 23, 223, 227, 337, 557, 577, 2333, 2357, 2377, 2557, 2777, 33377, 222337, 222557, 233357, 233777, 235577, 2222333, 2233337, 2235557, 3337777, 3355777, 5555777, 22222223, 22233577, 23333357, 23377777, 25577777, 222222227, 222222557, 222222577

Offset: 1

Mikk Heidemaa, May 01 2021

Intersection of A028864 and A062088.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A019546, A028864, A046704, A062088.

a[p_] := With[{dg = IntegerDigits@p}, PrimeQ@p && OrderedQ@dg && AllTrue[dg, PrimeQ] && PrimeQ@ Total@dg]; Cases[ Range[3*10^7], _?(a@# &)] (* or *)
upToDigitLen[k_] := Cases[ FromDigits@# & /@ Select[ Flatten[ Table[ Tuples[{2, 3, 5, 7}, {i}], {i, k}], 1], OrderedQ[#] &], _?(PrimeQ@# && PrimeQ@ Total@ IntegerDigits@# &)]; upToDigitLen[10]

from sympy import isprime
from sympy.utilities.iterables import multiset_combinations
def aupton(terms):
  n, digits, alst = 0, 1, []
  while len(alst) < terms:
    mcstr = "".join(d*digits for d in "2357")
    for mc in multiset_combinations(mcstr, digits):
      sd = sum(int(d) for d in mc)
      if not isprime(sd): continue
      t = int("".join(mc))
      if isprime(t): alst.append(t)
      if len(alst) == terms: break
    else: digits += 1
  return alst
print(aupton(35)) # Michael S. Branicky, May 01 2021

a(33) and beyond from Michael S. Branicky, May 01 2021

A343768 Primes p such that p^2 + 1 divides 3^(p+1) - 1.

Original entry on oeis.org

3, 5, 47, 1091, 172681

Offset: 1

Mikk Heidemaa, Apr 28 2021

a(n) != 13 (mod 20) and a(n) != 17 (mod 20). Proof: Write n=20*k + 13, and then show that n^2 + 1 == 0 (mod 5) and 3^(n+1) - 1 == 3 (mod 5). Because of its factor 5 the former cannot divide the latter. - Martin Ehrenstein, Jun 06 2021

a(6) > 10^14. - Martin Ehrenstein, Jun 18 2021

Cf. A001220, A014127.

a[n_Integer] := Select[ Prime@ Range@n, PowerMod[3, # + 1, #^2 + 1] == 1 &]; a[2*10^5] (* or *)
a[n_Integer] := If[ PrimeQ@n && Divisible[3^(n + 1) - 1, n^2 + 1] , Print@n]; Do[ a[n], {n, 2*10^5}]

from sympy import primerange
def afind(limit):
  for p in primerange(1, limit+1):
    if pow(3, p+1, p**2+1) == 1: print(p, end=", ")
afind(10**6) # Michael S. Branicky, May 03 2021

A331363 Pairs of coordinates of the corners in a counterclockwise triangular spiral.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, -2, -1, 3, -1, 0, 2, -4, -2, 5, -2, 0, 3, -6, -3, 7, -3, 0, 4, -8, -4, 9, -4, 0, 5, -10, -5, 11, -5, 0, 6, -12, -6, 13, -6, 0, 7, -14, -7, 15, -7, 0, 8, -16, -8, 17, -8, 0, 9, -18, -9, 19, -9, 0, 10, -20, -10, 21, -10, 0, 11

Offset: 1

Mikk Heidemaa, May 03 2020

Odd n yields the x- and even n the y-coordinates (i.e., x- and y-coordinates alternate in the sequence).

			X- and y-coordinates of the corners alternate in the sequence: 0, 0, 1, 0, 0, 1, -2,-1, 3, -1, ...
                      (0,4)
                     .     \
                    .       \
                   .         \
                      (0,3)   \
                     /     \   \
                    /       \   \
                   /         \   \
                  /   (0,2)   \   \
                 /   /     \   \   \
                /   /       \   \   \
               /   /         \   \   \
              /   /   (0,1)   \   \   \
             /   /   /     \   \   \   \
            /   /   /       \   \   \   \
           /   /   /         \   \   \   \
          /   /   /   (0,0)->(1,0)\   \   \
         /   /   /                 \   \   \
        /   /   /                   \   \   \
       /   /  (-2,-1)------------->(3,-1)\   \
      /   /                               \   \
     /   /                                 \   \
    /  (-4,-2)--------------------------->(5,-2)\
   /                                             \
  /                                               \
(-6,-3)------------------------------------------>(7,-3)

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Cf. A329116, A329972.

a[n_] := Ceiling[1/18*n*(Mod[2 - n, 6] + 4*Mod[n, -3] + 1)]; Table[ a[n], {n, 66}] (* or *)
LinearRecurrence[{0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -1}, {0, 0, 1, 0, 0, 1, -2, -1, 3, -1, 0, 2}, 66]

a(n) = ceiling(1/18*n*((2 - n) mod 6 + 4*n mod (-3) + 1)), for n >= 1.
x(n) = ceiling(n - 2/3*(n^2 + 1) mod 3), for n >= 1 (x-coordinates).
y(n) = floor(2*n/3)*((2 - n) mod (-3) + 1), for n >= 1 (y-coordinates).
From Colin Barker, May 03 2020: (Start)
G.f.: x^3*(1 + x^3 - 2*x^4 - x^5 + x^6 - x^7) / ((1 - x)^2*(1 + x)^2*(1 - x + x^2)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-6) - a(n-12) for n>12.
(End)
From Wolfdieter Lang, Jul 13 2020: (Start)
Bisection: x(k) = a(2*k-1). x(3+3*l) = 0, x(1+3*l) = -2*l, x(2+3*l) = 1+2*l, for l >= 0.
x(k) = (2*(k-1)*modp((k-4)^2,3) - (2*k-1)*modp((k-2)^2,3) + 1)/3, for k >= 1.
y(k) = a(2*k). y(3+3*k) = 1+l, y(1+3*k) = -l =  y(2+3*k), for l >= 0.
y(k) = ((k-1)*modp((k-1)^2,3) + (k-2)*modp((k+1)^2,3) - k*modp(k^2,3) -(k-3))/3, k >= 1.
G.f.s: Gx(t) = t^2*(1 - 2*t^2 + t^3)/(1 - t^3)^2, and Gy(t) = t^3*(1 - t - t^2) / (1 - t^3)^2.
This produces the g.f. G(x) = Gy(x^2) + Gx(x^2)/x given by Colin Barker.
(End)

A329972 Y-coordinate of a point moving in a triangular spiral.

Original entry on oeis.org

0, 0, 1, 0, -1, -1, -1, -1, -1, -1, 0, 1, 2, 1, 0, -1, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -1, 0, 1, 2, 3, 2, 1, 0, -1, -2, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -2, -1, 0, 1, 2, 3, 4, 3, 2, 1, 0, -1, -2, -3, -4

Offset: 0

Mikk Heidemaa, Nov 26 2019

A329116 gives x-coordinates for a point moving in counterclockwise triangular spiral.

			    y
     |
   4 |                         56
     |                           \
     |                            \
     |                             \
   3 |                         30  55
     |                         / \   \
     |                        /   \   \
     |                       /     \   \
   2 |                     31  12  29  54
     |                     /   / \   \   \
     |                    /   /   \   \   \
     |                   /   /     \   \   \
   1 |                 32  13   2  11  28  53
     |                 /   /   / \   \   \   \
     |                /   /   /   \   \   \   \
     |               /   /   /     \   \   \   \
   0 |             33  14   3   0---1  10  27  52
     |             /   /   /             \   \   \
     |            /   /   /               \   \   \
     |           /   /   /                 \   \   \
  -1 |         34  15   4---5---6---7---8---9  26  51
     |         /   /                             \   \
     |        /   /                               \   \
     |       /   /                                 \   \
  -2 |     35  16--17--18--19--20--21--22--23--24--25  50
     |     /                                             \
     |    /                                               \
     |   /                                                 \
  -3 | 36--37--38--39--40--41--42--43--44--45--46--47--48--49
     |
     +--------------------------------------------------------
   x:  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Index entries for sequences related to coordinates of 2D curves

Cf. A268038, A274923.

a[n_] := Table[Floor[Min[#*Max[0, 2*Mod[#, 2] - 2], -2*#*Mod[#, -1]] + Ceiling[-#/2]] &[Sqrt@ k], {k, 0, n}]; a[64]

a(n) = floor(min(s*max(((0, 2*s) mod 2) - 2), (-2*s*s) mod (-1)) + ceiling(-s/2)) where s=sqrt(n).

A329116 Successively count to (-1)^(n+1)*n (n = 0, 1, 2, ... ).

Original entry on oeis.org

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

Offset: 0

Mikk Heidemaa, Nov 13 2019

Also x-coordinates of a point moving in counterclockwise triangular spiral (A329972 gives the y-coordinates).

			   y
     |
   4 |                         56
     |                           \
     |                            \
     |                             \
   3 |                         30  55
     |                         / \   \
     |                        /   \   \
     |                       /     \   \
   2 |                     31  12  29  54
     |                     /   / \   \   \
     |                    /   /   \   \   \
     |                   /   /     \   \   \
   1 |                 32  13   2  11  28  53
     |                 /   /   / \   \   \   \
     |                /   /   /   \   \   \   \
     |               /   /   /     \   \   \   \
   0 |             33  14   3   0---1  10  27  52
     |             /   /   /             \   \   \
     |            /   /   /               \   \   \
     |           /   /   /                 \   \   \
  -1 |         34  15   4---5---6---7---8---9  26  51
     |         /   /                             \   \
     |        /   /                               \   \
     |       /   /                                 \   \
  -2 |     35  16--17--18--19--20--21--22--23--24--25  50
     |     /                                             \
     |    /                                               \
     |   /                                                 \
  -3 | 36--37--38--39--40--41--42--43--44--45--46--47--48--49
     |
     +--------------------------------------------------------
   x:  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7
We count as follows. Start at n=0 with 0.
Next step is to count to 1: so we have 0, 1.
Next step is to count to -2, so we have 0, 1, 0, -1, -2.
Next we have to go to +3, so we have 0, 1, 0, -1, -2, -1, 0, 1, 2, 3.
And so on.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Index entries for sequences related to coordinates of 2D curves

Cf. A053615, A196199, A339265 (first differences). Essentially the same as A255175.

a[n_] := Table[(-1)^(# + 1)*(-#^2 + # + k) &[Ceiling@ Sqrt@ k], {k, 0, n}]; a[64]

from math import isqrt
def A329116(n): return ((t:=1+isqrt(n-1))*(t-1)-n)*(-1 if t&1 else 1) if n else 0 # Chai Wah Wu, Aug 04 2022

a(n) = (-1)^t * (t^2 - t - n) where t=ceiling(sqrt(n)).
a(n) = (-1)^t * floor(t^2 - sqrt(n) - n) where t=ceiling(sqrt(n)).
A053615(n) = abs(a(n)).
abs(A196199(n)) = abs(a(n)).
A255175(n) = a(n+1).

A308553 Numbers k such that 5^(k+3) + 3^(k+2) + 2^(k+1) + 1 is prime.

Original entry on oeis.org

0, 4, 8, 524, 972, 3780, 7704

Offset: 1

Mikk Heidemaa, Jun 07 2019

Larger k values yield probable primes.

a(8) > 100000 (if it exists). - Michael S. Branicky, Nov 11 2024

Cf. A306573.

ParallelTable[If[PrimeQ[5^(n+3) + 3^(n+2) + 2^(n+1) + 1], n, Nothing], {n, 0, 10^4}]

cp's OEIS Frontend

User: Mikk Heidemaa

A371631 Primes whose product of nonzero digits divided by the sum of its digits is also prime.

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A359630 Primes p such that 10^p+3 or 10^p+9 is also prime.

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A357915 Concatenation of the decimal digits of {n, 1..n}.

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A359406 Integers k such that the concatenation of k consecutive primes starting at 31 is prime.

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A343834 Primes with digits in nondecreasing order, only primes, and with sum of digits also a prime.

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Python

Extensions

A343768 Primes p such that p^2 + 1 divides 3^(p+1) - 1.

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Python

A331363 Pairs of coordinates of the corners in a counterclockwise triangular spiral.

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Programs

Mathematica

Formula

A329972 Y-coordinate of a point moving in a triangular spiral.

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