A033846 -id:A033846 - cp's OEIS frontend

A382767 Smallest number k that is powerful but not a prime power that is also coprime to n.

36, 225, 100, 225, 36, 1225, 36, 225, 100, 441, 36, 1225, 36, 225, 196, 225, 36, 1225, 36, 441, 100, 225, 36, 1225, 36, 225, 100, 225, 36, 5929, 36, 225, 100, 225, 36, 1225, 36, 225, 100, 441, 36, 3025, 36, 225, 196, 225, 36, 1225, 36, 441, 100, 225, 36, 1225

Offset: 1

Michael De Vlieger, Apr 04 2025

Let p be the smallest prime that is coprime to n and let q be the second smallest prime that is coprime to n. Then a(n) = p^2 * q^2.

Records in this sequence are set by n in A002110.

			a(1) = 36 = (2*3)^2, since p = 2, q = 3.
a(2) = 225 = (3*5)^2, since p = 3, q = 5.
a(3) = 100 = (2*5)^2, since p = 2, q = 5.
a(4) = 225 = (3*5)^2, since p = 3, q = 5, a(2^i) = 225 for i > 0.
a(6) = 1225 = (5*7)^2, since p = 5, q = 7.
a(9) = 400 = (2*5)^2, since p = 2, q = 5, a(3^i) = 100 for i > 0.
a(10) = 441 = (3*7)^2, since p = 3, q = 7.
a(12) = 1225 = (5*7)^2, since p = 5, q = 7, a(k) = 1225 for n in A033845 (i.e., n such that rad(n) = 6), where rad = A007947.
a(20) = 441 = (3*7)^2, since p = 3, q = 7, a(k) = 441 for n in A033846 (i.e., n such that rad(n) = 10).
a(30) = 5929 = (7*11)^2, since p = 7, q = 11, etc.

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

Cf. A002110, A007947, A053669, A286708, A380539, A381805, A382248.

Table[c = 0; q = 2; Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^2]; q = NextPrime[q]; c++] ][[-1, 1]], {n, 120}]

a(n) = A053669(n)^2 * A380539(n)^2.
a(n) = A381805(n)^2.
a(n) = (A382248(n)/A020639(n))^2.
For k and m such that rad(k) = rad(m), a(k) = a(m), where rad = A007947.

A382960 Numbers k such that k < A053669(k)^2 * A380539(k)^2, i.e., k < A382767(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84

Offset: 1

Michael De Vlieger, Apr 14 2025

Numbers k whose reduced residue system does not intersect A286708 (i.e., powerful numbers that are not prime powers).
Let p = A053669(k), and let q = A380539(k). Thus, p and q are the smallest and second smallest primes, respectively, that do not divide k. Let m = p^2 * q^2 = A382767(k). Then this sequence is that of k such that k < m.
This sequence is finite following arguments akin to those in A051250 and A382659, with 626 terms.
Sequences A048597, A051250, and A382659 are proper subsets of this sequence.

			Let omega = A001221.
For omega = 0, we have the subset {1}. 1 is in the sequence since 1 < m, m = (2*3)^2 = 36.
For omega = 1, we have the subset {2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 64, 81, 128}.
  31 is in the sequence since 31 < m, m = (2*3)^2 = 36, but 37 is not a term since 37 > 36.
  25 is in the sequence since 25 < m, m = 36.
  49 is not a term since 49 > 36, and 243 is not a term since 243 > 100, 100 = (2*5)^2, etc.
For omega = 2, we have the squarefree numbers {6, 10, 14, 15, 22, 26, 34, 35, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218}.
  Intersection with A033845 = {k : rad(k) = 6} is {6, 12, 18, .., 1152}, since m = (5*7)^2 = 1225.
  Intersection with A033846 = {k : rad(k) = 10} is {10, 20, 40, ..., 400}, since m = (3*7)^2 = 441.
  Intersection with A033847 = {k : rad(k) = 14} is {14, 28, 56, ..., 224}, since m = (3*5)^2 = 225.
  Intersection with A033848 = {k : rad(k) = 15} is {15, 45, 75, 135}, since m = (2*7)^2 = 196, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..626

Cf. A048597 (k such that k < p^2), A051250 (k such that k < p*q), A053669, A286708, A380539, A382659 (k such that k < p^2*q), A382767.

Select[Range[510510], Function[n, c = 0; q = 2; n < Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^2]; q = NextPrime[q]; c++]][[-1, 1]] ] ]

A095678 Minimal sequence such that all triples of consecutive numbers have no common divisor greater than 1, but all three pairs within the triples are not coprime.

Original entry on oeis.org

6, 10, 15, 12, 20, 45, 18, 40, 75, 24, 50, 135, 36, 80, 225, 48, 100, 375, 54, 160, 405, 72, 200, 675, 96, 250, 1125, 108, 320, 1215, 144, 400, 1875, 162, 500, 2025, 192, 640, 3375, 216, 800, 3645, 288, 1000, 5625, 324, 1250, 6075, 384, 1280, 9375, 432

Offset: 1

Reinhard Zumkeller, Jul 04 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A003586, A003592, A003593, A006530, A020639, A033845, A033846, A033849.

seq1[p_, q_, lim_] := Sort[Flatten[Table[p^i * q^j, {i, 1, Log[p, lim]}, {j, 1, Log[q, lim/p^i]}]]];
seq[lim_] := Module[{s1 = seq1[2, 3, lim], s2 = seq1[2, 5, lim], s3 = seq1[3, 5, lim], ns}, ns = Length[s3]; Flatten[Transpose[{s1[[1;;ns]], s2[[1;;ns]], s3}]]]; seq[10^4] (* Amiram Eldar, Sep 29 2024 *)

gcd(a(n),a(n+1),a(n+2)) = 1, gcd(a(n),a(n+1)) > 1, gcd(a(n),a(n+2)) > 1 and gcd(a(n+1),a(n+2)) > 1.
A001221(a(n)) = 2; 2 <= A020639(a(n)) <= 3 <= A006530(a(n)) <= 5.
From Jianing Song, Jun 08 2022: (Start)
a(3n-2) = A033845(n) = 6*A003586(n);
a(3n-1) = A033846(n) = 10*A003592(n);
a(3n) = A033849(n) = 15*A003593(n). (End)
Sum_{n>=1} 1/a(n) = 7/8. - Amiram Eldar, Sep 29 2024

A296613 Smallest k such that either k >= n and k is a power of 2, or k >= 5n/3 and the prime divisors of k are precisely 2 and 5.

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 128, 128

Offset: 1

Eric M. Schmidt, Dec 16 2017

nonn

First disagreement with A062383(n-1) is at n = 129.

For n > 2, a(n) is not squarefree. - Iain Fox, Dec 17 2017

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
Bernadette Faye, Florian Luca, Pieter Moree, On the discriminator of Lucas sequences, arXiv:1708.03563 [math.NT], 2017, Theorem 1.

Cf. A033846.

a(n) = for(k=n, +oo, if(k == 2^valuation(k, 2) || (k >= 5*n/3 && factor(k)[, 1] == [2, 5]~), return(k))) \\ Iain Fox, Dec 17 2017

A363292 Numbers whose sum of (distinct) prime divisors (A008472) equals 7.

Original entry on oeis.org

7, 10, 20, 40, 49, 50, 80, 100, 160, 200, 250, 320, 343, 400, 500, 640, 800, 1000, 1250, 1280, 1600, 2000, 2401, 2500, 2560, 3200, 4000, 5000, 5120, 6250, 6400, 8000, 10000, 10240, 12500, 12800, 16000, 16807, 20000, 20480, 25000, 25600, 31250, 32000, 40000, 40960

Offset: 1

M. F. Hasler, Jul 20 2023

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A008472 (sopf), A000420 (7^n), A033846 (2^m*5^n), A362948 (A008472 = 5).

seq[max_] := Union[Join[7^Range[Floor[Log[7, max]]], Flatten@ Table[2^i*5^j, {i, 1, Log2[max]}, {j, 1, Log[5, max/2^i]}]]]; seq[40000] (* Amiram Eldar, Jul 27 2023 *)

select( {is_A363292(n)=vecsum(factor(n,0)[,1])==7}, [1..13^4]) \\ alternatively: [n | n<-[1..13^4], A008472(n)==7]

Union of A000420 = {7^k ; k > 0} and A033846 = {2^j*5^k ; j, k > 0}.

Sum_{n>=1} 1/a(n) = 5/12. - Amiram Eldar, Jul 27 2023

A370409 Numbers k = m * s, where s is composite and squarefree, rad(m) divides s, and 1 < m <= s, where rad() = A007947().

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 44, 45, 50, 52, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 98, 99, 100, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 196, 198, 204, 207, 208

Offset: 1

Michael De Vlieger, Feb 22 2024

nonn

A177492 is a proper subset.

Proper subset of A126706.

			Let T(j,k) = row j of A162306 and let s = A120944(n), n > 1.
This sequence contains finite sequences R(s) = s * T(s, 2..A010846(s)). The cardinality of R(s) is A010846(s)-1.
For s = 6, this sequence contains {12, 18, 24, 36},
  i.e., A033845(2..A010846(6)).
For s = 10, this sequence contains {20, 40, 50, 80, 100},
  i.e., A033846(2..A010846(10)).
For s = 14, this sequence contains {28, 56, 98, 112, 196},
  i.e., A033847(2..A010846(14)).
For s = 15, this sequence contains {45, 75, 135, 225},
  i.e., A033849(2..A010846(15)), etc.

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

Cf. A005117, A007947, A120944, A126706, A162306, A177492, A370266.

A287637 a(n) = A249125(n)/concatenation of prime factors of A249125(n).

Original entry on oeis.org

2, 4, 3, 8, 5, 9, 16, 7, 2, 32, 27, 4, 11, 25, 64, 13, 8, 81, 10, 128, 17, 49, 19, 16, 20, 256, 23, 125, 243, 32, 29, 31, 40, 512, 50, 121, 37, 64, 41, 43, 80, 1024, 729, 169, 47, 343, 100, 53, 625, 128, 59, 61, 160, 2048, 67, 289, 200, 71, 73, 79, 250, 256

Offset: 1

Michel Lagneau, May 28 2017

The squares of the sequence are, in increasing order: 4, 9, 16, 25, 49, 64, 81, 100, 121, 169, 256, 289, 361, 400, 625, 729, 1024, 4096,... including the squares of the prime numbers.
The numbers p^n, p prime and n = 1, 2, 3, 4,... are in the sequence.
The twin primes (a(m), a(m+1)) of the sequence are (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139),...
The numbers whose prime factors are 2 and 5 (A033846) are in the sequence.

			a(9)=2 because A249125(9) = 50 and the concatenation of the prime factors of 50 is 25. Hence, 50/25 = 2.

Cf. A033846, A077800, A084317, A249125.

with(numtheory):
for n from 2 to 10000 do:
  if type(n,prime)=false
   then
    x:=factorset(n):n0:=nops(x):
    d:=sum('length(x[i])', 'i'=1..n0):
    l:=sum('x[i]*10^sum('length(x[j])', 'j'=i+1..n0)', 'i'=1..n0):
    z:=n/l:
     if floor(z)=z
      then
      printf(`%d, `,z):
      else
     fi:
   fi:
od:

cf[n_] := FromDigits@ Flatten[ IntegerDigits /@ First /@ FactorInteger@n]; Reap[ Do[If[ CompositeQ[n] && IntegerQ[rz = n/cf[n]], Sow[rz]], {n, 6400}]][[2, 1]] (* Giovanni Resta, May 29 2017 *)

cp's OEIS Frontend

A382767 Smallest number k that is powerful but not a prime power that is also coprime to n.

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A382960 Numbers k such that k < A053669(k)^2 * A380539(k)^2, i.e., k < A382767(k).

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A095678 Minimal sequence such that all triples of consecutive numbers have no common divisor greater than 1, but all three pairs within the triples are not coprime.

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A296613 Smallest k such that either k >= n and k is a power of 2, or k >= 5n/3 and the prime divisors of k are precisely 2 and 5.

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PARI

A363292 Numbers whose sum of (distinct) prime divisors (A008472) equals 7.

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A370409 Numbers k = m * s, where s is composite and squarefree, rad(m) divides s, and 1 < m <= s, where rad() = A007947().

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A287637 a(n) = A249125(n)/concatenation of prime factors of A249125(n).

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