A077065 -id:A077065 - cp's OEIS frontend

A194593 Semiprimes s such that phi(s)/2 is prime.

9, 10, 14, 22, 46, 94, 118, 166, 214, 334, 358, 454, 526, 694, 718, 766, 934, 958, 1006, 1126, 1174, 1438, 1678, 1726, 1774, 1966, 2038, 2374, 2566, 2614, 2638, 2734, 2878, 2974, 3046, 3238, 3646, 3814, 4054, 4078, 4126, 4198, 4414, 4894, 4918, 5158, 5638, 5758, 5806, 5926, 5998

Offset: 1

Juri-Stepan Gerasimov, Aug 30 2011

nonn

For n > 2, A001221(a(n)) = A001221(A000010(a(n))) = 2, and A008683(a(n)) = A008683(A000010(a(n))) = 1. - Torlach Rush, Aug 23 2018

For n > 1, A000010(a(n)) = A077065(n-1). - Torlach Rush, Sep 11 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A001358, A005384, A005385, A079148, A065966.

[9] cat [2*p: p in PrimesUpTo(3000) | IsPrime((p - 1) div 2)]; // Vincenzo Librandi, Aug 25 2018

9, 10, op(select(s -> isprime(s/2) and isprime((s-2)/4), [seq(s,s=6..10000,8)])); # Robert Israel, Apr 06 2016

Select[Range@ 6000, PrimeOmega@ # == 2 && PrimeQ[EulerPhi[#]/2] &] (* Michael De Vlieger, Apr 06 2016 *)

isok(n) = (bigomega(n)== 2) && isprime(eulerphi(n)/2); \\ Michel Marcus, Apr 06 2016

a(n) = 2*A005385(n-1), n>1.

a(n) = 4*A005384(n-1) + 2, n > 1. - Michel Marcus, Apr 02 2020

Corrected by R. J. Mathar, Oct 13 2011

A369139 Numbers k such that Omega(k) = 1 + Omega(k + 1).

Original entry on oeis.org

4, 6, 8, 10, 20, 22, 45, 46, 50, 58, 68, 76, 80, 82, 92, 104, 105, 106, 110, 114, 117, 152, 154, 165, 166, 178, 182, 186, 189, 212, 226, 236, 246, 258, 260, 261, 262, 266, 273, 286, 290, 315, 318, 322, 325, 333, 338, 342, 344, 345, 346, 354, 357, 358, 370, 382, 385, 402, 406, 410, 412, 424, 426

Offset: 1

Zak Seidov and Robert Israel, Jan 14 2024

nonn

Numbers k that have one more prime divisor (counted by multiplicity) than k + 1.

			a(3) = 8 is a term because 8 = 2^3 has 3 prime divisors (counted by multiplicity) and 8 + 1 = 9 = 3^2 has 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001222, A045920, A076156. Contains A077065.

N:= 1000: # for terms <= N
V:= map(numtheory:-bigomega, [$1..N+1]):
select(t -> V[t] = 1 + V[t+1], [$1..N]);

s = {}; Do[If[PrimeOmega[k] == 1 + PrimeOmega[k + 1], AppendTo[s, k]], {k, 500}]; s

A317510 Numbers (4p+1)/3 where p is a Sophie Germain prime p > 3.

Original entry on oeis.org

7, 15, 31, 39, 55, 71, 111, 119, 151, 175, 231, 239, 255, 311, 319, 335, 375, 391, 479, 559, 575, 591, 655, 679, 791, 855, 871, 879, 911, 959, 991, 1015, 1079, 1215, 1271, 1351, 1359, 1375, 1399, 1471, 1631, 1639, 1719, 1879, 1919, 1935, 1975, 1999, 2015, 2079, 2111, 2135, 2311, 2415, 2519, 2535, 2575, 2631

Offset: 1

Hilko Koning, Jul 30 2018

nonn

It appears that this is a subsequence of A179882.
Define a set of consecutive positive odd numbers {1,......, (A077065(n)-1)} with n >= 3 and skip the number A077065(n)/2. Then the contraharmonic mean of that set gives the sequence. For example: ContraharmonicMean[{1, 3, 7, 9}] = 7, ContraharmonicMean[{1, 3, 5, 7, 9, 13, 15, 17, 19, 21}] = 15, ContraharmonicMean[{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 31, 33, 35, 37,39, 41, 43, 45}] = 31. - Hilko Koning, Aug 28 2018
Let p be a Sophie Germain prime and define h = 2p + 1 a safe prime. Then the contraharmonic mean of the totatives of h is given by: CHM(h) =  (Sum_{1 <= k < h, gcd(k, h) = 1} k^2) / (Sum_{1 <= k < h, gcd(k, h) = 1} k). Since h is prime, all integers k = 1, 2, ... , h - 1 are coprime to h. Then, CHM(h) = ((h - 1) * h * (2h-1) / 6) / ((h - 1) * h  / 2). Thus CHM = (2h-1) / 3 = (4p+1) / 3. These values are integers precisely when p == 2 mod 3, which holds for all Sophie Germain primes, p >= 5. The resulting values for the sequence A317510, which is therefore a subsequence of A179882. - Hilko Koning, Jun 17 2025

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Cf. A005384, A179882.

Subsequence of A004767, and of A004771.

a:=[];; for p in [3..2000] do if IsPrime(p) and IsPrime(2*p+1) then Add(a,(4*p+1)/3); fi; od; a; # Muniru A Asiru, Aug 28 2018

lst = {}; Do[If[PrimeQ[p] && PrimeQ[2 p + 1], AppendTo[lst, (4 p + 1)/3]], {p, 5, 2*10^3}]; lst
4 (Select[Prime@Range[3, 300], PrimeQ[2 # + 1] &] + 1)/3 - 1 (* Robert G. Wilson v, Jul 30 2018 *)

lista(nn) = {forprime (p=5, nn, if (isprime(2*p+1), print1((4*p+1)/3, ", ")););} \\ Michel Marcus, Aug 27 2018

A226540 Maximum of the proper divisors of the triangular numbers.

Original entry on oeis.org

1, 3, 5, 5, 7, 14, 18, 15, 11, 33, 39, 13, 35, 60, 68, 51, 57, 95, 105, 77, 23, 138, 150, 65, 117, 189, 203, 145, 155, 248, 264, 187, 119, 315, 333, 37, 247, 390, 410, 287, 301, 473, 495, 345, 47, 564, 588, 245, 425, 663, 689, 477, 495, 770, 798, 551, 59, 885

Offset: 2

Paolo P. Lava, Jun 10 2013

Solutions of A226540(n)=n are listed in A005383(n).

Solutions of A226540(n)=n+1 are listed in A005385(n).

			For n = 28 we have n*(n+1)/2 = 406 and its proper divisors are 1, 2, 7, 14, 29, 58, 203. Hence a(28) = 203.

Paolo P. Lava, Table of n, a(n) for n = 2..1000

Cf. A000217, A005383, A077065.

with(numtheory); A226540:=proc(q) local a,n;
for n from 2 to q do a:=sort([op(divisors(n*(n+1)/2))]);
print(a[nops(a)-1]); od; end: A226540(10^6);

Table[Divisors[(n(n+1))/2][[-2]],{n,2,60}] (* Harvey P. Dale, Apr 09 2021 *)

a(n)=if(n==2,return(1));my(p=factor(n/gcd(n,2))[1,1],q=factor((n+1)/gcd(n+1,2))[1,1]); binomial(n+1,2)/min(p,q) \\ Charles R Greathouse IV, Jun 10 2013

a(4n) = 4n^2 + n, 4n+1 <= a(4n+1) <= (8n^2 + 6n + 1)/3, 4n+3 <= a(4n+2) <= (8n^2 + 10n + 3)/3, a(4n+3) = 4n^2 + 7n + 3. - Charles R Greathouse IV, Jun 10 2013

A276983 Semiprimes n such that n-1 or n+1 is prime.

Original entry on oeis.org

4, 6, 10, 14, 22, 38, 46, 58, 62, 74, 82, 106, 158, 166, 178, 194, 226, 262, 278, 314, 346, 358, 382, 398, 422, 458, 466, 478, 502, 542, 562, 586, 614, 662, 674, 718, 734, 758, 838, 862, 878, 886, 982, 998, 1018, 1094, 1154, 1186, 1202, 1214, 1238, 1282, 1306, 1318, 1322

Offset: 1

Gary E. Davis, Sep 24 2016

nonn

Union of A077065 and A077068.

			a(3) = 10 = 2*5 is a product of 2 primes and 10+1 = 11 is prime.
a(4) = 14 = 2*7 is a product of 2 primes and 14-1 = 13 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A077065, A077068, A120628.

select(t -> isprime(t/2) and (isprime(t-1) or isprime(t+1)), [seq(i,i=2..10000,2)]); # Robert Israel, Sep 30 2016

func[n_] := PrimeOmega[n] == 2 && (PrimeQ[n + 1] || PrimeQ[n - 1])
Select[Range[1000], func[#] &]

isok(n) = (bigomega(n)==2) && (isprime(n-1) || isprime(n+1)); \\ Michel Marcus, Sep 24 2016

lista(nn) = forprime(p=2, nn, if(isprime(2*p+1) || isprime(2*p-1), print1(2*p, ", "))); \\ Altug Alkan, Sep 30 2016

from sympy import isprime, primerange
def aupto(N): return [t for t in (2*p for p in primerange(2, N//2+1)) if isprime(t-1) or isprime(t+1)]
print(aupto(1322)) # Michael S. Branicky, Aug 21 2022

a(n) = 2*A120628(n).

A343502 Numbers k such that tau(tau(k)) and tau(k+1) are both prime, where tau is the number of divisors function.

Original entry on oeis.org

2, 3, 4, 6, 8, 10, 15, 16, 22, 36, 46, 58, 82, 100, 106, 120, 166, 168, 178, 196, 210, 226, 256, 262, 270, 280, 312, 330, 346, 358, 378, 382, 408, 456, 462, 466, 478, 502, 520, 546, 562, 570, 586, 616, 640, 676, 690, 718, 728, 750, 760, 838, 858, 862, 886

Offset: 1

Claude H. R. Dequatre, Apr 17 2021

Considering the first 10^8 positive integers there are 1439855 terms in the sequence and only the first two (2,3) are prime, all the others are composite numbers of which only three are odd (15, 65535 and 4194303).
Conjecture: all members except 2 and 3 are composite.
Open question: is there a finite number of odd terms in this sequence?

			16 is a term because tau(16) = 5 and tau(5) = 2 and tau(17) = 2 and 2 is prime.
23 is not a term because tau(23) = 2 and tau(2) = 2 and tau(24) = 8 and 2 is prime but not 8.
98 is not a term because tau(98) = 6 and tau(6) = 4 and tau(99) = 6 and 4 and 6 are not prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000040, A010553. Includes A077065.

filter:= proc(n)
  isprime(numtheory:-tau(n+1)) and isprime(numtheory:-tau(numtheory:-tau(n)))
end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 02 2025

With[{t = DivisorSigma}, Select[Range[1000], And @@ PrimeQ[{t[0, t[0, #]], t[0, # + 1]}] &]] (* Amiram Eldar, May 27 2021 *)

for(k=1,1e4,if(isprime(numdiv(numdiv(k))) && isprime(numdiv(k+1)),print1(k", ")))

A353172 a(n) is the least k > 1 such that Omega(n) = Omega(n mod k), where Omega = A001222.

Original entry on oeis.org

2, 3, 4, 5, 3, 7, 4, 9, 5, 6, 3, 13, 5, 5, 9, 17, 3, 10, 4, 12, 11, 6, 3, 25, 7, 10, 15, 10, 3, 11, 4, 33, 9, 5, 13, 20, 5, 8, 5, 24, 3, 15, 4, 9, 25, 6, 3, 49, 5, 14, 9, 11, 3, 19, 7, 20, 12, 6, 3, 22, 7, 7, 11, 65, 11, 18, 4, 10, 5, 25, 3, 40, 5, 5, 19, 16

Offset: 1

Thomas Scheuerle, Apr 28 2022

It appears that a(m) = m*k/p if m = p*2^n ... . Are these formulas related to some well-known sequence of rational numbers?

			a(10) = 6 because 10 = 5*2 and 10 mod 6 = 4 = 2*2.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A003627, A029744, A061142, A077065, A095925.

a(n) = my(k=2); while(bigomega(n) != bigomega(max(n%k,1)), k++); k

from itertools import count
from sympy.ntheory.factor_ import primeomega
def A353172(n):
    a = primeomega(n)
    for k in count(2):
        if (m := n % k) > 0 and primeomega(m) == a:
            return k # Chai Wah Wu, Jun 20 2022

a(A029744(n)) = A029744(n) + 1.
a(A003627(n)) = 3.
a(A000040(n)) = A095925(n).
a(A077065(n)) = 6. For n > 2.
If a(n) = 10, then n mod 10 is in most cases 8 and seldom 6.
a(m) = m*3/5 if m = 5*2^n or m = 15. This formula is valid for all positive n because (5*2^n) mod (5*2^n)*(3/5) = 2^(n+1). If the sequence of solutions does not create powers of two in the modulo operation, it will be of finite length. See next two formulas:
a(m) = m*3/11 if m = 11, 22, 33 or 66.
a(m) = m*4/43 if m = 43*2^n for n < 4. This series of solutions terminates because of the next formula which replaces the powers of two:
a(m) = m*41/(43*2^4) if m = 43*2^4*2^n. This formula is valid for all positive n.
a(m) = m*5/9 if m = 9*2^n or m = 27 or 45. This formula is valid for all positive n.
For each k = a(p) if k < p and gcd(k, p) = 1 such a formula, of the form a(m) = m*k/p, if m = p*2^n ..., can be developed.

A380101 Numbers k such that omega(k-th triangular number) = 2, where omega = A001221.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 13, 16, 17, 18, 22, 25, 26, 31, 37, 46, 49, 53, 58, 61, 73, 81, 82, 97, 106, 121, 127, 157, 162, 166, 178, 193, 226, 241, 242, 250, 256, 262, 277, 313, 337, 346, 358, 361, 382, 397, 421, 457, 466, 478, 486, 502, 541, 562, 577, 586, 613

Offset: 1

Juri-Stepan Gerasimov, Jan 12 2025

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Supersequence of A077065 and of A178490.

Cf. A000217, A001221, A119663.

[k: k in [1..400] | #PrimeDivisors(k*(k+1) div 2) eq 2];

filter:= proc(n) local W1, n1, W2; uses numtheory;
     if n::odd then nops(factorset(n)) = 1 and nops(factorset((n+1)/2)) = 1
     else nops(factorset(n/2)) = 1 and nops(factorset(n+1)) = 1
     fi
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 12 2025

Select[Range[600], PrimeNu[#*(#+1)/2] == 2 &] (* Amiram Eldar, Jan 12 2025 *)

isok(k) = omega(k*(k+1)/2) == 2; \\ Michel Marcus, Jan 14 2025

A233973 a(n) = A232221(n)/4.

Original entry on oeis.org

0, 0, 1, 5, 9, 13, 32, 54, 82, 116, 159, 199, 227, 273, 331, 380, 429, 487, 536, 609, 679, 743, 816, 895, 953, 1008, 1042, 1067, 1104, 1180, 1253, 1332, 1429, 1532, 1626, 1675, 1733, 1812, 1921, 2045, 2169, 2317, 2420, 2535, 2656, 2756, 2850, 2953

Offset: 1

Omar E. Pol, Dec 18 2013

nonn

Cf. A001358, A077065, A077068, A232221.

a(n) = (Sum_{i=1..n} (A077068(i)-A077065(i)))/4.

A385720 Numbers k >= 1 such that k/A000005(k) + (k+1)/A000005(k+1) is an integer.

Original entry on oeis.org

1, 5, 6, 8, 10, 13, 22, 37, 45, 46, 58, 61, 62, 69, 73, 74, 77, 82, 89, 106, 114, 117, 126, 146, 149, 150, 154, 157, 166, 167, 178, 186, 193, 197, 198, 206, 221, 226, 233, 237, 258, 261, 262, 263, 266, 277, 278, 279, 280, 290, 293, 306, 309, 311, 312, 313

Offset: 1

Ctibor O. Zizka, Jul 07 2025

nonn

k/A000005(k) + (k+1)/A000005(k+1) = (3*k + 1)/4 for k >= 5 from A256072.
k/A000005(k) + (k+1)/A000005(k+1) = (3*k + 2)/4 for k >= 6 from A077065.
k/A000005(k) + (k+1)/A000005(k+1) =< (3*k + 2)/4 for k >= 5.

			For k = 6: 6/A000005(6) + 7/A000005(7) = 6/4 + 7/2 = 5, thus k = 6 is a term.

Cf. A000005, A005384, A005385, A077065, A256072.

Position[Plus @@@ Partition[Table[n/DivisorSigma[0, n], {n, 1, 320}], 2, 1], ?IntegerQ] // Flatten (* _Amiram Eldar, Jul 08 2025 *)

isok(k) = denominator(k/numdiv(k) + (k+1)/numdiv(k+1)) == 1; \\ Michel Marcus, Jul 08 2025

from itertools import count, islice
from sympy import divisor_count
def A385720_gen(startvalue=1): # generator of terms >= startvalue
    m = max(startvalue,1)
    a, b = divisor_count(m), divisor_count(m+1)
    for k in count(m):
        if not (k*b+(k+1)*a)%(a*b):
            yield k
        a, b = b, divisor_count(k+2)
A385720_list = list(islice(A385720_gen(),30)) # Chai Wah Wu, Jul 13 2025

cp's OEIS Frontend

A194593 Semiprimes s such that phi(s)/2 is prime.

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A369139 Numbers k such that Omega(k) = 1 + Omega(k + 1).

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A317510 Numbers (4p+1)/3 where p is a Sophie Germain prime p > 3.

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A226540 Maximum of the proper divisors of the triangular numbers.

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A276983 Semiprimes n such that n-1 or n+1 is prime.

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A343502 Numbers k such that tau(tau(k)) and tau(k+1) are both prime, where tau is the number of divisors function.

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A353172 a(n) is the least k > 1 such that Omega(n) = Omega(n mod k), where Omega = A001222.

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