A247820 -id:A247820 - cp's OEIS frontend

A247954 a(n) = sigma(sigma(2n-1)).

1, 7, 12, 15, 14, 28, 24, 60, 39, 42, 63, 60, 32, 90, 72, 63, 124, 124, 60, 120, 96, 84, 168, 124, 80, 195, 120, 195, 186, 168, 96, 210, 224, 126, 252, 195, 114, 224, 252, 186, 133, 224, 280, 360, 234, 248, 255, 360, 171, 392, 216, 210, 508, 280, 216, 300

Offset: 1

Jaroslav Krizek, Sep 28 2014

See A247821 - numbers k such that sigma(sigma(2k-1)) is a prime p.

			For n=2; a(2) = sigma(sigma(2*2-1)) = sigma(sigma(3)) = sigma(4) = 7.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A000203, A051027, A008438, A247820.

[SumOfDivisors(SumOfDivisors(2*n-1)): n in [1..1000]];

with(numtheory): A247954:=n->sigma(sigma(2*n-1)): seq(A247954(n), n=1..50); # Wesley Ivan Hurt, Oct 01 2014

Table[DivisorSigma[1, DivisorSigma[1, 2 n - 1]], {n, 50}] (* Wesley Ivan Hurt, Oct 01 2014 *)

vector(100,n,sigma(sigma(2*n-1))) \\ Derek Orr, Sep 29 2014

a(n) = A000203(A000203(2n-1)) = A000203(A008438(n-1)) = A051027(2n-1).

A247821 Numbers k such that sigma(sigma(2k-1)) is a prime p.

Original entry on oeis.org

2, 1334, 1969, 28669, 86006, 126961, 338603654, 536801281, 366479720500691270, 375344017599431990, 500461553802019261, 554079264075351985

Offset: 1

Jaroslav Krizek, Sep 28 2014

Numbers n such that A000203(A000203(2n-1)) = A000203(A008438(n-1)) = A051027(2n-1) is a prime p.
Corresponding values of primes p are 7, 8191, 8191, 131071, 524287, 524287, ... (= A247822). Conjecture: The primes p are Mersenne primes (A000668).
sigma(sigma(2*a(9)-1)) > 10^16.
If the above conjecture is true, the next terms are 366479720500691270, 375344017599431990, 500461553802019261, 554079264075351985, 98375588019240949991670086, ... . - Hiroaki Yamanouchi, Oct 01 2014
a(13) > 5*10^18. - Giovanni Resta, Feb 14 2020

			Number 1334 is in sequence because sigma(sigma(2*1334-1)) = sigma(sigma(2667)) = sigma(4096) = 8191, i.e., prime.

Cf. A000203, A008438, A247790, A247791, A247820, A247822, A247823, A247954.

[n: n in [1..10000000] | IsPrime(SumOfDivisors(SumOfDivisors(2*n-1)))]

Select[Range[10^6], PrimeQ[DivisorSigma[1, DivisorSigma[1, 2 # - 1]]] &] (* Robert Price, May 17 2019 *)

for(n=1,10^7,if(ispseudoprime(sigma(sigma(2*n-1))),print1(n,", "))) \\ Derek Orr, Sep 29 2014

a(n) = (A247838(n) +1) / 2.

a(n)-1 = numbers n such that sigma(sigma(2n+1)) is a prime p: 1, 1333, 1968, 28668, 86005, 126960, ...

a(7)-a(8) from Hiroaki Yamanouchi, Oct 01 2014

a(9)-a(12) from Giovanni Resta, Feb 14 2020

A247837 Primes p of the form sigma(2k-1) for a number k.

Original entry on oeis.org

13, 31, 307, 1093, 1723, 2801, 3541, 5113, 8011, 10303, 17293, 19531, 28057, 30103, 30941, 86143, 88741, 147073, 292561, 459007, 492103, 552793, 579883, 598303, 684757, 704761, 732541, 735307, 797161, 830833, 1191373, 1204507, 1353733, 1395943, 1424443, 1482307

Offset: 1

Jaroslav Krizek, Sep 24 2014

nonn

Supersequence of A247836.

The multiplicity of the sigma-function means that the 2k-1 are odd prime powers 3^2, 5^2, 17^2, 3^6, 41^2,... (A061345), and the fact that sigma(k)>=k means that a numerical search for any candidate p can be limited to the prime powers less than p. - R. J. Mathar, Jun 04 2016

			Prime 13 is in sequence because there is number 5 such that sigma(2*5-1) = sigma(9) = 13.

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

Cf. A000203, A247836, A247820.

Sort(b) where b is [a: n in [1..2500000] | IsPrime(a) where a is SumOfDivisors(2*n-1)];

isA247837 := proc(n)
    local i,opp;
    if isprime(n) then
        for i from 1 do
            opp := A061345(i) ;
            if numtheory[sigma](opp) = n then
                return true;
            elif opp > n then
                return false;
            end if;
        end do:
    else
        false;
    end if;
end proc:
for n from 2 do
    p := ithprime(n) ;
    if isA247837(p) then
        printf("%d,\n",p) ;
    end if;
end do: # R. J. Mathar, Jun 04 2016

for(n=1,10^7,if(isprime(sigma(2*n-1)),print1(sigma(2*n-1),", "))) \\ Derek Orr, Sep 25 2014. ***WARNING: This program prints the terms not in correct order. - M. F. Hasler, Nov 16 2014

a(n) = sigma(2*A247820(n)-1) = A000203(2*A247820(n)-1). ***WARNING: This formula is not correct for all n. - M. F. Hasler, Nov 16 2014

The first discrepancy in the above formula is at n=11, where a(11) = A000203(2*A247820(12)-1) while A000203(2*A247820(11)-1)=a(12). - Robert Israel, Mar 31 2020

Corrected and edited by Jaroslav Krizek, Nov 14 2014

A247822 Corresponding values of primes p from A247821 and A247838.

Original entry on oeis.org

7, 8191, 8191, 131071, 524287, 524287, 2147483647, 2147483647, 2305843009213693951, 2305843009213693951, 2305843009213693951, 2305843009213693951

Offset: 1

Jaroslav Krizek, Sep 28 2014

Conjecture: all terms are Mersenne primes (A000668).

Conjecture: next terms are 2305843009213693951, 2305843009213693951, 2305843009213693951, 2305843009213693951 and 618970019642690137449562111. - Jaroslav Krizek, Mar 25 2015

			a(2) = 8191 because sigma(sigma(2*A247821(2)-1)) = sigma(sigma(A247838(2))) = 8191.

Cf. A000203, A008438, A247790, A247791, A247820, A247821, A247823, A247954.

[SumOfDivisors(SumOfDivisors(n)): n in [A247838(n)]];

a(n) = sigma(sigma(2*A247821(n)-1)) = A000203(A000203(2*A247821(n)-1)) = A051027(2*A247821(n)-1).

a(n) = sigma(sigma(A247838(n))) = A000203(A000203(A247838(n))) = A051027(A247838(n)).

a(7)-a(8) from Jaroslav Krizek, Mar 25 2015

a(9)-a(12) from Giovanni Resta, Feb 14 2020

A247790 Primes p such that sigma(sigma(2p-1)) is a prime.

Original entry on oeis.org

2, 28669, 126961, 500461553802019261

Offset: 1

Jaroslav Krizek, Sep 28 2014

The next term, if it exists, must be greater than 5*10^7.
Primes p such that A247954(p) = A000203(A000203(2p-1)) = A000203(A008438(p-1)) = A051027(2p-1) is a prime q. The corresponding values of the primes q are: 7, 131071, 524287, ... (A247791). Conjecture: the primes q are Mersenne primes (A000668).
Conjecture: the next term is 500461553802019261 (see comment from Hiroaki Yamanouchi in A247821). - Jaroslav Krizek, Oct 08 2014
These are the primes in A247821. - M. F. Hasler, Oct 14 2014
No other terms up to 5*10^10. - Michel Marcus, Feb 11 2020
a(5) > 5*10^18. - Giovanni Resta, Feb 14 2020

			Prime 2 is in the sequence because sigma(sigma(2*2-1)) = sigma(sigma(3)) = sigma(4) = 7, i.e., prime.

Cf. A000203, A008438, A247791, A247820, A247821, A247822, A247823, A247954.

[p: p in PrimesUpTo(50000000) | IsPrime(SumOfDivisors(SumOfDivisors(2*p-1)))]

with(numtheory): A247790:=n->`if`(isprime(n) and isprime(sigma(sigma(2*n-1))),n,NULL): seq(A247790(n), n=1..130000); # Wesley Ivan Hurt, Oct 17 2014

forprime(p=1,10^7,if(ispseudoprime(sigma(sigma(2*p-1))),print1(p,", "))) \\ Derek Orr, Sep 29 2014

a(4) from Giovanni Resta, Feb 14 2020

A247791 Primes p such that there is a prime q for which sigma(sigma(2*q-1)) = p.

Original entry on oeis.org

7, 131071, 524287

Offset: 1

Jaroslav Krizek, Sep 28 2014

The next term, if it exists, must be greater than 5*10^7.
Primes p such that there is prime q for which sigma(sigma(2*q-1)) = A247954(q) = A000203(A000203(2*q-1)) = A000203(A008438(q-1)) = A051027(2*q-1) = p.
Corresponding values of primes q: 2, 28669, 126961, ... (A247790).
Conjecture: Subsequence of Mersenne primes.
Conjecture: the next term is 2305843009213693951 when 2305843009213693951 = sigma(sigma(2*500461553802019261-1)) where 500461553802019261 is prime (see comment of Hiroaki Yamanouchi in A247821). - Jaroslav Krizek, Oct 08 2014

			Prime 7 is in sequence because there is prime 2 such that sigma(sigma(2*2-1)) = sigma(sigma(3)) = sigma(4) = 7.

Cf. A000203, A008438, A247790, A247820, A247821, A247822, A247823, A247954.

[SumOfDivisors(SumOfDivisors(2*n-1)): n in [A247790(n)]];

[SumOfDivisors(SumOfDivisors(2*n-1)): n in[1..1000000] | IsPrime(SumOfDivisors(SumOfDivisors(2*n-1))) and IsPrime(n)];

forprime(p=1,10^7,if(ispseudoprime(sigma(sigma(2*p-1))),print1(sigma(sigma(2*p-1)),", "))) \\ Derek Orr, Sep 29 2014

A247823 Mersenne primes p such that there is a number k with sigma(sigma(2k-1)) = p.

Original entry on oeis.org

7, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111

Offset: 1

Jaroslav Krizek, Sep 28 2014

Mersenne primes p such that there is a number m such that sigma(sigma(m)) = p.
Distinct values attained by the A247822(n) function, in ascending order.
Mersenne primes p such that there are a numbers n and m such that sigma(sigma(2n-1)) = sigma(sigma(2*A247821(n)-1)) = A000203(A000203(2*A247821(n)-1)) = A051027(2*A247821(n)-1) = sigma(sigma(A247838(m))) = A000203(A000203(A247838(m))) = A051027(A247838(m)) where m = 2n-1.
The Mersenne prime 7 is the only prime p such that there is a prime q with sigma(sigma(q)) = p.

			Mersenne prime 8191 is in sequence because there are numbers n = 1334 and 1969 with sigma(sigma(2*n-1)) = 8191.

Cf. A000203, A008438, A247790, A247791, A247820, A247821, A247822, A247954.

Cf. A000668 (Mersenne primes).

Set(Sort([SumOfDivisors(SumOfDivisors(n)): n in [1..10000000] | IsPrime(SumOfDivisors(SumOfDivisors(n)))])) // Jaroslav Krizek, Mar 25 2015

a(5)-a(7) from Jaroslav Krizek, Mar 25 2015

A247789 Primes p such that sigma(2p-1) is a prime q.

Original entry on oeis.org

5, 13, 1201, 1741, 2521, 5101, 8581, 14281, 41761, 139921, 353641, 595141, 697381, 1460341, 1639861, 3723721, 3889261, 6093541, 7956061, 11143921, 14199121, 33874681, 46938361, 51521401, 56063461, 62395621, 67523821, 90706981

Offset: 1

Jaroslav Krizek, Sep 24 2014

nonn

Subsequence of A247820.

Corresponding values of primes q for a(n) are in A247836.

			Prime 13 is in sequence because sigma(2*13-1) = sigma(25) = 31 (prime).

Cf. A000203, A008438, A000203, A247820.

[p: p in PrimesUpTo(50000000) | IsPrime(SumOfDivisors(2*p-1))]

forprime(p=1,10^9,if(isprime(sigma(2*p-1)),print1(p,", "))) \\ Derek Orr, Sep 25 2014

A247836 Primes p of the form sigma(2q-1) where q is a prime.

Original entry on oeis.org

13, 31, 2801, 3541, 5113, 10303, 17293, 30941, 88741, 292561, 732541, 1191373, 1395943, 2922391, 3281533, 7450171, 7781311, 12190573, 15916111, 22292563, 28792661, 67757593, 93886411, 103052953, 112137511, 124802413, 135059263, 181427431

Offset: 1

Jaroslav Krizek, Sep 24 2014

nonn

Subsequence of A247837.

			Prime 13 is in sequence because there is prime 5 such that sigma(2*5-1) = sigma(9) = 13.

Cf. A000203, A247837, A247820.

Sort([a: p in PrimesUpTo(5000000) | IsPrime(a) where a is SumOfDivisors(2*p-1)]); // corrected by Jaroslav Krizek, Nov 16 2014

Select[DivisorSigma[1,2Prime[Range[3*10^6]]-1],PrimeQ] (* The program generates the first 23 terms of the sequence. *) (* Harvey P. Dale, Jul 17 2025 *)

forprime(p=1,10^9,if(isprime(sigma(2*p-1)),print1(sigma(2*p-1),", "))) \\ Derek Orr, Sep 25 2014

A247955 Primes p such that there is prime q with sigma(q+2) = p.

Original entry on oeis.org

7, 13, 31, 1093, 2801, 5113, 8011, 17293, 30103, 30941, 86143, 459007, 552793, 579883, 732541, 1191373, 3500201, 3730693, 4534771, 5168803, 5333791, 7450171, 10378063, 25646167, 25882657, 28792661, 30266503, 43553401, 48037081, 52265671, 56964757, 62433703, 65504743, 67856407, 76413823, 77572057

Offset: 1

Jaroslav Krizek, Sep 28 2014

nonn

Primes p such that there is prime q such that A000203(q+2) = p.

Primes p of the form sigma(A171130(n)) in increasing order.

Cf. A000203, A008438, A171130, A247791, A247820, A247821, A247822, A247823, A247954.

Sort[Select[DivisorSigma[1,#+2]&/@Prime[Range[5200000]],PrimeQ]] (* Harvey P. Dale, Apr 27 2022 *)

v=[];forprime(p=1,10^8,if(ispseudoprime(sigma(p+2)),v=concat(v,sigma(p+2))));v \\ Derek Orr, Oct 26 2014

More terms from Michel Marcus, Oct 02 2014

Corrected and extended by Harvey P. Dale, Apr 27 2022

cp's OEIS Frontend

A247954 a(n) = sigma(sigma(2n-1)).

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A247821 Numbers k such that sigma(sigma(2k-1)) is a prime p.

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A247837 Primes p of the form sigma(2k-1) for a number k.

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A247822 Corresponding values of primes p from A247821 and A247838.

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A247790 Primes p such that sigma(sigma(2p-1)) is a prime.

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A247791 Primes p such that there is a prime q for which sigma(sigma(2*q-1)) = p.

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A247823 Mersenne primes p such that there is a number k with sigma(sigma(2k-1)) = p.

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