A069562 -id:A069562 - cp's OEIS frontend

A072502 Numbers that are run sums (trapezoidal, the difference between two triangular numbers) in exactly 3 ways.

9, 18, 25, 36, 49, 50, 72, 98, 100, 121, 144, 169, 196, 200, 242, 288, 289, 338, 361, 392, 400, 484, 529, 576, 578, 676, 722, 784, 800, 841, 961, 968, 1058, 1152, 1156, 1352, 1369, 1444, 1568, 1600, 1681, 1682, 1849, 1922, 1936, 2116, 2209, 2304, 2312, 2704

Offset: 1

Ron Knott, Jan 27 2003

Also numbers that are the product of a power of 2 (A000079) and the square of an odd prime, or numbers having exactly 3 odd divisors: A001227(a(n)) = 3. - Reinhard Zumkeller, May 01 2012
Numbers n such that the symmetric representation of sigma(n) has 3 subparts. - Omar E. Pol, Dec 28 2016
Also numbers that can be expressed as the sum of k > 1 consecutive positive integers in exactly 2 ways. E.g., 2+3+4 = 9 and 4+5 = 9, 3+4+5+6 = 18 and 5+6+7 = 18. - Julie Jones, Aug 13 2018
Appears to be numbers n such that tau(2*n) = tau(n) + 3. - Gary Detlefs, Jan 22 2020
Column 3 of A266531. - Omar E. Pol, Dec 01 2020

			a(1)=9 is the smallest number with 3 run sums: 2+3+4 = 4+5 = 9.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ron Knott, Introducing Runsums - a sum of consecutive integers.
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article #99.1.6.

Not to be confused with A069562.

Cf. A001227, A001248, A038547, A038550, A085548, A237593, A266531, A279387.

import Data.Set (singleton, deleteFindMin, insert)
a072502 n = a072502_list !! (n-1)
a072502_list = f (singleton 9) $ drop 2 a001248_list where
   f s (x:xs) = m : f (insert (2 * m) $ insert x s') xs where
                (m,s') = deleteFindMin s
-- Reinhard Zumkeller, May 01 2012

Sum_{n>=1} 1/a(n) = 2 * Sum_{p odd prime} 1/p^2 = 2 * A085548 - 1/2 = 0.404494... - Amiram Eldar, Feb 18 2021

Extended by Ray Chandler, Dec 30 2011

A372724 Numbers k such that k = Sum_{j=2..k+2} L(k/prime(j)) where L(n/p) is the Legendre symbol. Fixed points of A372725.

Original entry on oeis.org

0, 9, 25, 36, 49, 81, 100, 121, 144, 169, 196, 289, 324, 361, 400, 484, 529, 576, 625, 676, 729, 784, 841, 961, 1156, 1296, 1369, 1444, 1600, 1681, 1849, 1936, 2116, 2209, 2304, 2401, 2500, 2704, 2809, 2916, 3136, 3364, 3481, 3721, 3844, 4489, 4624, 5041, 5184, 5329, 5476, 5776

Offset: 1

Peter Luschny, May 22 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 1..10000

Subsequence of A000290, and A069562 U {0}.

Cf. A372725, A336101, A373087.

L := (n, k) -> NumberTheory:-LegendreSymbol(n, ithprime(k)):
s := n -> local k; add(L(n, k), k = 2..n + 2):
select(m -> m = s(m), [seq(0..400)]);
# Alternative:
isA := k -> (k = 0) or (issqr(k) and
       nops(NumberTheory:-PrimeFactors(k/2^padic[ordp] (k, 2))) = 1):
select(isA, [seq(0..6000)]);

Join[{0}, Select[Range[100]^2, PrimeNu[#/2^IntegerExponent[#, 2]] == 1 &]] (* Paolo Xausa, Jul 10 2024 *)

isok(k) = k == sum(j=2, k+2, kronecker(k, prime(j))); \\ Michel Marcus, May 22 2024

A positive k is a term if k is a square and its odd part is divisible by exactly one prime.

A376218 Odd exponentially odd numbers.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 149

Offset: 1

Amiram Eldar, Sep 16 2024

First differs from its subsequence A182318 at n = 8318: a(8318) = 19683 = 3^9 = 3^(3^2) is not a term of A182318.
Numbers whose prime factorization contains only odd primes and odd exponents.
Numbers whose sum of coreful divisors (A057723) is odd (a coreful divisor d of a number k is a divisor that is divisible by every prime that divides k, see also A307958).
The even exponentially odd numbers are numbers of the form 2^k * m, where k is odd and m is a term of this sequence.
The asymptotic density of this sequence is (3/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (3/5) * A065463 = 0.42266532... .

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

Intersection of A005408 and A268335.
Cf. A057723, A065463, A182318, A307958.
Other numbers with an odd sum of divisors: A000079 (unitary divisors), A028982 (all divisors), A069562 (non-unitary divisors), A357014 (exponential divisors).

Select[Range[1, 150, 2], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

is(k) = k % 2 && vecprod(factor(k)[,2]) % 2;

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A072502 Numbers that are run sums (trapezoidal, the difference between two triangular numbers) in exactly 3 ways.

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A372724 Numbers k such that k = Sum_{j=2..k+2} L(k/prime(j)) where L(n/p) is the Legendre symbol. Fixed points of A372725.

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A376218 Odd exponentially odd numbers.

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