A008364 -id:A008364 - cp's OEIS frontend

A069556 Primes in which the k-th digit (counting from the right) is either a nonzero multiple of k or a divisor of k; furthermore the digit 1 is allowed only when k has no other divisors < 10.

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 347, 349, 367, 383, 389, 641, 643, 647, 661, 683, 929, 941, 947, 967, 983, 2341, 2347, 2381, 2383, 2389, 2621, 2647, 2663, 2683, 2687, 2689, 2927, 2963, 2969, 4327, 4349, 4363, 4621, 4643, 4649, 4663, 4943

Offset: 1

Amarnath Murthy, Mar 22 2002

			The prime 52627 is a member because 7 is a multiple of 1, 2
is a multiple of 2, 6 is a multiple of 3, 2 is a divisor of 4 and 5 is a multiple of 5.

Cf. A008364.

Edited by David Wasserman, Nov 03 2005

A094892 a(n) is the number of primes between n210 and (n+1)210.

Original entry on oeis.org

46, 35, 33, 32, 30, 29, 27, 31, 27, 27, 26, 25, 30, 26, 22, 27, 26, 27, 24, 24, 26, 23, 26, 26, 22, 24, 26, 27, 20, 25, 23, 25, 23, 24, 22, 23, 26, 21, 21, 24, 21, 26, 24, 23, 25, 22, 25, 20, 25, 22, 21, 22, 21, 22, 21, 18, 26, 22, 21, 26, 23, 24, 22, 19, 21, 24, 21, 17, 23

Offset: 0

Labos Elemer, Jun 16 2004

nonn

Arbitrarily long subsequences of consecutive 0's occur. a(n) is always <= 46. All values below 34 occur (see A095391); does 34?

			a(0) = 46 because there are 46 primes between 0*210 and 1*210.
a(1) = 35 because there are 35 primes between 1*210 and 2*210.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A008364, A038822, A078859, A095389, A098592.

[46] cat [#PrimesInInterval(210*n, 210*(n+1)): n in [1..80]]; // Vincenzo Librandi, Jul 08 2018

a[n_]:=PrimePi[210 (n + 1)] - PrimePi[210 n]; Table[a[n], {n, 0, 100}] (* Vincenzo Librandi, Jul 08 2018 *)

a(n) = primepi(210*(n+1)) - primepi(210*n); \\ Ruud H.G. van Tol, Oct 27 2024

a(n) = my(res = 0); forprime(p = n*210, (n+1)*210, isprime(p) && res++); res \\ David A. Corneth and Ruud H.G. van Tol, Oct 27 2024

Edited by Don Reble, Jun 16 2004

Examples corrected by Matthew Vandermast, Jun 17 2004

A235583 Numbers not divisible by 2, 5 or 7.

Original entry on oeis.org

1, 3, 9, 11, 13, 17, 19, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 57, 59, 61, 67, 69, 71, 73, 79, 81, 83, 87, 89, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 121, 123, 127, 129, 131, 137, 139, 141, 143, 149, 151, 153, 157, 159, 163, 167, 169, 171, 173, 177, 179, 181, 183

Offset: 1

Oleg P. Kirillov, Jan 12 2014

All primes, except 2, 5 and 7, are in this sequence. Any product of terms is also a term in the sequence. For example, a(2)a(4) = 3 * 11 = 33 = a(12). - Alonso del Arte, Jan 12 2014
In other words, numbers equivalent 1,3,9,...,69 modulo 70. This means the first differences of the sequence are 24-periodic. - Ralf Stephan, Jan 14 2014
Numbers coprime to 70. The asymptotic density of this sequence is 12/35. - Amiram Eldar, Oct 23 2020

			51 = 3 * 17, and gcd(51, 70) = 1, so it is in the sequence.
53 is prime, so it is in the sequence.
55 = 5 * 11, and gcd(55, 70) = 5, so it is not in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1).

Cf. A007775, A008364 (subsequence).

Select[Range[210], GCD[#, 70] == 1 &] (* Alonso del Arte, Jan 12 2014 *)
Select[Range[300], Mod[#, 2]>0 &&Mod[#, 5]>0 &&Mod[#, 7]>0&] (* Vincenzo Librandi, Feb 08 2014 *)

G.f.: x*(x^22 +3*x^21 +8*x^20 +7*x^19 +x^18-2*x^17 -x^16 +5*x^15 +10*x^14 +7*x^13 -x^12 -6*x^11 -x^10 +7*x^9 +10*x^8 +5*x^7 -x^6 -2*x^5 +x^4 +7*x^3 +8*x^2 +3*x +1) / ((x+1) *(x^2+1) *(x^2+x+1) *(x^4-x^2+1) *(x^4+1) *(x^8-x^4+1) *(x-1)^2). - Alois P. Heinz, Jan 12 2014

A078864 Smallest primes from A001359, each belonging to those different residue class of mod 210 which are listed in A078859. Arranged according to possible least positive residues mod 210.

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 419, 1427

Offset: 1

Labos Elemer, Dec 13 2002

			Several terms are equal to corresponding ones in A078859, while others are larger like: 1427=210.6+167 where r=167 is in A078859.

Cf. A001359, A008364, A078859-A078864.

f[x_] := Mod[Prime[x], 210] d[x_] := Prime[x+1]-Prime[x] t=Table[0, {210}]; Do[s=f[n]; If[Equal[d[n], 2]&&s<211&&t[[s]]==0, t[[s]]=Prime[n]], {n, 1, 10000}]; t

A158725 Non-repdigit composite numbers not divisible by 2, 3, 5 or 11.

Original entry on oeis.org

49, 91, 119, 133, 161, 169, 203, 217, 221, 247, 259, 287, 289, 299, 301, 323, 329, 343, 361, 371, 377, 391, 403, 413, 427, 437, 469, 481, 493, 497, 511, 527, 529, 533, 551, 553, 559, 581, 589, 611, 623, 629, 637, 667, 679, 689, 697, 703, 707, 713, 721, 731

Offset: 1

Lekraj Beedassy, Mar 24 2009

nonn

Non-repdigit composite numbers ending in 1, 3, 7 or 9, with digital root not a multiple of 3 and whose alternate digit sums do not differ by a multiple of 11.
The "compositeness" of larger entries of the sequence is not obvious right away or deducible by mere inspection, and hence these terms readily lend themselves to be (erroneously) suspected as primes to the casual glance.
This differs from the corresponding sequence without the repunit condition starting at a(1351) = 11123 rather than 11111. - Charles R Greathouse IV, Sep 08 2012

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

Cf. A007775, A053795, A008364.

is(n)=gcd(n,330)==1&&!isprime(n)&&n%(10^#Str(n)\9) \\ Charles R Greathouse IV, Sep 08 2012

a(n) ~ kn with k = 33/8. - Charles R Greathouse IV, Sep 08 2012

Corrected and extended by Ray Chandler, Mar 27 2009

A065823 Numbers k such that 6phi(k) = 5sigma(k).

Original entry on oeis.org

11, 527, 923, 36859, 40549, 55309, 88519, 120139, 138301, 280579, 293501, 313807, 529789, 719927, 2458859, 4864117, 6191413, 6811243, 7297877, 8402663, 8624107, 9487477, 10475821, 12356441, 12940957, 13624717, 13971229, 14869033, 15293137

Offset: 1

Labos Elemer, Nov 23 2001

nonn

Not all terms are squarefree: a(74) = 137640191 = 13^2 * 89 * 9151. - Charles R Greathouse IV, Nov 13 2015

Apart from the first term, no terms are divisible by 2, 3, 5, 7, or 11. - Charles R Greathouse IV, Nov 13 2015

Harry J. Smith and Charles R Greathouse IV, Table of n, a(n) for n = 1..651 (first 70 terms from Smith)

Subsequence of A008364.

Cf. A000010, A000203, A065818, A065819, A062699, A065822, A274535(5*sigma(n)).

n=0; for (m=1, 10^9, if (6*eulerphi(m) == 5*sigma(m), write("b065823.txt", n++, " ", m); if (n==70, return))) \\ Harry J. Smith, Nov 01 2009

is(n)=my(f=factor(n)); 6*eulerphi(f)==5*sigma(f) \\ Charles R Greathouse IV, Nov 13 2015

Terms a(16)-a(29) from Harry J. Smith, Nov 01 2009

A078860 Least positive residues [mod 210] representing those residue classes which can be lesser of prime pairs from A029710.

Original entry on oeis.org

7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 139, 163, 169, 187, 193

Offset: 1

Labos Elemer, Dec 13 2002

Cf. A029710, A008364, A078859.

t=Flatten[Position[Table[GCD[w, 210], {w, 1, 210}], 1]] t2=Intersection[t, t+4]-4

Intersection[RRS(210), 4+RRS{210)]-4 and {7}. RRS[210]=reduced residue system of 210=first 48=phi[210] terms of A008364; additional term 7 is a singular cases; 210k+r generates complete A029710 with suitable k and r taken from these 15+1 numbers.

A095391 a(n) is the least x such that A094892(x)=n.

Original entry on oeis.org

1751793, 235449, 60110, 10471, 17110, 8495, 6288, 3182, 2452, 1349, 331, 348, 446, 223, 249, 205, 111, 67, 55, 63, 28, 37, 14, 21, 18, 11, 10, 6, 551, 5, 4, 7, 3, 2

Offset: 0

Labos Elemer, Jun 16 2004

			a[0]=1751793 because there are no primes between 210*1751793 and 210*1751794.
a[1]=235449 because there is one prime between 210*235449 and 210*235450.

Cf. A078859, A095389, A008364.

ta=Table[0, {up}]; Do[{m=0};Do[s=210*k+r; s1=210*k+r+2; If[PrimeQ[s], m=m+1], {r, 1, 210}];ta[[k]]=m, {k, 1, up}] Table[Min[Flatten[Position[ta, j]]], {j, 1, 48}]

Edited by Don Reble, Jun 16 2004

A096489 Noncomposite numbers n such that number of decimal digits of n = number of divisors of n.

Original entry on oeis.org

1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Offset: 1

Labos Elemer, Jun 25 2004

Only 1 and primes with 2 decimal digits are here, so the sequence is finite: it consists of 1+25-4=22 terms. Part of A008364. Consists of the terms below 100 from A095862.

Cf. A008364, A095862.

{u=1, ta=Table[0, {25}]}; Do[s=Apply[Plus, IntegerDigits[n]];s1=Length[IntegerDigits[n]]; If[Equal[s1, DivisorSigma[0, n]], Print[n];ta[[u]]=n;u=u+1], {n, 1, 100}]
Select[Range[100],!CompositeQ[#]&&DivisorSigma[0,#]==IntegerLength[#]&] (* Harvey P. Dale, Jan 29 2024 *)

print1(1);forprime(p=9,99,print1(", "p)) \\ Charles R Greathouse IV, Apr 27 2011

A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 6, 15, 25, 11, 21, 13, 49, 35, 8, 17, 27, 19, 55, 77, 121, 23, 33, 65, 169, 39, 91, 29, 85, 31, 10, 143, 289, 119, 45, 37, 361, 221, 95, 41, 133, 43, 187, 115, 529, 47, 51, 161, 125, 323, 247, 53, 57, 209, 203, 437, 841, 59, 145, 61, 961

Offset: 1

Rémy Sigrist, Jan 27 2018

nonn

This sequence is a permutation of the natural numbers, with inverse A298882.
For any prime p and k > 0:
- if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number,
- then a(p * s_p(k)) = p * r_p(k),
- for example: a(11 * A051038(k)) = 11 * A008364(k).

			The first terms, alongside A006530(n), are:
  n     a(n)   gpf(n)
  --    ----   ------
   1      1      1
   2      2      2
   3      3      3
   4      4      2
   5      5      5
   6      9      3
   7      7      7
   8      6      2
   9     15      3
  10     25      5
  11     11     11
  12     21      3
  13     13     13
  14     49      7
  15     35      5
  16      8      2
  17     17     17
  18     27      3
  19     19     19
  20     55      5

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (with prime and semiprime values highlighted)
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A006530(n))
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A176506(k) when a(n) is the k-th semiprime)
Rémy Sigrist, PARI program for A298268
Index entries for sequences that are permutations of the natural numbers

Cf. A006530, A008364, A046022, A051038, A061395, A078899, A083140, A125624, A151800, A176506, A298268, A298882 (inverse).

PARI
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See Links section.
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a(1) = 1.
a(A125624(n, k)) = A083140(n, k) for any n > 0 and k > 0.
a(n) = A083140(A061395(n), A078899(n)) for any n > 1.
Empirically:
- a(n) = n iff n belongs to A046022,
- a(2^k) = 2 * k for any k > 0,
- a(2 * p) = p^2 for any prime p,
- a(3 * p) = p * A151800(p) for any odd prime p.

cp's OEIS Frontend

A069556 Primes in which the k-th digit (counting from the right) is either a nonzero multiple of k or a divisor of k; furthermore the digit 1 is allowed only when k has no other divisors < 10.

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A094892 a(n) is the number of primes between n*210 and (n+1)*210.

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A235583 Numbers not divisible by 2, 5 or 7.

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A078864 Smallest primes from A001359, each belonging to those different residue class of mod 210 which are listed in A078859. Arranged according to possible least positive residues mod 210.

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Mathematica

A158725 Non-repdigit composite numbers not divisible by 2, 3, 5 or 11.

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A065823 Numbers k such that 6*phi(k) = 5*sigma(k).

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A078860 Least positive residues [mod 210] representing those residue classes which can be lesser of prime pairs from A029710.

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Mathematica

Formula

A095391 a(n) is the least x such that A094892(x)=n.

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A094892 a(n) is the number of primes between n210 and (n+1)210.

A065823 Numbers k such that 6phi(k) = 5sigma(k).