A164977 Numbers m such that the set {1..m} has only one nontrivial decomposition into subsets with equal element sum.

3, 4, 5, 6, 10, 13, 22, 37, 46, 58, 61, 73, 82, 106, 157, 166, 178, 193, 226, 262, 277, 313, 346, 358, 382, 397, 421, 457, 466, 478, 502, 541, 562, 586, 613, 661, 673, 718, 733, 757, 838, 862, 877, 886, 982, 997, 1018, 1093, 1153, 1186, 1201, 1213, 1237, 1282

Offset: 1

Author

Alois P. Heinz, Sep 03 2009

Keywords

• nonn
• easy

Comments

Numbers m such that the triangular number T(m) = m*(m+1)/2 has exactly two divisors >= m.

Also numbers m such that m*(m+1)/2 is the product of two primes.

Contains all numbers in A005383. - Harry Richman, Jan 09 2025

Contains all numbers in A077065. - Alois P. Heinz, Jan 19 2025

Examples

			10 is in the sequence, because there is only one nontrivial decomposition of {1..10} into subsets with equal element sum: {1,10}, {2,9}, {3,8}, {4,7}, {5,6}; 11|55.
13 is in the sequence with decomposition of {1..13}: {1,12}, {2,11}, {3,10}, {4,9}, {5,8}, {6,7}, {13}; 13|91.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Crossrefs

Cf. A000217, A001222, A164978, A035470, A068443, A069904, A001358.

Cf. A005383, A077065 (distinct subsequences).

Programs

Maple

a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, 2, a(n-1))
      while not (andmap(isprime, [k, (k+1)/2]) or
                 andmap(isprime, [k+1, k/2]))
      do od; k
    end:
seq(a(n), n=1..100);

Mathematica

Select[Range@1304, PrimeOmega[#] + PrimeOmega[# + 1] == 3 &] (* Robert G. Wilson v, Jun 28 2010 and updated Sep 21 2018 *)

PARI

is(n)=if(isprime(n),bigomega(n+1)==2, isprime(n+1) && bigomega(n)==2) \\ Charles R Greathouse IV, Sep 08 2015

PARI

is(n)=if(n%2, isprime((n+1)/2) && isprime(n), isprime(n/2) && isprime(n+1)) \\ Charles R Greathouse IV, Mar 16 2022

PARI

list(lim)=my(v=List()); forprime(p=3,lim, if(isprime((p+1)/2), listput(v,p))); forprime(p=5,lim+1, if(isprime(p\2), listput(v,p-1))); Set(v) \\ Charles R Greathouse IV, Mar 16 2022

Formula

{ m : A035470(m) = 2 }.

{ m : A164978(m) = 2 }.

{ m : |{d|m*(m+1)/2 : d>=m}| = 2 }.

{ m : m*(m+1)/2 in {A068443} }.

{ m : m*(m+1)/2 in {A001358} }.

{ m : A069904(m) = 2 }.

{ m : A001222(n) + A001222(n+1) = 3 }. - Alois P. Heinz, Jan 08 2022

{ A005383 } union { A077065 }. - Alois P. Heinz, Jan 19 2025