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

1,1

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

Links

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

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Cf. A005383, A077065 (distinct subsequences).

Sequence in context: A218946 A174057 A335231 * A103033 A099561 A110300

Adjacent sequences: A164974 A164975 A164976 * A164978 A164979 A164980

Keyword

nonn,easy

Author

Alois P. Heinz, Sep 03 2009

Status

approved