A004086 -id:A004086 - cp's OEIS frontend

A337047 Numbers k such that A001414(k) and A001414(A004086(k)) are twin primes p, p+2.

405, 412, 850, 25315, 49419, 50127, 224315, 293394, 308700, 697136, 801350, 811910, 997425, 1118520, 1152000, 1177250, 1550520, 1659350, 1725332, 1739640, 1824500, 1976895, 2141150, 2580640, 2580831, 3530466, 3718376, 4050405, 4459455, 4536532, 4577732, 4832796, 5173100, 5510287, 5601570, 5603989, 5609439

Offset: 1

J. M. Bergot and Robert Israel, Aug 12 2020

			a(3)=850 is in the sequence because A001414(850)=2+5+5+17=29, A001414(58)=2+29=31, and (29,31) is a pair of twin primes.

Robert Israel, Table of n, a(n) for n = 1..150

Cf. A001097, A001414, A004086. Subsequence of A100118.

revdigs:= proc(n) local L,k;
  L:= convert(n,base,10);
  add(L[-k]*10^(k-1),k=1..nops(L))
end proc:
filter:= proc(n) local a,b;
  a:= convert(map(convert,ifactors(n)[2],`*`),`+`);
  if not isprime(a) then return false fi;
  b:= convert(map(convert,ifactors(revdigs(n))[2],`*`),`+`);
  b = a+2 and isprime(b)
end proc:
select(filter, [$1 .. 10^7]);

A338030 Primes p such that reverse(p), reverse(2p) and reverse(2reverse(p)) are all primes, where reverse = A004086.

Original entry on oeis.org

7, 17, 37, 71, 73, 167, 181, 191, 353, 373, 389, 761, 787, 797, 929, 983, 1753, 1879, 3571, 7057, 7177, 7507, 7717, 7879, 9349, 9439, 9781, 9787, 15053, 15227, 15307, 15451, 15551, 15667, 15679, 15791, 15919, 16061, 16073, 16453, 16547, 16561, 16747, 16883, 16979, 17471, 17909, 17971, 18427

Offset: 1

J. M. Bergot and Robert Israel, Oct 09 2020

			a(3) = 37 is a term because 37, reverse(37)=73, reverse(2*37)=47 and reverse(2*73)=641 are prime.

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

Cf. A004086, A007500.

rev:= proc(n) local L,k;
  L:= convert(n,base,10);
  add(L[-k]*10^(k-1),k=1..nops(L))
end proc:
filter:= proc(n) local r;
if not isprime(n) then return false fi;
  r:= rev(n);
isprime(r) and isprime(rev(2*n)) and isprime(rev(2*r))
end proc:
select(filter, [seq(i,i=3..20000,2)]);

With[{rev = IntegerReverse}, Select[Range[20000], AllTrue[{#, rev[#], rev[2*#], rev[2*rev[#]]}, PrimeQ] &]] (* Amiram Eldar, Oct 10 2020 *)

rev(n) = fromdigits(Vecrev(digits(n))); \\ A004086
isok(p) = if (isprime(p), my(r=rev(p)); isprime(r) && isprime(rev(2*p)) && isprime(rev(2*r))); \\ Michel Marcus, Oct 10 2020

A349709 Primes p such that if q is the next prime, A004086(pq) = A004086(p)A004086(q).

Original entry on oeis.org

2, 7, 11, 97, 101, 1021, 1201, 2003, 3001, 10103, 10111, 20011, 20021, 21001, 101111, 102001, 102101, 112103, 112111, 120103, 201011, 202001, 1000003, 1000211, 1010003, 1010201, 1011001, 1020011, 1100101, 1100311, 1111013, 1111021, 1112003, 1201001, 2011201, 2020001, 2100001, 2100011, 10000103

Offset: 1

J. M. Bergot and Robert Israel, Jan 05 2022

			a(6) = 1021 is a term because it is prime, the next prime is 1031, and the reverse of 1021*1031 = 1052651 is 1562501 = 1301*1201.

Robert Israel, Table of n, a(n) for n = 1..169

Cf. A004086.

revdigs:= proc(n) local L,i,m;
  L:= convert(n,base,10);
  m:= nops(L);
  add(L[i]*10^(m-i),i=1..m)
end proc:
R:= NULL: count:= 0:
q:= 2: qr:= 2:
while count < 50 do
  p:= q; pr:= qr;
  q:= nextprime(q); qr:= revdigs(q);
  if pr*qr = revdigs(p*q) then
     count:= count+1; R:= R, p;
  fi
od:
R;

seqQ[p_] := PrimeQ[p] && Module[{r = IntegerReverse, q = NextPrime[p]}, r[p*q] == r[p] * r[q]]; Select[Range[2.2*10^6], seqQ] (* Amiram Eldar, Jan 05 2022 *)

R(n) = fromdigits(Vecrev(digits(n))); \\ A004086
isok(p) = if (isprime(p), my(q=nextprime(p+1)); R(p*q) == R(p)*R(q)); \\ Michel Marcus, Jan 05 2022

from itertools import islice
from sympy import isprime, nextprime
def R(n): return int(str(n)[::-1])
def agen(): # generator of terms
    p, pr = 2, 2
    while True:
        q = nextprime(p)
        qr = R(q)
        if R(p*q) == pr * qr:
            yield p
        p, pr = q, qr
print(list(islice(agen(), 38))) # Michael S. Branicky, Jan 05 2022

A350401 Primes p such that if q is the next prime, p*q mod (A004086(p)+A004086(q)) is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 23, 29, 31, 53, 59, 71, 73, 83, 89, 101, 107, 109, 127, 137, 149, 163, 173, 181, 191, 193, 211, 223, 227, 233, 239, 257, 271, 277, 281, 283, 307, 317, 367, 373, 389, 409, 419, 431, 449, 461, 463, 467, 479, 491, 509, 521, 523, 547, 577, 587, 593, 607, 613, 631, 641, 643, 653, 659

Offset: 1

J. M. Bergot and Robert Israel, Dec 28 2021

			a(6) = 23 is a member because it is prime, the next prime is 29, and 23*29 mod (32+92) = 667 mod 124 = 47 is prime.

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

Cf. A004086.

revdigs:= proc(n) local L,i,m;
  L:= convert(n,base,10); m:= nops(L);
  add(L[i]*10^(m-i),i=1..m)
end proc:
q:= 2: qr:= 2:
R:= NULL: count:= 0:
while count < 100 do
  p:=q; pr:= qr;
  q:= nextprime(p); qr:= revdigs(q); s:= p*q mod (pr+qr);
  if isprime(s) then
     R:= R, p; count:= count+1;
  fi
od:
R;

A351728 Primes p such that if q is the next prime, p+A004086(q) and q+A004086(p) are prime.

Original entry on oeis.org

2, 61, 83, 433, 677, 2351, 2399, 2441, 4397, 4457, 4673, 6257, 6367, 6961, 8263, 8713, 8761, 20627, 21391, 21649, 22721, 22871, 23227, 23761, 25111, 25321, 25589, 25609, 25741, 25841, 26597, 26731, 26981, 27179, 27271, 27299, 27367, 27409, 27481, 27961, 28559, 29881, 40609, 40927, 40933, 42197

Offset: 1

J. M. Bergot and Robert Israel, Mar 20 2022

For each term p except 2, A013632(p) is divisible by 6.

			a(3) = 83 is a term because it is prime, the next prime is 89, and 83+98 = 181 and 38+89 = 127 are both prime.

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

Cf. A004086, A013632.

revdigs:= proc(n) local L,i; L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
Primes:= select(isprime, [2,seq(i,i=3..10000,2)]):
RPrimes:= map(revdigs,Primes):
Primes[select(i -> isprime(Primes[i]+RPrimes[i+1]) and isprime(RPrimes[i]+Primes[i+1]), [$1..nops(Primes)-1])]:

A354285 Numbers k such that one of k, k+1, k+2 is prime and the other two are semiprimes, and one of R(n), R(n+1), R(n+2) is prime and the other two are semiprimes, where R = A004086.

Original entry on oeis.org

4, 157, 177, 1381, 1437, 7417, 9661, 9901, 12757, 15297, 15681, 16921, 35961, 36901, 39777, 75741, 77277, 93097, 94441, 103317, 108201, 111261, 117541, 121377, 127597, 128461, 128901, 130197, 134677, 146841, 147417, 151377, 156601, 160077, 165441, 166861, 169177, 178537, 185901, 187881, 306541

Offset: 1

J. M. Bergot and Robert Israel, May 29 2022

All terms after the first == 1 (mod 4).

			a(3) = 177 is a term because 177 = 3*59 and 178 = 2*89 are semiprimes, 179 is prime, 771 = 3*257 and 871 = 13*67 are semiprimes and 971 is prime.

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

Cf. A004086.

revdigs:= proc(n) local i,L;
  L:= convert(n,base,10);
  add(10^(i-1)*L[-i],i=1..nops(L))
end proc:
f:= proc(n) uses numtheory;
      if not isprime((n+1)/2) then return false fi;
      if n mod 3 = 0 then if not(isprime(n/3) and isprime(n+2)) then return false fi
      elif n mod 3 = 2 then return false
      elif not(isprime(n) and isprime((n+2)/3)) then return false
      fi;
      sort(map(bigomega@revdigs,[n,n+1,n+2]))=[1,2,2]
end proc:
f(4):= true:
select(f, [4, seq(i,i=5..10^6,4)]);

Select[Range[300000], Sort[PrimeOmega[# + {0, 1, 2}]] == Sort[PrimeOmega[IntegerReverse[# + {0, 1, 2}]]] == {1, 2, 2} &] (* Amiram Eldar, May 29 2022 *)

A355617 a(1) = 1; a(2) = 2; for n > 2, a(n) = R(a(n-1)) if a(n-1) != R(a(n-2)) and R(a(n-1)) has not yet been used, where R is the digit reversal function A004086, otherwise a(n) is the smallest positive integer > a(n-1) that has not yet been used.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65, 66, 67, 76, 77, 78, 87, 88, 89, 98, 99, 100, 101, 102, 201, 202, 203, 302, 303, 304, 403, 404, 405, 504, 505, 506, 605, 606, 607, 706, 707, 708, 807, 808, 809, 908, 909

Offset: 1

Sylvia Zevi Abrams, Jul 09 2022

A regular pattern of rectangles, each 10 times larger than the last, appears on the scatter plot of this sequence. However, there are moments where the sequence deviates from its current rectangle to drop down to lower orders of magnitude or jump up to higher ones. Additionally, chaotic behavior can be observed at values of n in the approximate intervals [60000, 63000], [68000, 82000], and [85000, 100000].

Not all integers appear in this sequence. The smallest number that will never occur is 130, as the only way to achieve the number is through 129, which had already been used to jump to 921.

			a(11) = 11 because R(a(10)) = 1, which was already used as a(1).
a(13) = 21 because R(a(12)) = 21, which had not yet been used, and a(11) =! R(a(12)).
a(64) = 20 because although R(a(63)) = 91 and 91 has not yet been used, a(63) = R(a(62)).

Sylvia Zevi Abrams, Table of n, a(n) for n = 1..10000 [Corrected to remove leading 0's by _Sean A. Irvine_]
Sylvia Zevi Abrams, Scatter Plot, for n < 100000

Cf. A004086.

R(n) = fromdigits(Vecrev(digits(n))); \\ A004086
leastunused(va, m) = my(k=va[m]+1); while (#select(x->(x==k), va), k++); k;
lista(nn) = my(va = vector(nn)); va[1] = 1; va[2] = 2; for (n=3, nn, if ((va[n-1] != R(va[n-2])) && !#select(x->(x==R(va[n-1])), va), va[n] = R(va[n-1]), va[n] = leastunused(va, n-1));); va; \\ Michel Marcus, Jul 12 2022

A359173 Numbers whose square can be expressed as k * A004086(k) with non-palindromic k.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 200, 220, 252, 300, 330, 400, 403, 440, 500, 504, 550, 600, 660, 700, 770, 800, 816, 880, 900, 990, 1000, 1010, 1100, 1110, 1210, 1310, 1410, 1510, 1610, 1710, 1810, 1910, 2000, 2020, 2120, 2200, 2220, 2320, 2420, 2520, 2620, 2720, 2772

Offset: 1

Hugo Pfoertner, Dec 17 2022

If k is a term, then so is 10*k. - Robert Israel, Dec 23 2022

			a(1) = 10 because 100*1 = 10^2;
a(2) = 20: 200*2 = 20^2;
a(11) = 110: 1100*11 = 110^2;
a(14) = 252: 144*441 = 252^2;
a(28) = 816: 768*867 = 816^2.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A000290, A002113, A004086, A118959.

rev:= proc(n) local L,i; L:= convert(n,base,10); add(L[-i]*10^(i-1),i=1..nops(L)) end proc:
g:= proc(d,m) local r; r:= rev(d); r <> d and m = d*r end proc:
filter:= proc(n) ormap(g, numtheory:-divisors(n^2),n^2) end proc:
select(filter, [$1..3000]); # Robert Israel, Dec 23 2022

L=List(); for (k=1, 3*10^6, my (r=fromdigits(Vecrev(digits(k))), s); if (issquare(r*k, &s) && r!=k, if(s<3001, listput(L, s)))); Set(L)

from itertools import count, islice
from sympy import divisors
def A359173_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:any(d*int(str(d)[::-1])==n**2 for d in divisors(n**2,generator=True) if d != n),count(max(startvalue,1)))
A359173_list = list(islice(A359173_gen(),30)) # Chai Wah Wu, Dec 19 2022

A371034 For n >= 1, a(n) = A004086(n) if A055483(n) = 1, otherwise a(n) = n / A055483(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 31, 41, 5, 61, 71, 2, 91, 10, 7, 1, 32, 4, 52, 13, 3, 14, 92, 10, 13, 23, 1, 43, 53, 4, 73, 83, 13, 10, 14, 7, 34, 1, 5, 23, 74, 4, 94, 10, 17, 25, 35, 6, 1, 65, 19, 85, 95, 10, 16, 31, 7, 32, 56, 1, 76, 34, 23, 10, 17, 8, 37, 47

Offset: 1

Ctibor O. Zizka, Mar 31 2024

Also a(n) = R(n) if (n, R(n)) are coprime, otherwise a(n) = n / GCD(n, R(n)), where R(n) is the digit reversal of n. a(n) = 1 for n from the union of A011557 and A002113 and A001232 and A008918.

			n = 13: A004086(13) = 31, A055483(13) = 1 thus a(13) = 31.
n = 15: A004086(15) = 51, A055483(15) = 3 thus a(15) = 15/3 = 5.

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

Cf. A001232, A002113, A002275, A004086, A008918, A011557, A055483, A306273.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
f:= proc(n) local r,g;
  r:= rev(n);
  g:= igcd(n,r);
  if g = 1 then r else n/g fi
end proc;
map(f, [$1..100]); # Robert Israel, Jul 09 2024

a[n_] := Module[{r = IntegerReverse[n], g}, g = GCD[n, r]; If[g == 1, r, n/g]]; Array[a, 100] (* Amiram Eldar, Mar 31 2024 *)

a(A011557(k)) = 1, k >= 0.
a(A002113(k)) = 1, k >= 2.
a(A001232(k)) = 1, k >= 1.
a(A008918(k)) = 1, k >= 1.

A068158 a(n) = floor(n!/R(n)!), where R(n) = digit reversal of n (A004086).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 3628800, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1216451004088320000, 106661318400, 1, 0, 0, 0, 0, 0, 0, 0, 44208809968698509772718080000000, 1320509264105545113600000, 10178348544000, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Amarnath Murthy, Feb 24 2002

If n < R(n) then a(n) = 0.
If n = R(n) then n is a palindrome and a(n) = 1.
If n > R(n) then n!/R(n)! is an integer.

			a(21) = 21! / 12! = 51090942171709440000/479001600 = 106661318400.

Cf. A004086.

a(n) = n!\fromdigits(Vecrev(digits(n)))!; \\ Michel Marcus, Aug 19 2024

cp's OEIS Frontend

A337047 Numbers k such that A001414(k) and A001414(A004086(k)) are twin primes p, p+2.

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A338030 Primes p such that reverse(p), reverse(2*p) and reverse(2*reverse(p)) are all primes, where reverse = A004086.

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A349709 Primes p such that if q is the next prime, A004086(p*q) = A004086(p)*A004086(q).

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A350401 Primes p such that if q is the next prime, p*q mod (A004086(p)+A004086(q)) is prime.

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A351728 Primes p such that if q is the next prime, p+A004086(q) and q+A004086(p) are prime.

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A354285 Numbers k such that one of k, k+1, k+2 is prime and the other two are semiprimes, and one of R(n), R(n+1), R(n+2) is prime and the other two are semiprimes, where R = A004086.

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A355617 a(1) = 1; a(2) = 2; for n > 2, a(n) = R(a(n-1)) if a(n-1) != R(a(n-2)) and R(a(n-1)) has not yet been used, where R is the digit reversal function A004086, otherwise a(n) is the smallest positive integer > a(n-1) that has not yet been used.

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A359173 Numbers whose square can be expressed as k * A004086(k) with non-palindromic k.

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A338030 Primes p such that reverse(p), reverse(2p) and reverse(2reverse(p)) are all primes, where reverse = A004086.

A349709 Primes p such that if q is the next prime, A004086(pq) = A004086(p)A004086(q).