A008364 -id:A008364 - cp's OEIS frontend

A336669 a(n) is the number of n-digit terms in A336668 (assuming 0 has 0 digit).

1, 9, 25, 54, 93, 24, 192, 72, 464, 606, 40, 9, 302, 9, 88, 69, 464, 9, 1056, 9, 108, 117, 25, 9, 775, 24, 25, 606, 156, 9, 207, 9, 464, 54, 25, 87, 1166, 9, 25, 54, 479, 9, 255, 9, 93, 621, 25, 9, 775, 72, 40, 54, 93, 9, 1056, 24, 527, 54, 25, 9, 317, 9, 25

Offset: 0

Rémy Sigrist, Jul 29 2020

This sequence is bounded as the decimal representation of any term in A336668 is fully determined by at most 9 of its leading digits.

			For n = 2:
- let m be a two-digit term of A336668 (10 <= m <= 99),
- if m starts with an odd digit, say d = 1, 3, 5, 7 or 9, then m ends with d,
- if m starts with an even digit, say d = 2, 4, 6 or 8, then m ends with any even digit, say t = 0, 2, 4, 6 or 8,
- so a(2) = 5 + 4*5 = 25.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A336669

Cf. A008364, A336668.

PARI
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See Links section.
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a(n) = 9 iff n is 11-rough (A008364).
a(k*n) >= a(n) for any n >= 0 and k > 0.
Apparently, when n > 0, a(n) = a(gcd(n, 2^3 * 3^2 * 5 * 7)).

A358250 Numbers whose square has a number of divisors coprime to 210.

Original entry on oeis.org

1, 32, 64, 243, 256, 512, 729, 2048, 3125, 6561, 7776, 15552, 15625, 16384, 16807, 19683, 23328, 32768, 46656, 62208, 100000, 117649, 124416, 161051, 177147, 186624, 200000, 209952, 262144, 371293, 373248, 390625, 419904, 497664, 500000, 537824, 629856, 759375

Offset: 1

Michael De Vlieger, Dec 03 2022

nonn

210 is the product of the smallest 4 primes.
Numbers k such that gcd(d(k^2), 210) = 1, where d(k) is the number of divisors of k (A000005).
Also numbers with no exponents = 1 mod 3, 2 mod 5, or 3 mod 7; also numbers whose square has a number of divisors coprime to 105. - Charles R Greathouse IV, Dec 08 2022

Michael De Vlieger, Table of n, a(n) for n = 1..5000

Subsequence of A069492 and hence of A036967, A036966, and A001694.
Subsequence of other sequences of numbers k such that gcd(d(k^2), m) = 1: A350014 (m=6), A354179 (m=30).
Cf. A000005, A000290, A008364.

With[{nn = 2^20}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], CoprimeQ[DivisorSigma[0, #^2], 210] &]]

is(n,f=factor(n))=if(n<32, return(n==1)); my(t=f[,2]%105, N=19200959813818273241621521446046); for(i=1,#t, if(bittest(N,t[i]), return(0))); 1 \\ Charles R Greathouse IV, Dec 08 2022

Sum_{n>=1} 1/a(n) = Product_{p prime} (Sum_{k=2..210, gcd(k-1,210)=1} p^(k/2))/(p^105-1) = 1.05981355805... . - Amiram Eldar, Dec 06 2022

A052424 Numbers k with no single-digit factors (apart from 1 and k).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209, 211, 221, 223, 227, 229, 233, 239

Offset: 1

Henry Bottomley, Mar 14 2000

Cf. A008364.

isok(n) = {d = divisors(n); for (i = 2, #d - 1,if (length(Str(d[i])) == 1, return (0));); return (1);} \\ Michel Marcus, Jul 27 2013

A078861 Least positive residues [mod 210] representing those residue classes which can be smaller prime of a difference 6 taken from A031924.

Original entry on oeis.org

11, 13, 17, 23, 31, 37, 41, 47, 53, 61, 67, 73, 83, 97, 101, 103, 107, 121, 131, 137, 143, 151, 157, 163, 167, 173, 181, 187, 191, 193

Offset: 1

Labos Elemer, Dec 13 2002

Cf. A031924, A008364, A078859, A078860.

t=Flatten[Position[Table[GCD[w, 210], {w, 1, 210}], 1]] t2=Intersection[t, t+6]-6

Intersection[RRS(210), 6+RRS{210)]-6. RRS[210]=reduced residue system of 210=first 48=phi[210] terms of A008364; 210k+r generates complete A031924 with suitable k and r taken from these 30 numbers.

A129476 a(n) is the concatenation in increasing order of all single-digit divisors of n.

Original entry on oeis.org

1, 12, 13, 124, 15, 1236, 17, 1248, 139, 125, 1, 12346, 1, 127, 135, 1248, 1, 12369, 1, 1245, 137, 12, 1, 123468, 15, 12, 139, 1247, 1, 12356, 1, 1248, 13, 12, 157, 123469, 1, 12, 13, 12458, 1, 12367, 1, 124, 1359, 12, 1, 123468, 17, 125

Offset: 1

Colin Pitrat (colin.pitrat(AT)rez-gif.supelec.fr), May 29 2007

Sequence has period 2520 = 2^3 * 3^2 * 5 * 7.

a(n) = 1 iff n is a 11-rough number: not divisible by 2, 3, 5 or 7 (A008364). - Bernard Schott_, Dec 31 2020

			a(10)=125 because 1, 2 and 5 divides 10. 10 also divides 10 but it is not a digit so it doesn't appear.
a(2520) = 123456789. - Bernard Schott_, Dec 31 2020

Michel Marcus, Table of n, a(n) for n = 1..5040

Cf. A008364, A037278.

a:= n-> parse(cat(seq(`if`(irem(n, i)=0, i, [][]), i=1..9))):
seq(a(n), n=1..50);  # Alois P. Heinz, Dec 31 2020

Table[FromDigits[Select[Divisors[n],#<10&]],{n,50}] (* Harvey P. Dale, Jun 07 2015 *)

a(n) = fromdigits(select(x->(x<10), divisors(n))); \\ Michel Marcus, Dec 31 2020

def a(n): return int('1'+"".join(d for d in "23456789" if n%int(d) == 0))
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Dec 31 2020

Let n be the rank and result be the number for this rank let a1...ak be k digits (a1...ak in [0,9]) result=a1*10^(k-1)...ak*10^0 with (i|n) => i in {a1...ak}.

Editing and comment from Charles R Greathouse IV, Nov 02 2009
More terms from Harvey P. Dale, Jun 07 2015
Name edited by Joerg Arndt, Jan 01 2021

A298882 a(1) = 1, and for any n > 1, if n is the k-th number with least prime factor p, then a(n) is the k-th number with greatest prime factor p.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 16, 6, 32, 11, 64, 13, 128, 9, 256, 17, 512, 19, 1024, 12, 2048, 23, 4096, 10, 8192, 18, 16384, 29, 32768, 31, 65536, 24, 131072, 15, 262144, 37, 524288, 27, 1048576, 41, 2097152, 43, 4194304, 36, 8388608, 47, 16777216, 14, 33554432, 48

Offset: 1

Rémy Sigrist, Jan 28 2018

nonn

This sequence is a permutation of the natural numbers, with inverse A298268.
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 * r_p(k)) = p * s_p(k),
- for example: a(11 * A008364(k)) = 11 * A051038(k).

			The first terms, alongside A020639(n), are:
  n     a(n)    lpf(n)
  --    ----    ------
   1       1      1
   2       2      2
   3       3      3
   4       4      2
   5       5      5
   6       8      2
   7       7      7
   8      16      2
   9       6      3
  10      32      2
  11      11     11
  12      64      2
  13      13     13
  14     128      2
  15       9      3
  16     256      2
  17      17     17
  18     512      2
  19      19     19
  20    1024      2

Index entries for sequences that are permutations of the natural numbers

Cf. A008364, A020639, A046022, A051038, A055396, A078898, A083140, A125624, A298268 (inverse).

a(1) = 1.
a(A083140(n, k)) = A125624(n, k) for any n > 0 and k > 0.
a(n) = A125624(A055396(n), A078898(n)) for any n > 1.
Empirically:
- a(n) = n iff n belongs to A046022,
- a(2 * k) = 2^k for any k > 0,
- a(p^2) = 2 * p for any prime p,
- a(p * q) = 3 * p for any pair of consecutive odd primes (p, q).

A329100 Composite palindromes whose divisors > 1 are all nontrivial palindromes (i.e., palindromes with at least two digits).

Original entry on oeis.org

121, 1111, 1331, 1441, 1661, 1991, 3443, 3883, 7997, 10201, 12221, 13231, 14641, 15251, 15851, 18281, 19291, 31613, 35653, 37673, 37873, 38683, 112211, 113311, 115511, 116611, 124421, 125521, 134431, 136631, 139931, 145541, 146641, 157751, 167761, 169961, 176671

Offset: 1

Maxim Veselov, Nov 04 2019

This is the intersection of A062687 and A038511.
From Chai Wah Wu, Nov 08 2019 : (Start)
All terms start and end with the digits 1,3,7 or 9.
First term with 3 prime factors: 1331 = 11^3.
First term with 3 distinct prime factors: 145541 = 11*101*131.
First term with 4 prime factors: 14641 = 11^4.
First term with 5 prime factors: 1478741 = 11^4*101.
No term with more than 3 distinct prime factors or more than 5 prime factors among first 10000 terms.
(End)

			For k = 1331, its divisors > 1 are 11, 121 and 1331, all of which are palindromes with at least two digits, so 1331 is a term.
For k = 167761, its divisors > 1 are 11, 101, 151, 1111, 1661, 15251 and 167761, all of which are palindromes with at least two digits, so 167761 is a term.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A008364, A038511, A062687.

aQ[n_] := CompositeQ[n] && AllTrue[Rest @ Divisors[n], # > 10 && PalindromeQ @ IntegerDigits[#] &]; Select[Range[200000], aQ] (* Amiram Eldar, Nov 06 2019 *)

isA329100(n) = if((n>1) && !isprime(n) && gcd(n,210)==1, {d = divisors(n); rd = vector(#d, i, subst(Polrev(digits(d[i])), x, 10)); (d == rd); }, 0) \\ Jianing Song, Nov 06 2019, based on the program of A062687

More terms from Jianing Song, Nov 06 2019

A358001 Numbers whose number of divisors is coprime to 210.

Original entry on oeis.org

1, 1024, 4096, 59049, 65536, 262144, 531441, 4194304, 9765625, 43046721, 60466176, 241864704, 244140625, 268435456, 282475249, 387420489, 544195584, 1073741824, 2176782336, 3869835264, 10000000000, 13841287201, 15479341056, 25937424601, 31381059609, 34828517376

Offset: 1

Michael De Vlieger, Dec 03 2022

nonn

210 is the product of the smallest 4 primes.
Numbers k such that gcd(d(k), 210) = 1, where d(k) is the number of divisors of k (A000005).
The square roots of terms are in A001694.

Michael De Vlieger, Table of n, a(n) for n = 1..5000

Subsequence of other sequences of numbers k such that gcd(d(k), m) = 1: A000290 (m=2), A336590 (m=3), A352475 (m=6), A354178 (m=30).

Cf. A000005, A001694, A008364.

With[{nn = 200000}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], CoprimeQ[DivisorSigma[0, #^2], 210] &]^2]

a(n) = A358250(n)^2.

Sum_{n>=1} 1/a(n) = Product_{p prime} (Sum_{k=2..210, gcd(k-1,210)=1} p^k)/(p^210-1) = 1.001258995976... . - Amiram Eldar, Dec 06 2022

A078862 Smallest primes from A031924, each belonging to those different residue class of mod 210 which are listed in A078861. Arranged according to possible least positive residues mod 210.

Original entry on oeis.org

1901, 433, 647, 23, 31, 3607, 251, 47, 53, 61, 1117, 73, 83, 727, 941, 733, 947, 331, 131, 557, 353, 151, 157, 373, 167, 173, 601, 607, 3761, 1033

Offset: 1

Labos Elemer, Dec 13 2002

			Several terms are equal to corresponding ones in A078861, while others are larger like: 1033=4.210+193, where r=193 is in A078861.

Cf. A031924, A008364, A078861, 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], 6]&&s<211&&t[[s]]==0, t[[s]]=Prime[n]], {n, 1, 1000}]; t

A078863 Smallest primes from A029710, each belonging to those different residue class of mod 210 which are listed in A078860. Arranged according to possible least positive residues mod 210.

Original entry on oeis.org

7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 349, 163, 379, 397, 193

Offset: 1

Labos Elemer, Dec 13 2002

			Several terms are equal to corresponding ones in A078860, while others are larger like: 397=210.1+187 where r=187 is in A078860.

Cf. A031924, A008364, A078860, 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], 4]&&s<211&&t[[s]]==0, t[[s]]=Prime[n]], {n, 1, 10000}]; t

cp's OEIS Frontend

A336669 a(n) is the number of n-digit terms in A336668 (assuming 0 has 0 digit).

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A358250 Numbers whose square has a number of divisors coprime to 210.

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A052424 Numbers k with no single-digit factors (apart from 1 and k).

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A078861 Least positive residues [mod 210] representing those residue classes which can be smaller prime of a difference 6 taken from A031924.

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A129476 a(n) is the concatenation in increasing order of all single-digit divisors of n.

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A298882 a(1) = 1, and for any n > 1, if n is the k-th number with least prime factor p, then a(n) is the k-th number with greatest prime factor p.

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A329100 Composite palindromes whose divisors > 1 are all nontrivial palindromes (i.e., palindromes with at least two digits).

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Mathematica

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A358001 Numbers whose number of divisors is coprime to 210.

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