A030515 -id:A030515 - cp's OEIS frontend

A350150 a(1)=1; thereafter a(n+1) is the smallest unused number k such that d(k) and d(a(n)) are coprime, where d is the divisor counting function A000005.

1, 2, 4, 3, 9, 5, 16, 6, 25, 7, 36, 8, 49, 10, 64, 11, 81, 12, 625, 13, 100, 14, 121, 15, 144, 17, 169, 19, 196, 21, 225, 22, 256, 23, 289, 24, 324, 26, 361, 27, 400, 29, 441, 30, 484, 31, 529, 33, 576, 34, 676, 35, 729, 18, 1024, 20, 1296, 28, 2401, 32, 4096

Offset: 1

David James Sycamore, Dec 16 2021

nonn

A permutation of the positive integers. Identical to A137442 until a(18), with some terms in common thereafter. Numbers with the same number of divisors appear in their natural order, e.g. primes; d(p)=2, odd squarefree semiprimes; d(p*q) = 4, etc.
From Michael De Vlieger, Dec 16 2021: (Start)
a(2n+1) is square and d(a(2n+1)) odd. Let a(2n+1) constitute an "alpha" ray in scatterplot.
d(a(2n)) is even. Let a(2n) constitute a "beta" ray in scatterplot.
The occasion of d(a(2n)) = 6 induces a "flare" phase in the sequence, evident in scatterplot. The following term a(2n+1) is forced to have d(a(2n+1)) congruent to 1 or 5 (mod 6).
There are 5 flare-phases in the scatterplot associated with the occasion of d(a(2n)) = 6:
(I) a(18) = 12, a(19) = 625. The latter term interrupts what had been thereto and thereafter a series of square a(2n+1).
(II) a(54..61);
(III) 144..169 where a(k) with k in {152, 158, 164, 166} have d(a(k)) =/= 6, a characteristic common to subsequent phases;
(IV) 686..849;
(V) 11664..15515. (End)

			a(2) = 2 because d(1) = 1, d(2) = 2 and gcd(1,2) = 1.
a(3) cannot be 3 since d(2) = d(3) = 2, but gcd(d(2),d(4)) = gcd(2,3) = 1, so a(3) = 4.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log-log scatterplot of a(n) for n = 1..10^5 showing maxima in red and minima in blue. The golden line indicates the smallest missing number. We label terms that begin and end a flare phase among squares in alpha that comprise many records, and same in beta that comprise nearly all local minima.
Michael De Vlieger, Log-log scatterplot of a(n) for n = 1..512 annotating a(n) in black above and d(a(n)) in red below the point. The plot illustrates bifurcation of the sequence, with a(2n) such that d(a(2n)) is even in trajectory beta below, while square a(2n+1) such that d(a(2n+1)) is odd in trajectory alpha above. Terms a(2n) such that d(a(2n)) = 6 appear in blue, while a(2n+1) such that d(a(2n)) = 6 appear in red. The faint green line below represents d(a(n)) for reference.

Cf. A000005, A000027, A000040, A000290, A030515, A046388, A048691, A137442.

a[1] = 1; a[n_] := a[n] = Module[{k = 2, s = Array[a, n - 1], d = DivisorSigma[0, a[n - 1]]}, While[MemberQ[s, k] || ! CoprimeQ[d, DivisorSigma[0, k]], k++]; k]; Array[a, 100] (* Amiram Eldar, Dec 16 2021 *)

More terms from Amiram Eldar, Dec 16 2021

A119586 Triangle where T(n,m) = (n+1-m)-th positive integer with (m+1) divisors.

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 8, 16, 11, 49, 10, 81, 12, 13, 121, 14, 625, 18, 64, 17, 169, 15, 2401, 20, 729, 24, 19, 289, 21, 14641, 28, 15625, 30, 36, 23, 361, 22, 28561, 32, 117649, 40, 100, 48, 29, 529, 26, 83521, 44, 1771561, 42, 196, 80, 1024, 31, 841, 27

Offset: 1

Leroy Quet, May 31 2006

From Peter Munn, May 17 2023: (Start)
As a square array A(n,m), n, m >= 1, read by ascending antidiagonals, A(n,m) is the n-th positive integer with m+1 divisors.
Thus both formats list the numbers with m+1 divisors in their m-th column. For the corresponding sequences giving numbers with a specific number of divisors see the index entries link.
(End)

			Looking at the 4th row, 7 is the 4th positive integer with 2 divisors, 25 is the 3rd positive integer with 3 divisors, 8 is the 2nd positive integer with 4 divisors and 16 is the first positive integer with 5 divisors. So the 4th row is (7,25,8,16).
The triangle T(n,m) begins:
  n\m:    1     2     3     4     5     6     7
  ---------------------------------------------
   1 :    2
   2 :    3     4
   3 :    5     9     6
   4 :    7    25     8    16
   5 :   11    49    10    81    12
   6 :   13   121    14   625    18    64
   7 :   17   169    15  2401    20   729    24
  ...
Square array A(n,m) begins:
  n\m:     1      2      3       4      5  ...
  --------------------------------------------
   1 :     2      4      6      16     12  ...
   2 :     3      9      8      81     18  ...
   3 :     5     25     10     625     20  ...
   4 :     7     49     14    2401     28  ...
   5 :    11    121     15   14641     32  ...
  ...

Index entries for sequences related to divisors of numbers

Columns: A000040, A001248, A007422, A030514, A030515, A030516, A030626, A030627, A030628, ... (see the index entries link for more).
Cf. A073915.
Diagonals (equivalently, rows of the square array) start: A005179\{1}, A161574.
Cf. A091538.

t[n_, m_] := Block[{c = 0, k = 1}, While[c < n + 1 - m, k++; If[DivisorSigma[0, k] == m + 1, c++ ]]; k]; Table[ t[n, m], {n, 11}, {m, n}] // Flatten (* Robert G. Wilson v, Jun 07 2006 *)

More terms from Robert G. Wilson v, Jun 07 2006

A190267 Numbers n such that d(n-1) = d(n+1) = 6, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

19, 51, 243, 244, 424, 476, 604, 638, 723, 846, 926, 1683, 1774, 2008, 2524, 2526, 2528, 3124, 3176, 3178, 4204, 4476, 4526, 4924, 5824, 6726, 6812, 6963, 7300, 7443, 7676, 8426, 8958, 8974, 9458, 9926, 10052, 10083, 10468, 11674, 11710, 12428, 12483, 12592

Offset: 1

Juri-Stepan Gerasimov, May 06 2011

nonn

Nathaniel Johnston, Table of n, a(n) for n = 1..2500

Cf. A030515, A189974.

with(numtheory): A190267 := proc(n) option remember: local k: if(n=1)then return 19:else k:=procname(n-1)+1: do if(tau(k-1)=6 and tau(k+1)=6)then return k: fi: k:=k+1: od: fi: end: seq(A190267(n), n=1..44); # Nathaniel Johnston, May 06 2011

Mean/@SequencePosition[DivisorSigma[0,Range[15000]],{6,,6}] (* _Harvey P. Dale, Jan 10 2025 *)

A379569 Number of n-digit numbers that have exactly 6 divisors.

Original entry on oeis.org

0, 16, 94, 654, 4863, 38243, 313705, 2658846, 23073712, 203859889, 1826368510, 16544195786, 151222451513, 1392635179004, 12906366376283, 120260052661235

Offset: 1

Seiichi Manyama, Dec 26 2024

			For n = 2 the a(2) = 16 numbers are 12, 18, 20, 28, 32, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99.

Column k=6 of A284398. Cf. A030515 (Numbers with exactly 6 divisors).

from math import isqrt
from sympy import primerange, primepi, integer_nthroot
def _sum(N): return sum(primepi(N//(p * p)) for p in primerange(isqrt(N//2)+1)) - primepi(integer_nthroot(N, 3)[0]) + primepi(integer_nthroot(N, 5)[0])
def a379569(n): return sum(10**n) - _sum(10**(n-1)) # _David Radcliffe, Dec 29 2024

Sum_{i=1..n} a(i) = Sum_{p prime} PrimePi(10^n/p^2) - PrimePi(10^(n/3)) + PrimePi(10^(n/5)). - David Radcliffe, Dec 29 2024

a(10)-a(15) from David Radcliffe, Dec 29 2024

a(16) from David Radcliffe, Jan 01 2025

A065985 Numbers k such that d(k) / 2 is prime, where d(k) = number of divisors of k.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, 38, 39, 44, 45, 46, 48, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 125, 129, 133, 134

Offset: 1

Joseph L. Pe, Dec 10 2001

nonn

Numbers whose sorted prime signature (A118914) is either of the form {2*p-1} or {1, p-1}, where p is a prime. Equivalently, disjoint union of numbers of the form q^(2*p-1) where p and q are primes, and numbers of the form r * q^(p-1), where p, q and r are primes and r != q. - Amiram Eldar, Sep 09 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000005, A118914.
Numbers with exactly 2*p divisors: A030513 (p=2), A030515 (p=3), A030628 \ {1} (p=5), A030632 (p=7), A137485 (p=11), A137489 (p=13), A175744 (p=17), A175747 (p=19).
Subsequences: A006881, A030078, A050997, A054753, A095990, A096156, A138031, A178739, A179665, A189987.

Select[Range[1, 1000], PrimeQ[DivisorSigma[0, # ] / 2] == True &]

n=0; for (m=1, 10^9, f=numdiv(m)/2; if (frac(f)==0 && isprime(f), write("b065985.txt", n++, " ", m); if (n==1000, return))) \\ Harry J. Smith, Nov 05 2009

is(n)=n=numdiv(n)/2; denominator(n)==1 && isprime(n) \\ Charles R Greathouse IV, Oct 15 2015

A079836 First column of the triangle in which the n-th row contains n numbers with n divisors that lie between A079835(n) and A079835(n+1).

Original entry on oeis.org

1, 3, 9, 51, 81, 28577, 117649, 594823330, 595067236, 596971504

Offset: 1

Amarnath Murthy, Feb 15 2003

1
3 5
9 25 49
51 55 57 58
81 625 2401 14641 28561
...
The 4th row consists of 4 consecutive elements of A030513, the 5th row 5 consecutive elements of A030514, the 6th and 7th rows consecutive elements of A030515 and A030516, the 8th of A030626, the 9th of A030627 etc. - R. J. Mathar, Mar 29 2007

Cf. A079835, A079837.

Cf. A030513, A030514, A030515, A030516, A030626, A030627.

a(6)-a(7) from R. J. Mathar, Mar 29 2007
a(8)-a(10) from Lambert Herrgesell (zero815(AT)googlemail.com), Feb 08 2008
a(2) and a(9) corrected by Pontus von Brömssen, Jan 14 2024

A253388 Numbers n such that the number of divisors of n is the product of two distinct primes.

Original entry on oeis.org

12, 18, 20, 28, 32, 44, 45, 48, 50, 52, 63, 68, 75, 76, 80, 92, 98, 99, 112, 116, 117, 124, 144, 147, 148, 153, 162, 164, 171, 172, 175, 176, 188, 192, 207, 208, 212, 236, 242, 243, 244, 245, 261, 268, 272, 275, 279, 284, 292, 304, 316, 320, 324, 325, 332, 333

Offset: 1

Amritpal Singh, Dec 31 2014

nonn

n such that A000005(n) is in A006881.

n is either of the form p^k where p is prime and k+1 is in A006881 or p1^k1*p2^k2 where p1 and p2 are distinct primes and k1+1 and k2+1 are distinct primes. - Robert Israel, Dec 31 2014

			12 has 6 divisors, and 6 is the product of two distinct primes, 2 and 3.

Robert Israel, Table of n, a(n) for n = 1..10000
Amritpal Singh, c++ program for generating the sequence

Cf. A000005, A006881. Contains A030515.

filter:= proc(n) local F;
  F:= ifactors(numtheory:-tau(n))[2];
  nops(F)=2 and F[1,2]=1 and F[2,2]=1;
end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 31 2014

a253388Q[x_] := Block[{d = FactorInteger[DivisorSigma[0, x]]},
Length[d] == 2 && Max[Last@Transpose@d] == 1]; a253388[n_] := Select[Range@n, a253388Q]; a253388[333] (* Michael De Vlieger, Jan 02 2015 *)
fQ[x_] := PrimeOmega@ x == 2 == PrimeNu@ x; Select[ Range@ 250, fQ[ DivisorSigma[0, #]] &] (* Robert G. Wilson v, Jan 13 2015 *)

isok(n) = (nbd = numdiv(n)) && (omega(nbd) == 2) && (bigomega(nbd) == 2); \\ Michel Marcus, Feb 07 2015

A373560 a(n) is the smallest multiple of prime(n)^2 that starts a run of 5 consecutive integers with 6 divisors, or -1 if no such multiple exists.

Original entry on oeis.org

-1, -1, -1, 10093613546512321, -1, -1, 7700031346933907521, -1, 5344962129269790721, -1, 20453982425165652721, -1, 8163195338222675521, -1, 2467958104789157112721, -1, -1, -1, -1, 14666767069023896053921, 212170739123852995921, 287954235303137500060321, -1, 84769922583214545304321

Offset: 1

Jon E. Schoenfield, Jun 09 2024

sign

Terms were obtained using the b-file at A141621.
a(n) = -1 if prime(n) is not in A001132.
Conjecture: the converse is also true.

			a(1) = a(2) = a(3) = -1 because the first of five consecutive integers having six divisors is never a multiple of 2^2, 3^2, or 5^2.
a(4) = 10093613546512321 because it is the smallest term in A141621 that is a multiple of prime(4)^2 = 49.
a(9) = 5344962129269790721 because it is the smallest term in A141621 that is a multiple of prime(9)^2 = 23^2.

Cf. A001132, A030515, A042999, A054753, A141621.

cp's OEIS Frontend

A350150 a(1)=1; thereafter a(n+1) is the smallest unused number k such that d(k) and d(a(n)) are coprime, where d is the divisor counting function A000005.

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A119586 Triangle where T(n,m) = (n+1-m)-th positive integer with (m+1) divisors.

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A190267 Numbers n such that d(n-1) = d(n+1) = 6, where d(k) is the number of divisors of k (A000005).

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A379569 Number of n-digit numbers that have exactly 6 divisors.

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A065985 Numbers k such that d(k) / 2 is prime, where d(k) = number of divisors of k.

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A079836 First column of the triangle in which the n-th row contains n numbers with n divisors that lie between A079835(n) and A079835(n+1).

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A253388 Numbers n such that the number of divisors of n is the product of two distinct primes.

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Maple

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A373560 a(n) is the smallest multiple of prime(n)^2 that starts a run of 5 consecutive integers with 6 divisors, or -1 if no such multiple exists.

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