A003591 -id:A003591 - cp's OEIS frontend

A025664 Exponent of 7 (value of j) in n-th number of form 2^i*7^j.

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

Offset: 1

David W. Wilson

nonn

Cf. A003591.

A036312 Composite numbers whose prime factors contain no digits other than 2 and 7.

Original entry on oeis.org

4, 8, 14, 16, 28, 32, 49, 56, 64, 98, 112, 128, 196, 224, 256, 343, 392, 448, 454, 512, 554, 686, 784, 896, 908, 1024, 1108, 1372, 1454, 1568, 1589, 1792, 1816, 1939, 2048, 2216, 2401, 2744, 2908, 3136, 3178, 3584, 3632, 3878, 4096, 4432, 4802, 5089, 5488

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020459. - David A. Corneth, Oct 09 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 5340 terms from Robert Israel)
Index entries for sequences related to prime factors.

Cf. A003591, A020459, A036302-A036325.

dmax:= 4: # for terms < 2*10^dmax
P:= {2,7}:
L:= {7}:
for d from 1 to dmax-1 do
  L:= map(t -> 2*10^d+t, L) union map(t -> 7*10^d+t, L);
  P:= P union select(isprime,L);
od:
R:= {1}: N:= 2*10^dmax:
for p in P do
  R:= R union map(t -> seq(t*p^j,j=1..floor(log[p](N/t))), R)
od:
sort(convert(R minus P minus {1},list)); # Robert Israel, Aug 04 2020

Sum_{n>=1} 1/a(n) = Product_{p in A020459} (p/(p - 1)) - Sum_{p in A020459} 1/p - 1 = 0.7041098484... . - Amiram Eldar, May 18 2022

A365790 a(n) = number of k <= b(n) such that rad(k) | b(n), where rad(n) = A007947(n) and b(n) = A126706(n).

Original entry on oeis.org

8, 10, 8, 11, 8, 14, 11, 9, 8, 15, 12, 9, 16, 11, 26, 8, 10, 18, 9, 10, 14, 28, 11, 32, 10, 20, 13, 8, 15, 11, 21, 14, 10, 8, 36, 10, 33, 31, 12, 12, 27, 23, 10, 11, 41, 12, 8, 31, 18, 24, 11, 38, 8, 11, 8, 14, 44, 12, 11, 11, 25, 16, 36, 19, 33, 8, 14, 11, 26

Offset: 1

Michael De Vlieger, Sep 21 2023

nonn

Alternatively, position of A126706(n) in the list R(rad(n)) of k such that rad(k) | n, where rad(n) = A007947(n). Note that rad(b(n)) < b(n) for all n.

Let prime p divide n. The set R(rad(n)) is a list of numbers beginning with the empty product 1 and including all k such that p | k implies p | rad(n). For example, R(6) = A003586. All k in A003586 are such that no prime q coprime to 6 divides k.

			a(1) = 8 since rad(b(1)) = rad(12) = 6, and in the sequence R(6) = A003586 = {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, ...}, 12 is the 8th term.
a(2) = 10 since rad(b(2)) = rad(18) = 6, and 18 is the 10th term in R(6).
a(3) = 8 since rad(b(3)) = rad(20) = 10, and in the sequence R(10) = A003592 = {1, 2, 4, 5, 8, 10, 16, 20, ...}, 20 is the 8th term.
a(4) = 11 since rad(b(4)) = rad(24) = 6, and 24 is the 11th term in R(6).
a(5) = 8 since rad(b(5)) = rad(28) = 14, and in the sequence R(14) = A003591 = {1, 2, 4, 7, 8, 14, 16, 28, ...}, 28 is the 8th term, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A010846, A126706, A365783, A365791.

nn = 220;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
t = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
s = Map[f, t];
Map[Function[k, Set[r[k], Select[Range[nn], Divisible[k, f[#]] &]]], Union@ s];
Array[FirstPosition[r[s[[#]]], t[[#]] ][[1]] &, Length[t] ]

a(n) = A010846(A126706(n)).

A365791 a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A126706(n).

Original entry on oeis.org

2, 3, 2, 4, 2, 5, 3, 2, 2, 6, 4, 2, 7, 3, 2, 2, 2, 8, 3, 2, 5, 2, 3, 3, 2, 9, 4, 2, 6, 3, 10, 5, 2, 2, 4, 2, 3, 2, 4, 3, 2, 11, 3, 2, 5, 3, 2, 2, 7, 12, 2, 4, 2, 2, 2, 4, 6, 3, 2, 4, 13, 6, 3, 8, 2, 2, 4, 2, 14, 2, 7, 5, 2, 3, 3, 2, 7, 5, 2, 3, 3, 9, 5, 2, 2, 4

Offset: 1

Michael De Vlieger, Sep 21 2023

nonn

Alternatively, position of A126706(n) in the list k*{R(k)} containing m such that A007947(m) = k, where k = A007947(n).
The set R(k) is a list of numbers beginning with the empty product 1 and including all m such that p | m implies p | n. For example, R(6) = A003586. All k in A003586 are such that no prime q coprime to 6 divides k.
Then k*{R(k)} is the list of numbers beginning with k, followed by nonsquarefree k*m such that rad(k*m) = k.
The number k is composite and the only squarefree term in k*{R(k)} and appears in A120944; the rest of the list is in A126706.

			a(1) = 2 since rad(b(1)) = rad(12) = 6, and in the sequence k*{R(6)} = 6*{A003586} = {6, 12, 18, 24, 36, ...}, 12 is the 2nd term.
a(2) = 10 since rad(b(2)) = rad(18) = 6, and 18 is the 3rd term in k*{R(6)}.
a(3) = 2 since rad(b(3)) = rad(20) = 10, and in the sequence k*{R(10)} = 10*{A003592} = {10, 20, 40, 50, 80, ...}, 20 is the 2nd term.
a(4) = 4 since rad(b(4)) = rad(24) = 6, and 24 is the 4th term in k*{R(6)}.
a(5) = 2 since rad(b(5)) = rad(28) = 14, and in the sequence k*{R(14)} = 14*{A003591} = {14, 28, 56, 98, 112, ...}, 28 is the 2nd term, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A126706, A365783, A365792.

nn = 270;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
t = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
s = Map[f, t];
Map[Function[k, Set[r[k], k*Select[Range[nn/k], Divisible[k, f[#]] &]]], Union@ s];
Array[FirstPosition[r[s[[#]]], t[[#]] ][[1]] &, Length[t] ]

a(n) = A008479(A126706(n)).

a(n) > 1 for all n.

A025693 Index of 2^n within sequence of numbers of form 2^i*7^j.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 64, 71, 78, 86, 94, 102, 111, 120, 129, 139, 149, 159, 170, 181, 193, 205, 217, 230, 243, 256, 270, 284, 298, 313, 328, 343, 359, 375, 392, 409, 426, 444, 462, 480, 499, 518, 537, 557, 577, 597, 618, 639, 661, 683

Offset: 0

David W. Wilson

Positions of zeros in A025664. - R. J. Mathar, Jul 06 2025

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A003591.

a(n)=my(N=7<Charles R Greathouse IV, Jun 28 2011

a(n)=my(N=1); n+1+sum(i=1, n, logint(N<<=1, 7)); \\ Charles R Greathouse IV, Jan 11 2018

first(n)=my(s, N=1/2); vector(n+1, i, s+=logint(N<<=1, 7)+1) \\ Charles R Greathouse IV, Jan 11 2018

a(n) ~ kn^2 + O(n) with k = log(7)/log(2) - log(7)^2/log(2)^2. - Charles R Greathouse IV, Jun 28 2011

An incorrect g.f. was deleted by N. J. A. Sloane, Sep 16 2009

A036565 Triangle of numbers in which i-th row is {2^(i-j)*7^j, 0<=j<=i}; i >= 0.

Original entry on oeis.org

1, 2, 7, 4, 14, 49, 8, 28, 98, 343, 16, 56, 196, 686, 2401, 32, 112, 392, 1372, 4802, 16807, 64, 224, 784, 2744, 9604, 33614, 117649, 128, 448, 1568, 5488, 19208, 67228, 235298, 823543, 256, 896, 3136, 10976, 38416, 134456, 470596, 1647086, 5764801

Offset: 0

N. J. A. Sloane

			The triangle begins as:
  1;
  2,  7;
  4, 14, 49;
  8, 28, 98, 343;
  ...

Cf. A003591.

Cf. A000079 (1st column), A000420 (diagonal), A016130 (row sums).

row[n_] := Table[SeriesCoefficient[1/((1 - 2*x)(1 - 7*x*y)), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n}]; Array[row,9,0]//Flatten (* Stefano Spezia, Aug 19 2025 *)

G.f.: 1/((1 - 2*x)(1 - 7*x*y)). - Ilya Gutkovskiy, Jun 03 2017

A340268 Composite numbers k>1 such that (s-1) | (d-1) for each d | k, where s = lpf(k) = A020639(k).

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96

Offset: 1

Maxim Karimov, Jan 02 2021

nonn

Not a duplicate of A340058 because the complements A335902 and A340269 differ. - R. J. Mathar, Feb 16 2021

Cf. A000010, A000961, A020639, A340058, A335902, A340269 (complement).

Contains all composite terms of at least A003586, A003591, A003592, A003593, A003596.

n=300; % gives all terms of the sequence not exceeding n
A=[];
for i=2:n
    lpf=2;
    while mod(i,lpf)~=0
        lpf=lpf+1;
    end
    for d=1:floor(i/2)
        if mod(i,d)==0 && mod(d-1,lpf-1)~=0
            break
        elseif d==floor(i/2)
            A=[A i];
        end
    end
end

with(numtheory):
q:= n-> (f-> andmap(d-> irem(d-1, f)=0, divisors(n)))(min(factorset(n))-1):
select(not isprime and q, [$2..96])[];  # Alois P. Heinz, Feb 12 2021

Select[Range[2, 96], Function[{n, s}, And[! PrimeQ@ n, AllTrue[Divisors[n] - 1, Mod[#, s] == 0 &]]] @@ {#, FactorInteger[#][[1, 1]] - 1} &] (* Michael De Vlieger, Feb 12 2021 *)

isok(c) = if ((c>1) && !isprime(c), my(f=factor(c)[,1]); for (k=1, #f~, if ((f[k]-1) % (f[1]-1), return(0))); return(1)); \\ Michel Marcus, Jan 03 2021

A381803 Number of residues r in {0..n-1} that are not coprime to n and not in row n of A381801.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 3, 0, 4, 0, 1, 0, 4, 1, 0, 0, 6, 3, 0, 6, 8, 0, 4, 0, 11, 5, 8, 0, 9, 0, 0, 10, 13, 0, 7, 0, 9, 7, 11, 0, 17, 5, 3, 0, 12, 0, 6, 8, 21, 1, 0, 0, 17, 0, 25, 15, 26, 8, 15, 0, 24, 11, 12, 0, 29, 0, 0, 7, 17, 3, 22, 0, 32, 23

Offset: 1

Michael De Vlieger, Mar 24 2025

nonn

The intersection of row n of A038566 and row n of A381801 is {1} for n > 1. Therefore most of the terms in row n of A381801 are in row n of A121998 (reading n itself in row n of A121998 instead as n mod n = 0). Thus, a(n) is the number of terms n that are in row n of A121998 but not in A381801.

			Let R(n) = row n of A381801 and let S(n) = row n of A121998, where n in S(n) is instead taken mod n.
a(2) = 0 since S(2) = {} and R(2) = {0, 1}; R(2) \ S(2) is empty.
a(4) = 0 since S(4) = {0, 2} and R(4) = {0, 1, 2}; R(4) \ S(4) is empty.
a(6) = 0 since S(6) = {0, 2, 3, 4} and R(6) = {0, 1, 2, 3, 4} is empty.
a(8) = 1 since S(8) = {0, 2, 4, 6} and R(8) = {0, 1, 2, 4} = {6}.
a(9) = 1 since S(9) = {0, 3, 6} and R(6) = {0, 1, 3} = {6}.
a(10) = 0 since S(10) = {0, 2, 4, 5, 6, 8} and R(10) = {0, 1, 2, 4, 5, 6, 8} is empty.
  Therefore in base 10, numbers k such that rad(k) | 10 (i.e., k in A003592) may end in any number that is not coprime to 10. (Except 1 ends in the digit one, which is coprime to 10).
a(12) = 1 since S(12) = {0, 2, 3, 4, 6, 8, 9, 10} and R(12) = {0, 1, 2, 3, 4, 6, 8, 9} = {10}.
  Therefore in base 12, numbers k such that rad(k) | 12 (i.e., k in A003586) never end in digit 10.
a(14) = 3 since S(14) = {0, 2, 4, 6, 7, 8, 10, 12} and R(14) = {0, 1, 2, 4, 7, 8} = {6, 10, 12}.
  Therefore in base 14, numbers k such that rad(k) | 14 (i.e., k in A003591) never end in digits 6, 10, or 12.
a(16) = 4 since S(16) = {0, 2, 4, 6, 8, 10, 12, 14} and R(14) = {0, 1, 2, 4, 8} = {6, 10, 12, 14}, etc.
  Therefore in hexadecimal, numbers k such that powers of 2 (i.e., A000079) never end in digits 6, 10, 12, or 14.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000010, A038566, A051953, A121998, A381800, A381802.

f[x_] := Block[{c, ff, m, r, p, s, w},
  c[_] := True; ff = FactorInteger[x][[All, 1]]; w = Length[ff];
  s = {1};
  Do[Set[p[i], ff[[i]]], {i, w}];
  Do[Set[s, Union@ Flatten@ Join[s, #[[-1, 1]]]] &@ Reap@
    Do[m = s[[j]];
      While[Sow@ Set[r, Mod[m*p[i], x]];
        c[r],
        c[r] = False;
        m *= p[i]],
       {j, Length[s]}],
    {i, w}]; s ];
{0}~Join~Table[1 + n - EulerPhi[n] - Length@ f[n], {n, 2, 120}]

a(n) = 1 + n - phi(n) - A381800(n)
     = 1 + n - A000010(n) - A381800(n)
     = 1 + A051953(n) - A381800(n)
     = A381802(n) - phi(n) - 1.
a(p) = 0.
a(p^m) = p^(m-1) - m.

cp's OEIS Frontend

A025664 Exponent of 7 (value of j) in n-th number of form 2^i*7^j.

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A036312 Composite numbers whose prime factors contain no digits other than 2 and 7.

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A365790 a(n) = number of k <= b(n) such that rad(k) | b(n), where rad(n) = A007947(n) and b(n) = A126706(n).

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A365791 a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A126706(n).

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A025693 Index of 2^n within sequence of numbers of form 2^i*7^j.

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A036565 Triangle of numbers in which i-th row is {2^(i-j)*7^j, 0<=j<=i}; i >= 0.

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A340268 Composite numbers k>1 such that (s-1) | (d-1) for each d | k, where s = lpf(k) = A020639(k).

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A381803 Number of residues r in {0..n-1} that are not coprime to n and not in row n of A381801.

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