A164977 - OEIS

%I #66 Jan 19 2025 18:21:26

%S 3,4,5,6,10,13,22,37,46,58,61,73,82,106,157,166,178,193,226,262,277,

%T 313,346,358,382,397,421,457,466,478,502,541,562,586,613,661,673,718,

%U 733,757,838,862,877,886,982,997,1018,1093,1153,1186,1201,1213,1237,1282

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

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

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

%C Contains all numbers in A005383. - _Harry Richman_, Jan 09 2025

%C Contains all numbers in A077065. - _Alois P. Heinz_, Jan 19 2025

%H Alois P. Heinz, <a href="/A164977/b164977.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

%F { m :  A035470(m) = 2 }.

%F { m :  A164978(m) = 2 }.

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

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

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

%F { m :  A069904(m) = 2 }.

%F { m :  A001222(n) + A001222(n+1) = 3 }. - _Alois P. Heinz_, Jan 08 2022

%F { A005383 } union { A077065 }. - _Alois P. Heinz_, Jan 19 2025

%e 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.

%e 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.

%p a:= proc(n) option remember; local k;

%p       for k from 1+ `if`(n=1, 2, a(n-1))

%p       while not (andmap(isprime, [k, (k+1)/2]) or

%p                  andmap(isprime, [k+1, k/2]))

%p       do od; k

%p     end:

%p seq(a(n), n=1..100);

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

%o (PARI) is(n)=if(isprime(n),bigomega(n+1)==2, isprime(n+1) && bigomega(n)==2) \\ _Charles R Greathouse IV_, Sep 08 2015

%o (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

%o (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

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

%Y Cf. A005383, A077065 (distinct subsequences).

%K nonn,easy

%O 1,1

%A _Alois P. Heinz_, Sep 03 2009