A285800 -id:A285800 - cp's OEIS frontend

A131651 Positive integers obtained as the difference of two triangular numbers in exactly 4 ways.

15, 21, 27, 30, 33, 35, 39, 42, 51, 54, 55, 57, 60, 65, 66, 69, 70, 77, 78, 84, 85, 87, 91, 93, 95, 102, 108, 110, 111, 114, 115, 119, 120, 123, 125, 129, 130, 132, 133, 138, 140, 141, 143, 145, 154, 155, 156, 159, 161, 168, 170, 174, 177, 182, 183, 185, 186, 187

Offset: 1

John W. Layman, Sep 10 2007

nonn

It appears that terms of the sequence are all given by a power of 2 times the cube of an odd prime or a power of 2 times a product of two distinct odd primes. (This has been verified for a(n) <= 10000.)
Apparently the integers that have exactly 4 odd divisors. (Verified for a(n) <= 187.) - Philippe Beaudoin, Oct 24 2013
Also numbers that can be expressed as the sum of k > 1 consecutive positive integers in exactly 3 ways; e.g., 7+8 = 15, 4+5+6 = 15 and 1+2+3+4+5 = 15. - Julie Jones, Aug 13 2018

			15 is in the sequence because 15 = 15 - 0 = 21 - 6 = 36 - 21 = 120 - 105, where all operands are triangular, and in no other way.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A001227, A038550, A072502, A285800.

isok(n) = {v = vector(n, i, i*(i+1)/2); nb = 0; for (i=1, n, if (ispolygonal(i*(i+1)/2 - n, 3), nb++; if (nb > 4, return (0)););); nb == 4;} \\ Michel Marcus, Jan 14 2014

A285801 Numbers having at most one odd prime factor to an odd power in their prime factorization.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 56, 58, 59, 61, 62, 63, 64, 67, 68, 71, 72, 73, 74, 75, 76, 79, 80, 81, 82, 83, 86, 88, 89

Offset: 1

M. F. Hasler, Apr 26 2017

The sequence is of asymptotic density zero. It seems to grow faster than n*(log_10(n)-1), which is a fair approximation in the range 10^3 .. 10^6 or beyond, cf. examples.

			A285800(1) = 15 = 3*5 is the smallest positive integer to have two odd prime factors to an odd power (here 1) in its factorization, therefore it's the first number not in this sequence.
A285800(2) = 21 = 3*7, A285800(3) = 30 = 2*A285800(1) and A285800(3) = 33 = 3*11 are the next three numbers not in this sequence.
a(10) = 10, a(100) = 137, a(10^3) = 2066, a(10^4) = 29996, a(10^5) = 402878, a(10^6) = 5083823.

N. J. A. Sloane, Table of n, a(n) for n = 1..10001

Complement of A285800.

s800:=[]; s801:=[1];
for n from 2 to 1000 do
c:=0;
t2:=ifactors(n)[2];
for t3 in t2 do if t3[1]>2 and (t3[2] mod 2 = 1) then c:=c+1; fi; od:
if c <= 1 then s801:=[op(s801),n]; else s800:=[op(s800),n]; fi;
od:
s800; # A285800
s801; # A285801 - N. J. A. Sloane, Sep 30 2017

is(n)=2>#select(t->bittest(t,0),factor(n>>valuation(n,2))[,2])

A340380 Numbers whose odd part is a squarefree semiprime (A006881); numbers of the form 2^k * p * q, with k >= 0, and distinct odd primes p and q.

Original entry on oeis.org

15, 21, 30, 33, 35, 39, 42, 51, 55, 57, 60, 65, 66, 69, 70, 77, 78, 84, 85, 87, 91, 93, 95, 102, 110, 111, 114, 115, 119, 120, 123, 129, 130, 132, 133, 138, 140, 141, 143, 145, 154, 155, 156, 159, 161, 168, 170, 174, 177, 182, 183, 185, 186, 187, 190, 201, 203, 204, 205, 209, 213, 215, 217, 219, 220, 221, 222, 228, 230

Offset: 1

Antti Karttunen, Jan 06 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..24567; terms < 2^16

Cf. A000265, A006881, A280710, A340370 (characteristic function).

Subsequence of A285800, from which this differs for the first time at n=25, where a(25) = 110, while A285800(25) = 105, which is missing from this sequence.

semiQ[n_] := FactorInteger[n][[;;,2]] == {1, 1}; Select[Range[230], semiQ[#/2^IntegerExponent[#, 2]] &] (* Amiram Eldar, Jan 03 2022 *)

isA340380(n) = A340370(n); \\ Uses the program given in A340370.

Sum_{n>=1} 1/a(n)^s = (2^s/(2^s-1)) * ((1/2)*(P(s)^2 - P(2*s)) + 1/4^s - P(s)/2^s), for s>1, where P is the prime zeta function. - Amiram Eldar, Jan 03 2022

A359641 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have exactly n odd prime factors, all with exponent 1.

Original entry on oeis.org

307, 8929, 992263, 229658167, 28674536239

Offset: 2

Hugo Pfoertner, Jan 17 2023

			a(3) = 8929: 8930 = 2*5*19*47, 8931 = 3*13*229, 8932 = 2^2*7*11*29;
a(6) = 28674536239: a(6)+1 = 2^4*5*7*31*43*107*359, a(6)+2 = 3*13*23*151*269*787, a(6)+3 = 2*11*17*19*37*191*571.

Cf. A001222, A001359, A087436, A285800, A359640.

obi(x,m=0) = {my (x2=x>>valuation(x,2), o=omega(x2)); if (o2, for (j=pp+1, p-1, my (mo=obi(j)); if (mo

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A131651 Positive integers obtained as the difference of two triangular numbers in exactly 4 ways.

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A285801 Numbers having at most one odd prime factor to an odd power in their prime factorization.

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A340380 Numbers whose odd part is a squarefree semiprime (A006881); numbers of the form 2^k * p * q, with k >= 0, and distinct odd primes p and q.

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A359641 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have exactly n odd prime factors, all with exponent 1.

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