A070258 -id:A070258 - cp's OEIS frontend

A365865 Starts of runs of 3 consecutive integers that are divisible by the square of their least prime factor.

423, 475, 1323, 1375, 1519, 2007, 2223, 2275, 2871, 3123, 3175, 3211, 3283, 3479, 3575, 3751, 3771, 4023, 4075, 4475, 4923, 4959, 4975, 5047, 5535, 5823, 5875, 6723, 6775, 6811, 7299, 7623, 7675, 8107, 8379, 8523, 8575, 8955, 9423, 9475, 10323, 10339, 10375, 10467

Offset: 1

Amiram Eldar, Sep 21 2023

Numbers k such that k, k+1 and k+2 are all terms of A283050.
Numbers of the form 4*k+2 are not terms of A283050. Therefore, there are no runs of 4 or more consecutive integers, and all the terms of this sequence are of the form 4*k+3.
The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are 2, 40, 429, 4419, 44352, 444053, 4441769, 44421000, 444220814, ... . Apparently, the asymptotic density of this sequence exists and equals 0.004442... .

			423 is a term since 3 is the least prime factor of 423 and 423 is divisible by 3^2 = 9, 2 is the least prime factor of 424 and 424 is divisible by 2^2 = 4, and 5 is the least prime factor of 425 and 425 is divisible by 5^2 = 25.

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

Cf. A067029.

Subsequence of A004767, A070258, A283050 and A365864.

Select[4 * Range[2700] + 3, AllTrue[# + {0, 1, 2}, FactorInteger[#1][[1, -1]] >= 2 &] &]
SequencePosition[Table[If[Divisible[n,FactorInteger[n][[1,1]]^2],1,0],{n,11000}],{1,1,1}][[;;,1]] (* Harvey P. Dale, Aug 05 2024 *)

is(n) = factor(n)[1,2] >= 2;
lista(kmax) = forstep(k = 3, kmax, 4, if(is(k) && is(k+1) && is(k+2), print1(k, ", ")));

A114180 Numbers n with mu(n) = mu(n+1) = mu(n+2).

Original entry on oeis.org

29, 33, 41, 48, 85, 93, 98, 101, 124, 137, 141, 201, 213, 217, 229, 242, 243, 281, 301, 342, 350, 393, 423, 429, 433, 445, 475, 548, 603, 617, 633, 641, 645, 697, 724, 741, 774, 821, 844, 845, 846, 869, 921, 969, 1021, 1024, 1041, 1085, 1129, 1137, 1189

Offset: 1

Franklin T. Adams-Watters, Feb 03 2006

nonn

Any sequence of 4 or more consecutive numbers with the same value for mu must all have mu(n)=0 (n divisible by a proper square) since at least one of every 4 consecutive numbers is divisible by 4.

A261890(a(n)) = 0. - Reinhard Zumkeller, Sep 05 2015

			mu(n)=1 for 33,34,35; 85,86,87; 93,94,95; ...
mu(n)=-1 for 29,30,31; 41,42,43; 101,102,103; ...
mu(n)=0 for 48,49,50; 98,99,100; 124,125,126; ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Union of A070258, A063838 and A070268. Cf. A008683, A070284.

Cf. A261890.

a114180 n = a114180_list !! (n-1)
a114180_list = filter ((== 0) . a261890) [1..]
-- Reinhard Zumkeller, Sep 05 2015

SequencePosition[MoebiusMu[Range[1200]],{x_,x_,x_}][[;;,1]] (* Harvey P. Dale, Jul 23 2023 *)

A372694 Numbers k such that A372692(k) = A372692(k+1) = A372692(k+2) > 1.

Original entry on oeis.org

17784450, 28873348, 38990474, 44534923, 48780675, 85648274, 95438475, 100982924, 111100050, 157757850, 184508323, 188754075, 225621674, 240956324, 251073450, 308820148, 318937274, 334271924, 365595074, 371139523, 378806848, 391046850, 437704650, 505568474, 511112923

Offset: 1

Amiram Eldar, May 10 2024

nonn

Can 4 consecutive integers have the same value of A372692? There are none below 2*10^10.

Amiram Eldar, Table of n, a(n) for n = 1..1200

Subsequence of A070258 and A372693.

Cf. A334021, A324310, A372692.

f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], ?(# == 0 &)])); s[1] = 1; s[n] := s[n] = Times @@ (Flatten@ (f @@@ FactorInteger[n]) + 1);
Select[Range[10^8], (s1 = s[#]) > 1 && s1 == s[# + 1] == s[# + 2] &]

s(n) = {my(f = factor(n), k); prod(i = 1, #f~, k = apply(x -> 1 - x, binary(f[i, 2])); prod(j = 1, #k, if(k[j], f[i, 1]^(2^(#k-j)) + 1, 1)));}
lista(kmax) = {my(s1 = s(1), s2 = s(2), s3); for(k = 3, kmax, s3 = s(k); if(s1 > 1 && s1 == s2 && s2 == s3, print1(k - 2, ", ")); s1 = s2; s2 = s3);}

A376469 Starts of runs of 3 consecutive integers in which each member of the run has at least one divisor of the form p^e with p <= e, where p is a prime.

Original entry on oeis.org

71874, 109375, 156248, 181250, 228123, 265624, 409374, 446875, 493748, 518750, 565623, 603124, 746874, 784375, 831248, 856250, 903123, 940624, 1084374, 1121875, 1168748, 1193750, 1240623, 1278124, 1421874, 1459375, 1506248, 1531250, 1578123, 1615624, 1759374, 1796875

Offset: 1

Amiram Eldar, Sep 23 2024

nonn

The start of the least run of 4 (and also 5) consecutive integers with this property is 3988418748.

The numbers of terms that do not exceed 10^k, for k = 5, 6, ..., are 1, 18, 178, 1783, 17845, 178458, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00001784... .

			71874 = 2 * 3^3 * 11^3 is a term since it is divisible by 3^3, 71875 = 5^5 * 23 is divisible by 5^5, and 71876 = 2^2 * 7 * 17 * 151 is divisible by 2^2.

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

Subsequence of A100716, A070258 and A129254.

q[n_] := q[n] = AnyTrue[FactorInteger[n], First[#] <= Last[#] &]; Select[Range[2*10^6], q[#] && q[#+1] && q[#+2] &]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i,1] <= f[i,2], return(1))); 0;}
lista(kmax) = {my(is1 = 0, is2 = 0, is3); for(k = 3, kmax, is3 = is(k); if(is1 && is2 && is3, print1(k-2, ", ")); is1 = is2; is2 = is3);}

A268270 Least prime that is at distance > n from the nearest squarefree number.

Original entry on oeis.org

2, 17, 727, 47527, 29002021, 494501773, 44633461423, 7489886099881

Offset: 0

Juri-Stepan Gerasimov and M. F. Hasler, Jan 01 2016

			a(0)=2 is the least prime and it is at distance 1 from the nearest squarefree numbers (1 and/or 3).
a(1)=17 is the least prime that has no squarefree neighbor: both 16 and 18 are divisible by a square; the nearest squarefree numbers, 15 and 19, are both at distance 2.
a(2)=727 is the least prime p such that p-2 and p+1 are (two consecutive terms) in A068781, namely A068781(75..76).
a(3)=47527 is the least prime p such that p-3 and p+1 are (two consecutive terms) in A070258, namely A070258(878..879).
a(4)=29002021 is the least prime p such that p-4 and p+1 are (two consecutive terms) in A070284.
a(5)=494501773 is the least prime p such that p-5 and p+1 are (two consecutive terms) in A078144.
Similarly, for n = 6, 7, 8 and 9, a(n) is the least prime p such that p-n and p+1 are (two consecutive terms) in A049535, A077640, A077647 and A078143, respectively.

Chris Thompson, What is the next one? [primes isolated from squarefrees], SeqFan list.

Cf. A023186, A045882.

a(n)=forprime(p=n,,for(s=1,n,(issquarefree(p-s)||issquarefree(p+s)) && next(2)); return(p))

a(4) corrected and a(5) computed by Christopher E. Thompson, Jan 20 2016

a(6)-a(7) from Bert Dobbelaere, Jan 28 2019

A359839 Numbers k such that k, k + 1 and k + 2 are 3 consecutive Niven (Harshad) numbers that are also divisible by a square.

Original entry on oeis.org

2023, 4912, 12103, 17575, 23273, 51424, 52675, 60399, 78650, 80800, 87723, 93624, 100303, 112624, 117962, 121224, 122875, 182182, 193075, 200752, 228175, 235024, 245725, 245726, 249500, 263275, 306963, 320704, 333475, 373490, 403675, 416583, 421072, 444624, 448000

Offset: 1

Bernard Schott, Jan 15 2023

Equivalently, smallest of 3 consecutive numbers each divisible by a square and also divisible by the sum of their digits (Niven numbers).

			2023 = 7 * 17^2 = 289 * (2+0+2+3); 2024 = 506 * 2^2 = 253 * (2+0+2+4) and 2025 = 81 * 5^2 = 225 * (2+0+2+5) hence 2023 is a term.

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 48, p. 18, Ellipses, Paris, 2008.
Jean-Marie De Koninck, Those Fascinating Numbers, Entry 110, p. 36, American Mathematical Society, 2009.

Curtis Cooper and Robert E. Kennedy, On consecutive Niven numbers, Fibonacci Quarterly, Vol. 21, No. 2 (1993), pp. 146-151.
Giovanni Resta, Harshad numbers.
Wikipedia, Harshad number.

Intersection of A070258 and A154701.
Cf. A013929, A068781, A235578.
Cf. A005349, A060159, A141769, A330927, A330928, A330929, A330930.

q[n_] := Divisible[n, Total@IntegerDigits[n]] && ! SquareFreeQ[n]; tri = q /@ Range[3]; seq = {}; Do[tri = Join[Rest[tri], {q[k]}]; If[And @@ tri, AppendTo[seq, k - 2]], {k, 3, 5*10^5}]; seq (* Amiram Eldar, Jan 15 2023 *)

More terms from Amiram Eldar, Jan 15 2023

cp's OEIS Frontend

A365865 Starts of runs of 3 consecutive integers that are divisible by the square of their least prime factor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A114180 Numbers n with mu(n) = mu(n+1) = mu(n+2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A372694 Numbers k such that A372692(k) = A372692(k+1) = A372692(k+2) > 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A376469 Starts of runs of 3 consecutive integers in which each member of the run has at least one divisor of the form p^e with p <= e, where p is a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A268270 Least prime that is at distance > n from the nearest squarefree number.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A359839 Numbers k such that k, k + 1 and k + 2 are 3 consecutive Niven (Harshad) numbers that are also divisible by a square.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions