A278911 -id:A278911 - cp's OEIS frontend

A193070 Odd numbers N for which sigma(N^2) is prime.

3, 5, 17, 27, 41, 49, 59, 71, 89, 101, 125, 131, 167, 169, 173, 289, 293, 383, 529, 677, 701, 729, 743, 761, 773, 827, 839, 841, 857, 911, 1091, 1097, 1163, 1181, 1193, 1217, 1373, 1427, 1487, 1559, 1583, 1709, 1811, 1847, 1849, 1931, 1973, 2129, 2197, 2273, 2309

Offset: 1

M. F. Hasler, Jul 15 2011

nonn

The function sigma(n) (=A000203(n)) takes odd values when n is a square or twice a square. Thus, odd numbers n for which sigma(n) is prime (i.e. which are in A023194) must be odd squares. This sequence consists exactly of the square roots of these terms.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A023194, A195382, A278911.

Select[Range[1,2401,2],PrimeQ[DivisorSigma[1,#^2]]&] (* Harvey P. Dale, Mar 07 2015 *)

forstep(N=1, 1e7, 2, isprime(sigma(N^2)) && print1(N", "))

a(n) = A278911(n)^(1/2). - Robert Israel, Jan 22 2019

A278914 a(n) is the smallest odd number k with prime sum of divisors such that tau(k) = n-th prime.

Original entry on oeis.org

9, 2401, 729, 9765625, 531441, 45949729863572161, 5559917313492231481, 1471383076677527699142172838322885948765175969, 10264895304762966931257013446474591264089923314972889033759201, 230466617897195215045509519405933293401

Offset: 2

Jaroslav Krizek, Nov 30 2016

nonn

tau(n) = A000005(n) = the number of divisors of n.

For n >= 7; a(n) > A023194(10000) = 5896704025969.

			a(2) = 9 because 9 is the smallest odd number with prime values of sum of divisors (sigma(9) = 13) such that tau(9) = 3 = 2nd prime.

Robert G. Wilson v, Table of n, a(n) for n = 2..66 (first 49 terms from Davin Park)

Cf. A000005, A000203, A278911, A278913.

A278914:=func; [A278914(n): n in[2..6]];

A278914[n_] := NestWhile[NextPrime, 3, ! PrimeQ[Cyclotomic[Prime[n], #]] &]^(Prime[n] - 1); Array[A278914, 10, 2] (* Davin Park, Dec 28 2016 *)

a(n) = {my(k=1); while(! (isprime(sigma(k)) && isprime(p=numdiv(k)) && (primepi(p) == n)), k+=2); k;} \\ Michel Marcus, Dec 03 2016

a(n) = A101636(n)^(prime(n)-1). - Davin Park, Dec 10 2016

More terms from Davin Park, Dec 11 2016

A273459 Even numbers such that the sum of the odd divisors is a prime p and the sum of the even divisors is 2p.

Original entry on oeis.org

18, 50, 578, 1458, 3362, 4802, 6962, 10082, 15842, 20402, 31250, 34322, 55778, 57122, 59858, 167042, 171698, 293378, 559682, 916658, 982802, 1062882, 1104098, 1158242, 1195058, 1367858, 1407842, 1414562, 1468898, 1659842, 2380562, 2406818, 2705138, 2789522

Offset: 1

Michel Lagneau, May 30 2016

nonn

a(n) is of the form 2q^2 where q is an odd numbers for which sigma(q^2) is prime (A193070).
The corresponding primes p are 13, 31, 307, 1093, 1723, 2801, 3541, 5113, 8011, 10303, 19531, 17293, 28057, 30941, 30103, 88741, 86143, 147073, 292561, 459007, 492103, 797161, 552793, 579883, 598303, 684757, 704761, 732541, 735307, 830833, 1191373, 1204507, ...
We observe an interesting property: each prime p is element of A053183 (primes of the form m^2 + m + 1 when m is prime) or element of A247837 (primes of the form sigma(2m-1) for a number m) or element of both A053183 and A247837.
Examples:
The numbers 13, 31, 307, 1723, 3541, 5113,... are in A053183;
The numbers 13, 31, 307, 1093, 1723, 2801, 3541,...are in A247837;
The numbers 13, 31, 307, 1723, 3541,... are in A053183 and A247837.

			18 is in the sequence because the divisors of 18 are {1, 2, 3, 6, 9, 18}. The sum of the odd divisors is 1 + 3 + 9 = 13 and the sum of the even divisors is 2 + 6 + 18 = 26 = 2*13.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002384, A053183, A193070, A247837, A278911.

with(numtheory):
for n from 2 by 2  to 500000 do:
   y:=divisors(n):n1:=nops(y):s0:=0:s1:=0:
     for k from 1 to n1 do:
       if irem(y[k], 2)=0
        then
        s0:=s0+ y[k]:
        else
        s1:=s1+ y[k]:
      fi:
     od:
     ii:=0:
        if isprime(s1) and s0=2*s1
        then
        printf(`%d, `, n):
         else fi:
     od:

Select[Range[2, 3000000, 2], And[PrimeQ[Total@ Select[#, EvenQ]/2], PrimeQ@ Total@ Select[#, OddQ]] &@ Divisors@ # &] (* Michael De Vlieger, May 30 2016 *)
sodpQ[n_]:=Module[{d=Divisors[n],s},s=Total[Select[d,OddQ]];PrimeQ[ s] && Total[ Select[d,EvenQ]]==2s]; Select[Range[2,279*10^4,2],sodpQ] (* Harvey P. Dale, Dec 01 2020 *)
2 * Select[Range[1, 1200, 2]^2, PrimeQ@DivisorSigma[1, #] &] (* Amiram Eldar, Jul 19 2022 *)

is(n)=my(t); n%4==2 && issquare(n/2,&t) && isprime(n/2+t+1) \\ Charles R Greathouse IV, Jun 08 2016

a(n) >> n^2. - Charles R Greathouse IV, Jun 08 2016

a(n) = 2 * A278911(n) = 2 * A193070(n)^2. - Amiram Eldar, Jul 19 2022

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A193070 Odd numbers N for which sigma(N^2) is prime.

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A278914 a(n) is the smallest odd number k with prime sum of divisors such that tau(k) = n-th prime.

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A273459 Even numbers such that the sum of the odd divisors is a prime p and the sum of the even divisors is 2p.

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Maple

Mathematica

PARI

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