A070915 -id:A070915 - cp's OEIS frontend

A000977 Numbers that are divisible by at least three different primes.

30, 42, 60, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 198, 204, 210, 220, 222, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258, 260, 264, 266, 270, 273, 276, 280, 282, 285

Offset: 1

N. J. A. Sloane

a(n+1)-a(n) seems bounded and sequence appears to give n such that the number of integers of the form nk/(n+k) k>=1 is not equal to Sum_{ d | n} omega(d) (i.e., n such that A062799(n) is not equal to A063647(n)). - Benoit Cloitre, Aug 27 2002

The first differences are bounded: clearly a(n+1) - a(n) <= 30. - Charles R Greathouse IV, Dec 19 2011

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A000961, A007774, A033992, A033993, A051270.

Complement of A070915.

a000977 n = a000977_list !! (n-1)
a000977_list = filter ((> 2) . a001221) [1..]
-- Reinhard Zumkeller, May 03 2013

A000977 := proc(n)
if (nops(numtheory[factorset](n)) >= 3) then
   RETURN(n)
fi: end:  seq(A000977(n), n=1..500); # Jani Melik, Feb 24 2011

DeleteCases[Table[If[Count[PrimeQ[Divisors[i]], True] >= 3, i, 0], {i, 1, 274}], 0]
Select[Range[300], PrimeNu[#] >= 3 &] (* Paolo Xausa, Mar 28 2024 *)

is(n)=omega(n)>2 \\ Charles R Greathouse IV, Dec 19 2011

a(n) = n + O(n log log n / log n). - Charles R Greathouse IV, Dec 19 2011 A001221(a(n)) > 2. - Reinhard Zumkeller, May 03 2013

A033992 UNION A033993 UNION A051270 UNION A074969 UNION A176655 UNION ... - R. J. Mathar, Dec 05 2016

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 17 2002

A085726 Numbers n such that n-th Lucas number is a semiprime.

Original entry on oeis.org

3, 10, 14, 20, 23, 26, 29, 32, 38, 43, 49, 56, 62, 64, 67, 68, 73, 76, 80, 83, 86, 89, 97, 107, 121, 128, 136, 137, 157, 164, 167, 172, 178, 197, 202, 211, 223, 229, 284, 293, 307, 311, 328, 373, 389, 397, 458, 487, 521, 541, 557, 577, 586, 619, 673, 857, 914, 929, 947, 1082, 1151, 1249, 1277, 1279, 1306, 1318, 1493, 1499, 1667

Offset: 1

Jason Earls, Jul 20 2003

nonn

From results on the divisibility of generalized Fibonacci sequences (2nd order recurrences with various integer initial values), it follows that if n is such that n-th Lucas number is a semiprime, it is necessary but not sufficient that n have at most two distinct prime factors (A070915). That is: A000204(n) an element of A001358 implies n an element of A070915. - Jonathan Vos Post, Sep 22 2005
All numbers in this sequence have the form 2^r p^s, where p is an odd prime and r and s are not both zero. It appears that s=2 for only p=7 and 11, otherwise s=0 or 1. - T. D. Noe, Nov 29 2005
Sequence continues as 1831?, 1877?, 1901, 1951, ... where ? mark uncertain terms. - Max Alekseyev, Aug 18 2013

Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A000204.

Cf. A072381 (n such that Fibonacci(n) is a semiprime).

IsSemiprime:=func; [n: n in [2..300] | IsSemiprime(Lucas(n))]; // Vincenzo Librandi, Feb 12 2016

a = 1; b = 3; Do[c = a + b; If[Plus@@Last/@FactorInteger[c] == 2, Print[n]]; a = b; b = c, {n, 3, 200}] (* Ryan Propper, Jun 28 2005 *)
Select[Range[400], PrimeOmega[LucasL[#]] == 2 &] (* Vincenzo Librandi, Feb 12 2016 *)

isok(n) = bigomega(fibonacci(n+1)+fibonacci(n-1)) == 2; \\ Michel Marcus, Feb 12 2016

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 25 2004
More terms from Ryan Propper, Jun 28 2005
More terms from T. D. Noe, Nov 29 2005
a(60)-a(62) from Max Alekseyev, Aug 18 2013
a(63)-a(69) from Sean A. Irvine, Feb 11 2016

A292786 a(n) = psi(n) - phi(n).

Original entry on oeis.org

0, 2, 2, 4, 2, 10, 2, 8, 6, 14, 2, 20, 2, 18, 16, 16, 2, 30, 2, 28, 20, 26, 2, 40, 10, 30, 18, 36, 2, 64, 2, 32, 28, 38, 24, 60, 2, 42, 32, 56, 2, 84, 2, 52, 48, 50, 2, 80, 14, 70, 40, 60, 2, 90, 32, 72, 44, 62, 2, 128, 2, 66, 60, 64, 36, 124, 2, 76, 52, 120, 2, 120, 2, 78, 80, 84, 36, 144

Offset: 1

Altug Alkan, Sep 23 2017

Even numbers that are not the terms of this sequence are 12, 102, 114, 130, ...

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A001615, A051612, A069359, A070915, A088245, A291784.

psi[n_] := If[n < 1, 0, n Sum[ MoebiusMu[d]^2/d, {d, Divisors@ n}]]; Array[psi@# - EulerPhi@# &, 87] (* Robert G. Wilson v, Sep 23 2017 *)

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
a(n) = a001615(n) - eulerphi(n); \\ after Charles R Greathouse IV at A001615

a(n) = A001615(n) - A000010(n).
a(n) = 2 iff n is prime.
a(n) = 2*A069359(n) iff n is in A070915.
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c =  9/(2*Pi^2) = 0.455945... (A088245). - Amiram Eldar, Dec 05 2023

A379096 Waterproof numbers >= 60.

Original entry on oeis.org

61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121

Offset: 1

Peter Luschny, Dec 16 2024

nonn

All nonnegative numbers less than 60 are waterproof.
Zero and one are waterproof numbers by convention. Numbers that admit a prime factorization are waterproof if their water capacity is 0. (The water capacity of a number is defined in A275339.)
If the factors p_i^e_i in the canonical prime factorization of n are weakly ascending or weakly descending, then n is waterproof.
A number is waterproof if and only if it equals its waterproof hull (A379098). The waterproof hull h(n) of n is the smallest waterproof number that n divides.
Numbers that are not waterproof are listed in A379097.

			Numbers having at most two distinct prime factors (A070915) are waterproof. The primorials (A002110) are waterproof.
48300 has a water capacity of 17 and so is not waterproof. The waterproof hull of 48300 is 1014300.

Cf. A275339, A379094, A379095, A379097, A379098.

# The function 'water_capacity' is defined in A275339.
is_waterproof := n -> ifelse(n < 2, true, is(water_capacity(n) = 0)):
select(is_waterproof, [seq(60..121)]);

# The function 'WaterCapacity' is defined in A275339.
print([n for n in range(60, 122) if WaterCapacity(n) == 0])

A309015 2-highly composite numbers: 3-smooth numbers (A003586) k with d(k) > d(j) for all 3-smooth numbers j < k, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 72, 144, 216, 288, 432, 576, 864, 1296, 1728, 2592, 3456, 5184, 6912, 10368, 15552, 20736, 31104, 41472, 62208, 82944, 93312, 124416, 186624, 248832, 373248, 497664, 746496, 995328, 1119744, 1492992, 2239488, 2985984, 4478976, 5971968

Offset: 1

Amiram Eldar, Jul 06 2019

nonn

Also numbers with record numbers of divisors among the numbers with at most 2 distinct prime factors (A070915).
Bessi and Nicolas proved that there exists a constant c such that the number of 2-highly composite numbers smaller than x is larger than c*(log(x))^(4/3).
In general, k-highly composite numbers (defined by Nicolas, 2005) are numbers with a record number of divisors where only p(k)-smooth numbers are being considered. Equivalently only numbers with at most k distinct prime factors can be considered.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Pascal Alessandri and Valérie Berthé, Three distance theorems and combinatorics on words, Enseignement Mathématique, Vol. 44 (1998), pp. 103-132.
Gérard Bessi, Etude des nombres 2-hautement composés, Séminaire de Théorie des nombres de Bordeaux, Vol. 4 (1975), pp. 1-22.
Gérard Bessi and J. L. Nicolas, Nombres 2-hautement composés, J. Math. pures et appl., Vol. 56 (1977), pp. 307-326.
Amiram Eldar, Table of n, a(n), A000005(a(n)) for n = 1..10000
Jean-Louis Nicolas, Some open questions, The Ramanujan Journal, Vol. 9 (2005), pp. 251-264.

Cf. A000005, A002182, A003586, A070915.

dmax = 0; s = {}; Do[If[EulerPhi[6n] == 2n, d = DivisorSigma[0, n]; If[d > dmax, dmax = d; AppendTo[s, n]]], {n, 1, 10^4}]; s (* after Artur Jasinski at A003586 *)
Block[{n = 10^7, s, t}, s = Union@ Flatten@ Table[2^a * 3^b, {a, 0, Log2@ n}, {b, 0, Log[3, n/(2^a)]}]; t = DivisorSigma[0, s]; Map[s[[FirstPosition[t, #][[1]] ]] &, Union@ FoldList[Max, t]]] (* Michael De Vlieger, Jul 09 2019 *)

A306369 a(n) = A000010(n) + A069359(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 14, 13, 15, 16, 16, 17, 21, 19, 22, 22, 23, 23, 28, 25, 27, 27, 30, 29, 39, 31, 32, 34, 35, 36, 42, 37, 39, 40, 44, 41, 53, 43, 46, 48, 47, 47, 56, 49, 55, 52, 54, 53, 63, 56, 60, 58, 59, 59, 78, 61, 63, 66, 64, 66, 81, 67, 70, 70, 83, 71, 84, 73, 75, 80

Offset: 1

Torlach Rush, Feb 10 2019

nonn

a(n) = A291784(n) iff A001221(n) < 3, that is, iff n is in A070915.

			1 is a term because A000010(1) + A069359(1) = 1 + 0.
7 is a term because A000010(6) + A069359(6) = 2 + 5 = 7 = 6 + 1 = A000010(7) + A069359(7).

Cf. A000010, A001221, A069359, A070915, A291784.

Cf. A059956, A085548.

A069359[n_] := n * Plus @@ (1/FactorInteger[n][[;; , 1]]); A069359[1] = 0; a[n_] := A069359[n] + EulerPhi[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2023 *)

a(n) = eulerphi(n) + n*sumdiv(n, d, isprime(d)/d); \\ Michel Marcus, Feb 12 2019

a(n) = A000010(n) + A069359(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A059956 + A085548 = 1.0601745... . - Amiram Eldar, Dec 05 2023

A381873 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) while containing at most two distinct prime factors.

Original entry on oeis.org

1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 20, 22, 11, 33, 27, 36, 26, 13, 39, 45, 25, 35, 28, 32, 34, 17, 51, 48, 38, 19, 57, 54, 40, 44, 46, 23, 69, 63, 49, 56, 50, 52, 58, 29, 87, 72, 62, 31, 93, 75, 55, 65, 80, 64, 68, 74, 37, 111, 81

Offset: 1

Scott R. Shannon, Mar 09 2025

Like the EKG sequence A064413 for the terms studied the primes appear in their natural order, although unlike A064413 some primes p are preceded by 3*p and followed by 2*p. The first time this occurs is a(682) = 1401, a(683) = 467, a(684) = 934, although as n increases this becomes more common.
The primes are all contained in the lowest line of values which has up upward curvature - see the attached image. This leads to it crossing the line a(n) = n and creating the fixed point a(15527). The only other fixed points are 1, 2, 8 and 40, and it is likely no more exist.
No further fixed points through a(8*10^5). - Michael S. Branicky, Mar 10 2025

			a(23) = 36 = 2^2*3^3 as a(22) = 27 and 36 is unused and shares a factor with 27 while containing two distinct prime factors. Note that 30 = 2*3*5 cannot be chosen as it contains three distinct prime factors; this is the first term to differ from A064413.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 30000 terms. The green line is a(n) = n.

Cf. A070915, A027748, A000977, A064413.

from math import gcd
from sympy import factorint
from functools import cache
from itertools import count, islice
@cache
def omega(n): return len(factorint(n))
def agen(): # generator of terms
    yield 1
    aset, an, m = {1}, 2, 3
    while True:
        yield an
        aset.add(an)
        an = next(k for k in count(m) if k not in aset and gcd(an, k) > 1 and omega(k) <= 2)
        while m in aset or omega(m) > 2: m += 1
print(list(islice(agen(), 66))) # Michael S. Branicky, Mar 09 2025

cp's OEIS Frontend

A000977 Numbers that are divisible by at least three different primes.

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A085726 Numbers n such that n-th Lucas number is a semiprime.

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A292786 a(n) = psi(n) - phi(n).

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A379096 Waterproof numbers >= 60.

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A309015 2-highly composite numbers: 3-smooth numbers (A003586) k with d(k) > d(j) for all 3-smooth numbers j < k, where d(k) is the number of divisors of k (A000005).

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A306369 a(n) = A000010(n) + A069359(n).

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A381873 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) while containing at most two distinct prime factors.

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