A235578 -id:A235578 - cp's OEIS frontend

A070258 Smallest of 3 consecutive numbers each divisible by a square.

48, 98, 124, 242, 243, 342, 350, 423, 475, 548, 603, 724, 774, 844, 845, 846, 1024, 1250, 1274, 1323, 1375, 1420, 1448, 1519, 1664, 1674, 1680, 1681, 1682, 1848, 1862, 1924, 2007, 2023, 2056, 2106, 2150, 2223, 2275, 2348, 2366, 2523, 2527, 2574, 2644

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 09 2002

nonn

The sequence includes an infinite family of arithmetic progressions. Such AP's can be constructed to each term, with large differences [like e.g. square of primorials, A061742]. It is necessary to solve suitable systems of linear Diophantine equations. E.g.: subsequences of triples of terms = {900a+548, 900a+549, 900a+550} = {4(225f+137), 9(100f+61), 25(36f+22)}; starting terms in this sequence = {548, 1448, 2348, ...}; difference = A002110(3)^2. - Labos Elemer, Nov 25 2002
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 2, 16, 180, 1868, 18649, 186335, 1863390, 18634236, 186340191, ... . Apparently, the asymptotic density of this sequence exists and equals 0.01863... . - Amiram Eldar, Jan 18 2023
The asymptotic density of this sequence is 1 - 3/zeta(2) + 3 * Product_{p prime} (1 - 2/p^2) - Product_{p prime} (1 - 3/p^2) = 1 - 3 * A059956 + 3 * A065474 - A206256 = 0.018634010349844827414... . - Amiram Eldar, Sep 12 2024

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 48, p. 18, Ellipses, Paris, 2008.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Subsequence of A013929 and A068781.
Cf. A002110, A061742, A235578.
Cf. A059956, A065474, A206256.

f[n_] := Union[ Transpose[ FactorInteger[n]] [[2]]] [[ -1]]; a = 0; b = 1; Do[c = f[n]; If[a> 1 && b > 1 && c > 1, Print[n - 2]]; a = b; b = c, {n, 3, 10^6}]
Flatten[Position[Partition[SquareFreeQ/@Range[3000],3,1],?(Union[#] == {False}&),{1},Heads->False]] (* _Harvey P. Dale, May 24 2014 *)
f@n_ := Flatten@  Position[Partition[SquareFreeQ /@ Range@2000, n, 1], Table[False, {n}]]; f@3 (* Hans Rudolf Widmer, Aug 30 2022 *)

a(n) = A235578(n) - 1. - Amiram Eldar, Feb 09 2021

More terms from Jason Earls and Robert G. Wilson v, May 10 2002

Offset corrected by Amiram Eldar, Feb 09 2021

A281192 Numbers with no squarefree neighbors.

Original entry on oeis.org

17, 19, 26, 49, 51, 53, 55, 89, 91, 97, 99, 125, 127, 149, 151, 161, 163, 170, 197, 199, 233, 235, 241, 243, 244, 249, 251, 269, 271, 293, 295, 305, 307, 337, 339, 341, 343, 349, 351, 362, 377, 379, 413, 415, 424, 449, 451, 476, 485, 487, 489, 491, 521, 523, 530, 549, 551, 557, 559, 577

Offset: 1

Juri-Stepan Gerasimov, Jan 16 2017

Includes all k == 17 or 19 (mod 36), also 2*p^2-1 and 2*p^2+1 for odd primes p. - Robert Israel, Jan 17 2017
This sequence has density around 0.106.... - Charles R Greathouse IV, Jan 23 2017
More accurately, the asymptotic density of this sequence is 1 - 2/zeta(2) + Product_{p prime} (1 - 2/p^2) = 1 - 2 * A059956 + A065474 = 0.1067798952... - Amiram Eldar, Feb 25 2021

			17 is in this sequence because 17 - 1 = 16 = 2^4 and 17 + 1 = 18 = 2*3^2 are not squarefree.

Robert Israel, Table of n, a(n) for n = 1..10000

Supersequence of A075432 and A235578.

Cf. A005117, A059956, A065474, A280892.

[n: n in [2..600] | not IsSquarefree(n-1) and not IsSquarefree(n+1)];

select(t -> not numtheory:-issqrfree(t-1) and not numtheory:-issqrfree(t+1), [$1..1000]); # Robert Israel, Jan 17 2017

Select[Range[600], !SquareFreeQ[# - 1] && !SquareFreeQ[# + 1] &] (* Vincenzo Librandi, Jan 17 2017 *)

is(n)=!issquarefree(n-1) && !issquarefree(n+1) \\ Charles R Greathouse IV, Jan 23 2017

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

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A070258 Smallest of 3 consecutive numbers each divisible by a square.

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A281192 Numbers with no squarefree neighbors.

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

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