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A164977
Numbers m such that the set {1..m} has only one nontrivial decomposition into subsets with equal element sum.
11
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 m*(m+1)/2 has exactly two divisors >= m.
Also numbers m such that m*(m+1)/2 is the product of two primes.
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
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 (isprime(k) and isprime((k+1)/2)
or isprime(k+1) and isprime(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
KEYWORD
nonn,easy
AUTHOR
Alois P. Heinz, Sep 03 2009
STATUS
approved