A062700 -id:A062700 - cp's OEIS frontend

A100382 Record values of A062700.

3, 7, 13, 31, 127, 307, 1093, 1723, 2801, 3541, 8191, 10303, 19531, 28057, 30941, 131071, 147073, 524287, 797161, 830833, 1191373, 1204507, 1353733, 1395943, 1424443, 1482307, 1886503, 2037757, 2212657, 2432041, 2507473, 2922391

Offset: 1

Jorge Coveiro, Dec 30 2004

nonn

Take sequence A062700: 3, 7, 13, 31, 31, 127, 307, 1093, 1723, 2801, 3541, 8191, 5113, 8011, 10303, .... Then eliminate terms so that each term of the sequence is larger than the preceding one: 3, 7, 13, 31, 127, 307, 1093, 1723, 2801, 3541, 8191, 10303, ....

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A062700 (terms of A000203 that are prime), A000203 (sigma(n), sum of divisors of n), A034885 (record values of sigma(n)).

S:=[]; a:=0; for n in [1..3000000] do c:=SumOfDivisors(n); if IsPrime(c) and a lt c then Append(~S,c); a:=c; end if; end for; S; // Klaus Brockhaus, Oct 21 2009

More terms from Ryan Propper, Jul 13 2005

Edited, corrected and extended by Klaus Brockhaus, Oct 21 2009

A071167 a(n) = A023194 - A062700(n). Negative values of A071166(m) = m-A006530(A000203(m)) differences. In these cases m is square number from A023194.

Original entry on oeis.org

-1, -3, -4, -15, -6, -63, -18, -364, -42, -400, -60, -4095, -72, -90, -102, -3906, -132, -168, -2380, -174, -65535, -5220, -294, -384, -262143, -12720, -678, -702, -265720, -744, -762, -774, -828, -840, -25260, -858, -912, -1092, -1098, -1164, -1182, -1194, -1218, -1374, -1428, -1488, -1560

Offset: 1

Labos Elemer, May 15 2002

sign

			m=29929=173, sigma[29929]=1+173+29929=30103 and 29929-30103=-174, the 20th term here.

Cf. A000203, A006530, A062700, A023194, A071166.

ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=n-ma[DivisorSigma[1, n]]; If[Equal[Sign[s], -1], Print[s]], {n, 2, 10000000}]

for(n=1,1e3,if(isprime(s=sigma(n^2)),print1(n^2-s", "))) \\ Charles R Greathouse IV, Feb 19 2013

Values of m - A006530(A000203(m)) differences, when m < A006530(A000203(m)).

A055638 Numbers k for which sigma(k^2) is prime.

Original entry on oeis.org

2, 3, 4, 5, 8, 17, 27, 41, 49, 59, 64, 71, 89, 101, 125, 131, 167, 169, 173, 256, 289, 293, 383, 512, 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

Offset: 1

Robert G. Wilson v, Jun 07 2000

nonn

sigma(n) is the sum of the divisors of n (A000203).
If sigma(x) is prime, then x=2 or x=p^(2m), an even power of a prime, cf. A023194. This sequence lists the values n = p^m such that sigma(n^2) is prime, i.e., sqrt( A023194 \ {2} ). The corresponding primes sigma(n^2)=A062700(n) are 1+p+...+p^(2m) = (p^(2m+1)-1)/(p-1), and any prime of that form (cf. A023195) corresponds to a term p^m is in this sequence. - M. F. Hasler, Oct 14 2014
This is a subsequence of A000961, see A248963 for its complement therein. - M. F. Hasler, Oct 19 2014
a(n) nearly always has digitsum of the form 2 mod 3. Specifically, 99.8% of the first 33733 entries examined conformed. The first exceptions are 3, 4, 27, 49, 64, 169, 256, 289, 529, 729. The exceptions (examined) appear to be integer powers themselves excepting the initial 3. Similarly, except for the initial 3, all entries of A023195 appear to have digitsum = 1 mod 3. - Bill McEachen, Mar 05 2017, Mar 20 2025
Number of terms < 10^k: 5, 13, 36, 137, 735, 4730, 33732, 253393, ..., . Robert G. Wilson v, Mar 09 2017
Primes in the sequence are A053182. - Thomas Ordowski, Nov 18 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 4730 terms from T. D. Noe)

Cf. A023194 (sigma(n) is prime).
Cf. A023195 (primes of the form sigma(n)), A062700 (in order of appearance).
Cf. A000961, A248963.

[n: n in [1..2000] | IsPrime(SumOfDivisors(n^2))]; // Vincenzo Librandi, Oct 18 2014

Select[Range[2000], PrimeQ[DivisorSigma[1, #^2]] &]

for(n=1,9999,isprime(sigma(n^2))&&print1(n",")) \\ M. F. Hasler, Oct 18 2014

a(n) = sqrt(A023194(n+1)).

Equal to A000961 \ A248963. - M. F. Hasler, Oct 19 2014

Minor edits by M. F. Hasler, Oct 18 2014

A065403 Primes of the form sigma(m^2) where m is a composite number ordered by values m.

Original entry on oeis.org

31, 127, 1093, 2801, 8191, 19531, 30941, 131071, 88741, 524287, 292561, 797161, 732541, 3500201, 5229043, 12207031, 25646167, 28792661, 39449441, 48037081, 305175781, 262209281, 917087137, 2147483647, 1394714501, 2666986681

Offset: 1

Labos Elemer, Nov 06 2001

nonn

There are 46 cases below 10^12.
All Mersenne primes are here: sigma((2^((p-1)/2))^2) = sigma(2^(p-1)) = -1 + 2^p, for suitable p.
m is of the form p^(2*e) for some prime p and e > 1 as sigma is multiplicative and m is composite. Terms are sorted by values of m. The sequence isn't monotonic. - David A. Corneth, Jul 18 2020

			19531 is in the sequence as for the composite m = 125 we have sigma(m^2) = 19531. - _David A. Corneth_, Jul 18 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 100 terms from Harry J. Smith, terms 101-500 from Amiram Eldar)

Cf. A062700, A000203, A001348, A065404, A065405.

Do[s=DivisorSigma[1, n]; If[PrimeQ[s]&&!PrimeQ[Sqrt[n]], Print[{n, Sqrt[n], s}]], {n, 1, 20000000}]

{ n=0; for (m=1, 10^9, if (isprime(m), next); x=sigma(m^2); if (isprime(x), write("b065403.txt", n++, " ", x); if (n==100, return)) ) } \\ Harry J. Smith, Oct 18 2009

upto(n) = {res = List(); forstep(e = 4, logint(n, 2), 2, forprime(p = 2, sqrtnint(n, e), c = (p^(e + 1) - 1)/(p - 1); if(isprime(c), listput(res, [p^e, c]) ) ) ); listsort(res); vector(#res, i, res[i][2]) } \\ David A. Corneth, Jul 18 2020

Name corrected by David A. Corneth, Jul 18 2020

A065405 Composite numbers k such that the sum of the divisors of k^2 is a prime.

Original entry on oeis.org

4, 8, 27, 49, 64, 125, 169, 256, 289, 512, 529, 729, 841, 1849, 2197, 3125, 4913, 5329, 6241, 6889, 15625, 16129, 29791, 32768, 37249, 51529, 57121, 69169, 76729, 113569, 117649, 128881, 139129, 157609, 192721, 208849, 226981, 229441, 253009

Offset: 1

Labos Elemer, Nov 06 2001

nonn

All these composite numbers k should be prime powers because if k=a*b with gcd(a,b)=1, then sigma(aabb) = sigma(aa)*sigma(bb) cannot be a prime; 46 of the 236 prime powers below 1000000 are here.

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..70 from Harry J. Smith)

Cf. A062700, A000203, A065403, A065404, A053182, A053183, A028982.

Select[ Range[3 10^5], ! PrimeQ[ # ] && PrimeQ[ DivisorSigma[1, #^2]] & ]

isok(k) = { !isprime(k) && isprime(sigma(k^2)) } \\ Harry J. Smith, Oct 18 2009

sigma(a(n)^2) = sigma(A065404(n)) = A065403(n) is prime.

A065404 Squares of composite numbers k such that sigma(k) (sum of divisors of k, A000203) is a prime.

Original entry on oeis.org

16, 64, 729, 2401, 4096, 15625, 28561, 65536, 83521, 262144, 279841, 531441, 707281, 3418801, 4826809, 9765625, 24137569, 28398241, 38950081, 47458321, 244140625, 260144641, 887503681, 1073741824, 1387488001, 2655237841

Offset: 1

Labos Elemer, Nov 06 2001

nonn

			46 cases below 10^12; for M a Mersenne prime, (M+1)/2 is here: M=8191, 4096=(M+1)/2.

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..100 from Harry J. Smith)

Cf. A062700, A000203 (sigma), A065403, A065405, A053182, A053183, A028982.

{ n=0; for (m=1, 10^9, if (isprime(m), next); x=sigma(m^2); if (isprime(x), write("b065404.txt", n++, " ", m^2); if (n==100, return)) ) } \\ Harry J. Smith, Oct 18 2009

sigma(a(n)) = A065403(n).

A152677 Subsequence of odd terms in A000203 (sum-of-divisors function sigma), in the order in which they occur and with repetitions.

Original entry on oeis.org

1, 3, 7, 15, 13, 31, 39, 31, 63, 91, 57, 93, 127, 195, 121, 171, 217, 133, 255, 403, 363, 183, 399, 465, 403, 399, 511, 819, 307, 847, 549, 381, 855, 961, 741, 1209, 931, 1023, 553, 1651, 921, 781, 1815, 1281, 1143, 1093, 1767, 1953, 871, 2223, 2821, 993, 1995

Offset: 1

Omar E. Pol, Dec 10 2008

Equivalently: subsequence of A000203 (sigma) with indices equal to a square or twice a square (A028982).

See A060657 for the set of odd values in the range of the sigma function, i.e., the list of odd values in ordered by increasing size and without repetitions.

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

Cf. A000203 (sigma = sum-of-divisors function), A152678 (even terms in A000203), A028982 (squares and twice the squares).
See A062700 and A023195 for the subsequence resp. subset of primes; A023194 for the indices of A000203 which yield these primes.
Cf. A002117.

[d:k in [1..1000]|IsOdd(d) where d is DivisorSigma(1,k)]; // Marius A. Burtea, Jan 09 2020

Select[DivisorSigma[1, Range[1000]], OddQ[#] &] (* Giovanni Resta, Jan 08 2020 *)
With[{max = 1000}, DivisorSigma[1, Union[Range[Sqrt[max]]^2, 2*Range[Sqrt[max/2]]^2]]] (* Amiram Eldar, Nov 28 2023 *)

A152677_upto(lim)=apply(sigma,vecsort(concat(vector(sqrtint(lim\1), i, i^2), vector(sqrtint(lim\2), i, 2*i^2)))) \\ Gives [a(n) = sigma(k) with k = A028982(n) <= lim]. - Charles R Greathouse IV, Feb 15 2013, corrected by M. F. Hasler, Jan 08 2020

a(n) = A000203(A028982(n)). - R. J. Mathar, Dec 12 2008

Sum_{k=1..n} a(k) ~ c * n^3, where c = (16-10*sqrt(2))*zeta(3)/Pi^2 = 0.226276... . - Amiram Eldar, Nov 28 2023

Extended by R. J. Mathar, Dec 12 2008

Edited and definition reworded by M. F. Hasler, Jan 08 2020

A065061 Numbers k such that sigma(k) - tau(k) is a prime.

Original entry on oeis.org

3, 8, 162, 512, 1250, 8192, 31250, 32768, 41472, 663552, 2531250, 3748322, 5120000, 6837602, 7558272, 8000000, 15780962, 33554432, 35701250, 42762752, 45334242, 68024448, 75031250, 78125000, 91125000, 137149922, 243101250, 512000000, 907039232, 959570432

Offset: 1

Jason Earls, Nov 06 2001

nonn

From Kevin P. Thompson, Jun 20 2022: (Start)
Terms greater than 3 must be twice a square (see A064205).
No terms are congruent to 4 or 6 (mod 10) (see A064205).
(End)

			162 is a term since sigma(162) - tau(162) = 363 - 10 = 353, which is prime.

Amiram Eldar, Table of n, a(n) for n = 1..5000 (terms 1..265 from Kevin P. Thompson)

I.e., A065608(n) is prime. Cf. A064205.

Cf. A000005, A000203, A023194, A038344, A055813, A062700, A115919, A141242, A229264, A229265, A229266, A229268.

Do[ If[ PrimeQ[ DivisorSigma[1, n] - DivisorSigma[0, n]], Print[n]], {n, 1, 10^7}]

{ n=0; for (m=1, 10^9, if (isprime(sigma(m) - numdiv(m)), write("b065061.txt", n++, " ", m); if (n==100, return)) ) } \\ Harry J. Smith, Oct 05 2009

from itertools import count, islice
from sympy import isprime, divisor_sigma as s, divisor_count as t
def agen(): # generator of terms
    yield 3
    yield from (k for k in (2*i*i for i in count(1)) if isprime(s(k)-t(k)))
print(list(islice(agen(), 30))) # Michael S. Branicky, Jun 20 2022

a(17)-a(28) from Harry J. Smith, Oct 05 2009

a(29)-a(30) from Kevin P. Thompson, Jun 20 2022

A229264 Primes in A065387 in the order of their appearance.

Original entry on oeis.org

2, 19, 19, 79, 103, 113, 257, 523, 509, 1151, 1279, 1193, 1579, 2273, 3061, 2389, 2693, 2843, 5003, 4831, 5119, 7411, 5693, 5623, 8623, 6323, 10139, 8933, 18401, 14957, 20411, 20479, 21191, 20123, 29683, 28211, 36833, 55021, 57203, 68743, 48761, 66533, 62423

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			Third term of A038344 is 9 and sigma(9) + phi(9) = 13 + 6 = 19 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A065387, A115919, A141242, A229265, A229266, A229267, A229268.

with(numtheory); P:=proc(q) local a, n; for n from 1 to q do a:=sigma(n)+phi(n);
if isprime(a) then print(a); fi; od; end: P(10^6);

Select[Table[DivisorSigma[1,n]+EulerPhi[n],{n,30000}],PrimeQ] (* Harvey P. Dale, Apr 30 2018 *)

lista(kmax) = {my(f, s); for(k = 1, kmax, f = factor(k); s= sigma(f) + eulerphi(f); if(isprime(s), print1(s, ", ")));} \\ Amiram Eldar, Nov 19 2024

Name corrected by Amiram Eldar, Nov 19 2024

A229268 Primes of the form sigma(k) - tau(k), where sigma(k) = A000203(k) and tau(k) = A000005(k).

Original entry on oeis.org

2, 11, 353, 1013, 2333, 16369, 58579, 65519, 123733, 1982273, 7089683, 5778653, 12795053, 10500593, 22586027, 19980143, 24126653, 67108837, 72494713, 90781993, 106199593, 203275951, 164118923, 183105421, 320210549, 259997173, 794091653, 1279963973

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			Second term of A065061 is 8 and sigma(8) - tau(8) = 15 - 4 = 11 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A065608, A141242, A229264, A229266

with(numtheory); P:=proc(q) local a,n; a:= sigma(n)-tau(n); for n from 1 to q do
if isprime(a) then print(a); fi; od; end: P(10^6);

Join[{2}, Select[(DivisorSigma[1, #] - DivisorSigma[0, #]) & /@ (2*Range[20000]^2), PrimeQ]] (* Amiram Eldar, Dec 06 2022 *)

a(n) = A000203(A065061(n)) - A000005(A065061(n)). - Michel Marcus, Sep 21 2013

a(n) = A065608(A065061(n)). - Amiram Eldar, Dec 06 2022

More terms from Michel Marcus, Sep 21 2013

cp's OEIS Frontend

A100382 Record values of A062700.

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A071167 a(n) = A023194 - A062700(n). Negative values of A071166(m) = m-A006530(A000203(m)) differences. In these cases m is square number from A023194.

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A055638 Numbers k for which sigma(k^2) is prime.

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A065403 Primes of the form sigma(m^2) where m is a composite number ordered by values m.

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A065405 Composite numbers k such that the sum of the divisors of k^2 is a prime.

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A065404 Squares of composite numbers k such that sigma(k) (sum of divisors of k, A000203) is a prime.

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A152677 Subsequence of odd terms in A000203 (sum-of-divisors function sigma), in the order in which they occur and with repetitions.

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