A023194 -id:A023194 - cp's OEIS frontend

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

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

A225534 Numbers whose sum of cubed digits is prime.

Original entry on oeis.org

11, 101, 110, 111, 113, 115, 122, 124, 128, 131, 139, 142, 146, 148, 151, 155, 164, 166, 182, 184, 193, 199, 212, 214, 218, 221, 223, 227, 232, 236, 238, 241, 245, 254, 256, 263, 265, 269, 272, 278, 281, 283, 287, 289, 296, 298, 311, 319, 322, 326, 328, 335

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 10 2013

Note that 11 is the only two-digit number in the sequence.

a(n) ~ n. For 414 < n < 10000, 6.38*n - 528 provides an estimate of a(n) to within 6%.

			139 is in the sequence because 1^3 + 3^3 + 9^3 = 757, which is prime.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A197039, A055012, A046459, A225535.
Cf. A165330, A046197.
Cf. A046704, A023194.

Select[Range[350],PrimeQ[Total[IntegerDigits[#]^3]]&] (* Harvey P. Dale, Mar 16 2016 *)

digcubesum<-function(x) sum(as.numeric(strsplit(as.character(x),split="")[[1]])^3); library(gmp);
which(sapply(1:1000,function(x) isprime(digcubesum(x))>0))

A347877 Numbers k for which A003415(sigma(k)) is odd.

Original entry on oeis.org

2, 4, 5, 9, 10, 13, 16, 17, 20, 25, 26, 29, 32, 34, 37, 40, 41, 45, 52, 53, 58, 61, 64, 68, 72, 73, 74, 80, 82, 89, 90, 97, 98, 101, 104, 106, 109, 113, 116, 117, 122, 128, 136, 137, 146, 148, 149, 153, 157, 160, 162, 164, 173, 178, 180, 181, 193, 194, 196, 197, 200, 202, 208, 212, 218, 226, 229, 232, 233, 234

Offset: 1

Antti Karttunen, Sep 19 2021

nonn

Numbers k such that A342925(k) == 1 (mod 2).

Squares present in this sequence are terms of A347885 squared. (There are no even squares present, see A235991 for the explanation).

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A003415, A023194 (subsequence), A235991, A342925, A347872, A347873, A347885, A347878 (complement).

Cf. A347870 (characteristic function), A349909 (its partial sums).

ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[234], OddQ[ad[DivisorSigma[1, #]]] &] (* Amiram Eldar, Sep 19 2021 *)

For all n >= 1, A349909(a(n)) = n. - Antti Karttunen, Feb 23 2022

A371418 The largest aliquot divisor of the sum of divisors of n; a(1) = 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 4, 5, 1, 9, 6, 14, 7, 12, 12, 1, 9, 13, 10, 21, 16, 18, 12, 30, 1, 21, 20, 28, 15, 36, 16, 21, 24, 27, 24, 13, 19, 30, 28, 45, 21, 48, 22, 42, 39, 36, 24, 62, 19, 31, 36, 49, 27, 60, 36, 60, 40, 45, 30, 84, 31, 48, 52, 1, 42, 72, 34, 63, 48, 72

Offset: 1

Amiram Eldar, Mar 23 2024

Carmichael (1921) defined this arithmetic function for the purpose of studying periodic chains that are formed by repeatedly applying the mapping x -> a(x) starting at a given positive integer. This results in a sequence that is analogous to an aliquot sequence.
Periodic chains of cycle 1 are the fixed points of this sequence. 1 and the even perfect numbers (the even terms of A000396) are fixed points. Are there any other numbers k such that a(k) = k?
If a(k) = k and k is even, then a(k) is even and so is sigma(k), and therefore sigma(k) = 2*k and k is an even perfect number. If k is odd, then it is an odd multiperfect number, and no odd multiperfect number above 1 is known.
More specifically, if a(k) = k and k is odd, then k must be a square, and an m-multiperfect number (number k such that sigma(k) = m * k), with m being an odd prime number that is the least prime factor of sigma(k). For example, if there is an odd triperfect number (A005820) then it is a fixed point of this sequence.
Periodic chains of cycle 2 are amicable pairs (A371419 and A371420). Are there any longer cycles?

			The sum of the divisors of 3 is 1 + 3 = 4. The divisors of 4 are 1, 2, 4. 2 is the largest aliquot divisor of 4. Therefore a(3) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.
Eric Weisstein's World of Mathematics, Aliquot Sequence.
Eric Weisstein's World of Mathematics, Multiperfect Number.
Wikipedia, Aliquot sequence.

Cf. A000203, A000396, A005820, A023194, A028982, A028983, A032742, A071189, A371419, A371420, A371421, A371422, A371423.

r[n_] := n/FactorInteger[n][[1, 1]]; a[n_] := r[DivisorSigma[1, n]]; Array[a, 100]

a(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}

a(n) = A032742(A000203(n)).
a(n) = A000203(n)/A071189(n).
a(n) = A000203(n)/2 if n is in A028983 (i.e., n is not in A028982).
a(k) = 1 if and only if k = 1 or k is in A023194.

A063783 Numbers k such that the sum of the cubes of divisors of k is a prime.

Original entry on oeis.org

4, 9, 121, 36481, 72361, 146689, 259081, 654481, 683929, 786769, 1985281, 2036329, 3193369, 3636649, 3798601, 4583881, 5031049, 5470921, 5555449, 6135529, 6713281, 7284601, 7778521, 16589329, 20403289, 21557449, 22915369, 26739241, 27426169, 30261001, 30591961

Offset: 1

Labos Elemer, Aug 17 2001

nonn

Solutions to sigma_3(x) = prime.

			All these terms are squares of primes {2, 3, 11, 191, 269, 383, 509, 809, 827, 887, 1409, 1427, 1787, 1907, 1949, 2141, 2243, 2339, 2357, 2477, 2591, 2699, 2789, ...}, so their sigma_3(p^2) = p^6 + p^3 + 1 has polynomial of degree 6.
sigma_3(9) = 1 + 27 + 729 = 757 is a prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A001158, A023194, A066100.

Select[Prime[Range[500]]^2, PrimeQ@ DivisorSigma[3, #] &] (* Michael De Vlieger, Jul 16 2017 *)

{ n=0; p=0; for (m=1, 10^9, p=nextprime(p+1); if(isprime(p^6 + p^3 + 1), write("b063783.txt", n++, " ", p^2); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 31 2009

a(n) = A066100(n)^2. - Amiram Eldar, Aug 16 2024

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

A105402 Positive integers k such that the prime factors of sigma(k) are a subset of the prime factors of k.

Original entry on oeis.org

1, 6, 28, 30, 42, 66, 84, 102, 120, 138, 186, 210, 270, 282, 318, 330, 364, 420, 426, 462, 496, 510, 546, 570, 642, 672, 690, 714, 762, 840, 868, 870, 924, 930, 966, 1080, 1092, 1122, 1146, 1302, 1320, 1410, 1428, 1488, 1518, 1590, 1638, 1722, 1770, 1782, 1890

Offset: 1

Walter Kehowski, May 01 2005

Also numbers k such that k^k/sigma(k) is integral. - Vicente Izquierdo Gomez, Jan 04 2013.

Pollack and Pomerance call these numbers "prime-deficient numbers". - Amiram Eldar, Jun 02 2020

			102 is a term since 102 = 2*3*17 and sigma(102) = 2^3*3^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.

Cf. A000203, A027598, A058063, A054027, A014567, A023194, A175200.

A:=select(proc(z) numtheory[factorset](sigma(z)) subset numtheory[factorset](z) end,[$1..100000]); has 716 members.

Select[Range[2000],IntegerQ[#^#/DivisorSigma[1,#]] &] (* Vicente Izquierdo Gomez, Jan 04 2013 *)

Extended by R. J. Mathar, Dec 08 2008

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

Original entry on oeis.org

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

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

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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A225534 Numbers whose sum of cubed digits is prime.

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A347877 Numbers k for which A003415(sigma(k)) is odd.

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A371418 The largest aliquot divisor of the sum of divisors of n; a(1) = 1.

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A063783 Numbers k such that the sum of the cubes of divisors of k is a prime.

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

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A105402 Positive integers k such that the prime factors of sigma(k) are a subset of the prime factors of k.

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