A007691 -id:A007691 - cp's OEIS frontend

A341606 Square array A(n,k) = A017666(A246278(n,k)), read by falling antidiagonals; denominator of abundancy index as applied onto prime shift array A246278.

2, 4, 3, 1, 9, 5, 8, 5, 25, 7, 5, 27, 35, 49, 11, 3, 21, 125, 77, 121, 13, 7, 15, 55, 343, 143, 169, 17, 16, 11, 175, 13, 1331, 221, 289, 19, 6, 81, 65, 539, 187, 2197, 323, 361, 23, 10, 75, 625, 119, 1573, 247, 4913, 437, 529, 29, 11, 63, 245, 2401, 209, 2873, 391, 6859, 667, 841, 31

Offset: 1

Antti Karttunen, Feb 16 2021

See also comments and examples in A341605.

			The top left corner of the array:
   k=  1    2    3      4    5      6    7       8      9     10   11      12
  2k=  2    4    6      8   10     12   14      16     18     20   22      24
    |
----+--------------------------------------------------------------------------
  1 |  2,   4,   1,     8,   5,     3,   7,     16,     6,    10,  11,      2,
  2 |  3,   9,   5,    27,  21,    15,  11,     81,    75,    63,  39,      9,
  3 |  5,  25,  35,   125,  55,   175,  65,    625,   245,   275,  85,    875,
  4 |  7,  49,  77,   343,  13,   539, 119,   2401,   121,    91, 133,   3773,
  5 | 11, 121, 143,  1331, 187,  1573, 209,  14641,  1859,  2057, 253,  17303,
  6 | 13, 169, 221,  2197, 247,  2873, 299,  28561,  3757,  3211, 377,   2197,
  7 | 17, 289, 323,  4913, 391,  5491, 493,  83521,  6137,  6647, 527,  93347,
  8 | 19, 361, 437,  6859, 551,  8303, 589, 130321, 10051, 10469,  37, 157757,
  9 | 23, 529, 667, 12167, 713, 15341, 851, 279841, 19343, 16399, 943, 352843,
etc.
Arrays A341607 and A341608 give the largest prime factor (A006530) and the number of prime factors with multiplicity (A001222) of these terms. There are nonmonotonicities in both, for example, in columns 11, 12 and 14. This is illustrated below:
For column 11, with successive prime shifts of 22, we obtain:
     n sigma(n)             sigma(n)/n in lowest terms,
                            A017665(n)/A017666(n)
---------------------------------------------------------------------------
    22   36 = (2^2 * 3^2),        18/11  = (2 * 3^2)/11
    39   56 = (2^3 * 7),          56/39  = (2^3 * 7)/(3 * 13)
    85  108 = (2^2 * 3^3),       108/85  = (2^2 * 3^3)/(5 * 17)
   133  160 = (2^5 * 5),         160/133 = (2^5 * 5)/(7 * 19)
   253  288 = (2^5 * 3^2),       288/253 = (2^5 * 3^2)/(11 * 23)
   377  420 = (2^2 * 3 * 5 * 7), 420/377 = (2^2 * 3 * 5 * 7)/(13 * 29)
   527  576 = (2^6 * 3^2),       576/527 = (2^6 * 3^2)/(17 * 31)
   703  760 = (2^3 * 5 * 19),     40/37  = (2^3 * 5)/37 <-- A001222 drops!
   943 1008 = (2^4 * 3^2 * 7),  1008/943 = (2^4 * 3^2 * 7)/(23 * 41)
-
On the second last row, the denominator of 760/703 (= 40/37) has only one prime factor (instead of two), namely 37, because sigma(703) has 19 as its divisor, which otherwise would be present in the denominator.
-
For column 12, with successive prime shifts of 24, we obtain:
      n sigma(n)                        sigma(n)/n
---------------------------------------------------------------------------
     24     60 = (2^2 * 3 * 5),            5/2     = (5)/(2)
    135    240 = (2^4 * 3 * 5),           16/9     = (2^4)/(3^2)
    875   1248 = (2^5 * 3 * 13),        1248/875   = (2^5 * 3 * 13)/(5^3 * 7)
   3773   4800 = (2^6 * 3 * 5^2),       4800/3773  = (2^6 * 3 * 5^2)/(7^3 * 11)
  17303  20496 = (2^4 *3 *7 *61),      20496/17303 = (2^4 *3 *7 *61)/(11^3 * 13)
  37349  42840 = (2^3 *3^2 *5 *7 *17),  2520/2197  = (2^3 * 3^2 *5 *7)/(13^3) !!
  93347 104400 = (2^4 *3^2 *5^2 *29), 104400/93347 = (2^4 *3^2 *5^2 *29)/(17^3 *19)
-
On the second last row, the denominator of 42840/37349 (= 2520/2197) has no prime factor 17 (which would be otherwise present), because sigma(37349) has it as its divisor.
-
For column 14, with successive prime shifts of 28, we obtain:
     n sigma(n)               sigma(n)/n
---------------------------------------------------------------------------
    28   56 = (2^3 * 7),             2/1,
    99  156 = (2^2 * 3 * 13),       52/33   = (2^2 * 13)/(3 * 11)
   325  434 = (2 * 7 * 31),        434/325  = (2 * 7 * 31)/(5^2 * 13)
   833 1026 = (2 * 3^3 * 19),     1026/833  = (2 * 3^3 * 19)/(7^2 * 17)
  2299 2660 = (2^2 * 5 * 7 * 19),  140/121  = (2^2 * 5 * 7)/(11^2) <-- !!
  3887 4392 = (2^3 * 3^2 * 61),   4392/3887 = (2^3 * 3^2 * 61)/(13^2 * 23)
On the second last row, the denominator of 2660/2299 (= 140/121) has no prime factor 19 (which would be otherwise present), because sigma(2299) has it as its divisor.
Note that if A006530 does not grow, then certainly A001222 drops.

Cf. A017666, A246278.
Cf. A341605 (numerators), A341626 (numerators of the columnwise first quotients of A341605/A341606), A341627 (and their denominators), A355925, A355927.
Cf. A341607 (the largest prime factor in this array), A341608 (the number of prime factors, with multiplicity).
Cf. also A007691, A341523, A341524.

up_to = 105;
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A017666(n) = denominator(sigma(n)/n);
A341606sq(row,col) = A017666(A246278sq(row,col));
A341606list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A341606sq(col,(a-(col-1))))); (v); };
v341606 = A341606list(up_to);
A341606(n) = v341606[n];

A(n, k) = A017666(A246278(n, k)).

A(n, k) = A246278(n, k) / A355925(n, k). - Antti Karttunen, Jul 22 2022

A054030 Sigma(n)/n for n such that sigma(n) is divisible by n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 4, 3, 4, 4, 2, 4, 4, 3, 4, 3, 2, 5, 5, 4, 3, 4, 2, 4, 4, 5, 4, 5, 5, 4, 5, 5, 4, 4, 4, 5, 4, 4, 2, 5, 4, 5, 6, 5, 5, 5, 5, 5, 5, 6, 5, 5, 4, 5, 6, 5, 4, 4, 5, 4, 5, 4, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 5, 6, 5, 6, 6, 5, 4, 4, 5, 4, 4, 5, 6, 5, 5, 4, 6, 4, 4, 6, 5, 6, 6, 6, 6, 6, 6, 6, 5, 6

Offset: 1

Asher Auel, Jan 19 2000

The graph supports the conjecture that all numbers except 2 appear only a finite number of times. Sequences A000396, A005820, A027687, A046060 and A046061 give the n for which the abundancy sigma(n)/n is 2, 3, 4, 5 and 6, respectively. See A134639 for the number of n having abundancy greater than 2. - T. D. Noe, Nov 04 2007

T. D. Noe, Table of n, a(n) for n = 1..1600 (using Flammenkamp's data)
Achim Flammenkamp, The Multiply Perfect Numbers Page
Eric Weisstein's World of Mathematics, Abundancy

Cf. A000203, A007691, A054024, A065997, A219545.

with(numtheory): for i while i < 33000 do
if sigma(i) mod i = 0 then print(sigma(i)/i) fi od;

for(n=1,1e7,if(denominator(k=sigma(n,-1))==1, print1(k", "))) \\ Charles R Greathouse IV, Mar 09 2014

a(n) = sigma(A007691(n))/A007691(n)

More terms from Jud McCranie, Jul 09 2000

More terms from David Wasserman, Jun 28 2004

A336849 a(n) = A003961(n) / gcd(A003961(n), sigma(A003961(n))), where A003961 is the prime shift towards larger primes.

Original entry on oeis.org

1, 3, 5, 9, 7, 5, 11, 27, 25, 21, 13, 15, 17, 11, 35, 81, 19, 75, 23, 63, 55, 39, 29, 9, 49, 17, 125, 33, 31, 35, 37, 243, 65, 57, 77, 225, 41, 23, 85, 189, 43, 55, 47, 9, 175, 29, 53, 135, 121, 49, 19, 17, 59, 125, 13, 99, 115, 93, 61, 105, 67, 111, 275, 729, 119, 65, 71, 171, 29, 77, 73, 135, 79, 41, 245, 69, 143

Offset: 1

Antti Karttunen, Aug 06 2020

If there are no more 1's in this sequence after the initial one, then there are no odd terms of A007691 (multiply perfect numbers) larger than one.

Denominator of the ratio A003973(n) / A003961(n), also denominator of the ratio (A341528(n)/A341529(n)) / (n / sigma(n)). - Antti Karttunen, Feb 16 2021

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A003961, A003973, A007691, A017666, A336848, A336850, A341528, A341529, A341530.

Cf. A341525 (numerators).

f[p_, e_] := NextPrime[p]^e; g[1] = 1; g[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := (gn = g[n])/GCD[gn, DivisorSigma[1, gn]]; Array[a, 100] (* Amiram Eldar, Feb 17 2021 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336849(n) = { my(u=A003961(n)); (u/gcd(u, sigma(u))); };
\\ Or alternatively as:
A336849(n) = { my(u=A003961(n)); denominator(sigma(u)/u); };

a(n) = A003961(n) / A336850(n) = A003961(n) / gcd(A003961(n), A003973(n)).

a(n) = A017666(A003961(n)).

A127724 k-imperfect numbers for some k >= 1.

Original entry on oeis.org

1, 2, 6, 12, 40, 120, 126, 252, 880, 2520, 2640, 10880, 30240, 32640, 37800, 37926, 55440, 75852, 685440, 758520, 831600, 2600640, 5533920, 6917400, 9102240, 10281600, 11377800, 16687440, 152182800, 206317440, 250311600, 475917120, 715816960, 866829600

Offset: 1

T. D. Noe, Jan 25 2007

For prime powers p^e, define a multiplicative function rho(p^e) = p^e - p^(e-1) + p^(e-2) - ... + (-1)^e. A number n is called k-imperfect if there is an integer k such that n = k*rho(n). Sequence A061020 gives a signed version of the rho function. As with multiperfect numbers (A007691), 2-imperfect numbers are also called imperfect numbers. As shown by Iannucci, when rho(n) is prime, there is sometimes a technique for generating larger imperfect numbers.
Zhou and Zhu find 5 more terms, which are in the b-file. - T. D. Noe, Mar 31 2009
Does this sequence follow Benford's law? - David A. Corneth, Oct 30 2017
If a term t has a prime factor p from A065508 with exponent 1 and does not have the corresponding prime factor q from A074268, then t*p*q is also a term. - Michel Marcus, Nov 22 2017
For n >= 1, the least n-imperfect numbers are 1, 2, 6, 993803899780063855042560. - Michel Marcus, Feb 13 2018
For any m > 0, if n*p^(2m-1) is k-imperfect, q = rho(p^(2m)) is prime and gcd(pq,n) = 1, then n*p^(2m)*q is also k-imperfect. - M. F. Hasler, Feb 13 2020

			126 = 2*3^2*7, rho(126) = (2-1)*(9-3+1)*(7-1) = 42.  3*42 = 126, so 126 is 3-imperfect. - _Jud McCranie_ Sep 07 2019

R. K. Guy, Unsolved Problems in Theory of Numbers, Springer, 1994, B1.

Giovanni Resta, Table of n, a(n) for n = 1..50 (terms < 10^13, a(1)-a(39) from T. D. Noe (from Iannucci, Zhou, and Zhu), a(40)-a(44) from Donovan Johnson)
David A. Corneth, Conjectured to be the terms up to 10^28
Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.
Donovan Johnson, 43 terms > 2*10^11
Andrew Lelechenko, 4-imperfect numbers, Apr 19 2014.
Michel Marcus, More 4-imperfect numbers, Nov 07 2017.
Michel Marcus, Least known integers with small denominator-fractional k's, Feb 13 2018.
Allan Wechsler, Some progress in k-imperfect numbers (A127724), Seqfan, Feb 13 2020. Gives a first instance of a 5-imperfect number.
Weiyi Zhou and Long Zhu, On k-imperfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 9 (2009), #A01.

Cf. A127725 (2-imperfect numbers), A127726 (3-imperfect numbers), A127727 (related primes), A309806 (the k values).
Cf. A061020 (signed version of rho function), A206369 (the rho function).
Cf. A065508, A074268.

f[p_,e_]:=Sum[(-1)^(e-k) p^k, {k,0,e}]; rho[n_]:=Times@@(f@@@FactorInteger[n]); Select[Range[10^6], Mod[ #,rho[ # ]]==0&]

isok(n) = denominator(n/sumdiv(n, d, d*(-1)^bigomega(n/d))) == 1; \\ Michel Marcus, Oct 28 2017

upto(ulim) = {res = List([1]); rhomap = Map(); forprime(p = 2, 3, for(i = 1, logint(ulim, p), mapput(rhomap, p^i, rho(p^i)); iterate(p^i, mapget(rhomap, p^i), ulim))); listsort(res, 1); res}
iterate(m, rhoo, ulim) = {my(c); if(m / rhoo == m \ rhoo, listput(res, m); my(frho = factor(rhoo)); for(i = 1, #frho~, if(m%frho[i, 1] != 0, for(e = 1, logint(ulim \ m, frho[i, 1]), if(mapisdefined(rhomap, frho[i, 1]^e) == 0, mapput(rhomap, frho[i, 1]^e, rho(frho[i, 1]^e))); iterate(m * frho[i, 1]^e, rhoo * mapget(rhomap, frho[i, 1]^e), ulim)); next(2))))}
rho(n) = {my(f = factor(n), res = q = 1); for(i=1, #f~, q = 1; for(j = 1, f[i, 2], q = -q + f[i, 1]^j); res * =q); res} \\ David A. Corneth, Nov 02 2017

A127724_vec=concat(1, select( {is_A127724(n)=!(n%A206369(n))}, [1..10^5]*2))
  /* It is known that the least odd term > 1 is > 10^49. This code defines an efficient function is_A127724, but A127724_vec is better computed with upto(.) */
  A127724(n)=A127724_vec[n] \\ Used in other sequences. - M. F. Hasler, Feb 13 2020

Small correction in name from Michel Marcus, Feb 13 2018

A351458 Numbers k for which k * gcd(sigma(k), A276086(k)) is equal to sigma(k) * gcd(k, A276086(k)), where A276086 is the primorial base exp-function, and sigma gives the sum of divisors of its argument.

Original entry on oeis.org

1, 10, 56, 9196, 9504, 56160, 121176, 239096, 354892, 411264, 555520, 716040, 804384, 904704, 1063348, 1387386, 1444352, 1454112, 1884800, 2708640, 3317248, 3548920, 4009824, 4634784, 6179712, 6795360, 7285248, 14511744, 16328466, 28377216, 29855232, 31940280, 37444736, 42711552, 49762944, 52815744

Offset: 1

Antti Karttunen, Feb 13 2022

nonn

Numbers k such that k * A324644(k) = A000203(k) * A324198(k).
Numbers k such that gcd(A064987(k), A324580(k)) = gcd(A064987(k), A351252(k)).
Numbers k such that their abundancy index [sigma(k)/k] is equal to A324644(k)/A324198(k). See A364286.
A324644 gives odd values for even numbers and for the odd squares. A324198 is odd on all arguments, therefore on odd squares the above equation reduces to odd * odd = odd * odd, and on odd nonsquares as odd * even = even * odd. It is an open question whether there are any odd terms after the initial a(1)=1.
If k is even, but not a multiple of 3, then A276086(k) is a multiple of 3, but not even (i.e., is an odd multiple of 3). If for such k also sigma(k) = 3*k, then A007949(A324644(k)) = min(A007949(sigma(k)), A007949(A276086(k))) = 1, while A007949(A324198(k)) = min(A007949(k), A007949(A276086(k))) = 0, therefore all such k's do occur in this sequence, for example, the two known terms of A005820 (3-perfect numbers) that are not multiples of three: 459818240, 51001180160, but also any hypothetical term of A005820 of the form 4u+2, where 2u+1 is not multiple of 3, and which by necessity is then also an odd perfect number.
Similarly, of the 65 known 5-multiperfect numbers (A046060), those 20 that are not multiples of five are included in this sequence. Note that all 65 are multiples of six.
It is conjectured that the intersection of this sequence with the multiperfect numbers (A007691) gives A323653, see comments in the latter.
For all even terms k of this sequence, A007814(A000203(k)) = A007814(k), sigma preserves the 2-adic valuation, and A007949(A000203(k)) >= A007949(k), i.e., does not decrease the 3-adic valuation. The condition is equivalence (=) when k is a multiple of 6. With odd terms, any hypothetical odd perfect number x would yield a one greater 2-adic valuation for sigma(x) than for x, but would satisfy the main condition of this sequence. - Corrected Feb 17 2022
If k is a nonsquare positive odd number (in A088828), then it must be a term of A191218. - Antti Karttunen, Mar 10 2024

Cf. A000203, A005820, A007691, A007814, A007949, A046060, A088828, A191218, A276086, A323653, A324198, A324644, A327938, A364286, A351459 (intersection with a similar sequence, A349745).

Cf. also A351549.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA351458(n) = { my(s=sigma(n), z=A276086(n)); (n*gcd(s,z))==(s*gcd(n,z)); };

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]); \\ Works OK with rationals also!
isA351458(n) = { my(orgn=n, s=sigma(n), abi=s/n, p=2, q=A006530(abi), d, e1, e2); while((1!=abi)&&(p<=q), d = n%p; e1 = min(d, valuation(s, p)); e2 = min(d, valuation(orgn, p)); d = e1-e2; if(valuation(abi,p)!=d, return(0), abi /= (p^d)); n = n\p; p = nextprime(1+p)); (abi==1); }; \\ (This implementation does not require the construction of largish intermediate numbers, A276086, but might still be slower and return a few false positives on the long run, so please check the results with the above program). - Antti Karttunen, Feb 19 2022

A019294 Number (> 0) of iterations of sigma (A000203) required to reach a multiple of n when starting with n.

Original entry on oeis.org

1, 2, 4, 2, 5, 1, 5, 2, 7, 4, 15, 3, 13, 3, 2, 2, 13, 4, 12, 5, 2, 13, 16, 2, 17, 4, 9, 1, 78, 7, 10, 4, 17, 11, 6, 5, 28, 22, 4, 7, 39, 2, 16, 16, 16, 10, 32, 5, 13, 17, 9, 3, 58, 11, 19, 5, 13, 67, 97, 2, 23, 5, 16, 2, 4, 8, 101, 21, 19, 11, 50, 4, 20, 20, 23, 14, 21, 10, 36, 5, 15

Offset: 1

N. J. A. Sloane

Let sigma^m(n) be result of applying sum-of-divisors function m times to n; sequence gives m(n) = min m such that n divides sigma^m(n).
Perfect numbers require one iteration.
It is conjectured that the sequence is finite for all n.
See also the Cohen-te Riele links under A019276.
a(A111227(n)) > A111227(n). - Reinhard Zumkeller, Aug 02 2012
a(659) > 870. - Michel Marcus, Jan 04 2017

			If n = 9 the iteration sequence is s(9) = {9, 13, 14, 24, 60, 168, 480, 1512, 4800, 15748, 28672} and Mod[s(9), 9] = {0, 4, 5, 6, 6, 6, 3, 0, 3, 7, 7}. The first iterate which is a multiple of 9 is the 7th = 1512, so a(9) = 7. For n = 67, the 101st iterate is the first, so a(67) = 101. Usually several iterates are divisible by the initial value. E.g., if n = 6, then 91 of the first 100 iterates are divisible by 6.
A difficult term to compute: a(461) = 557. - _Don Reble_, Jun 23 2005

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.

Don Reble, Table of n, a(n) for n = 1..1578 (Terms a(1..400) from T. D. Noe, Nov 2007; a(401..659) from Michel Marcus, Jan 02 2017), Feb 20 2022.
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100. See Table 2, p. 95.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Carl Pomerance, On the composition of the functions sigma and phi, Colloq. Math., 59 (1989), 11-15.
Wikipedia, Iterated function, as of Jan 02 2020.
Zeraoulia Rafik, On congruence of the iterated form sigma^k(m) = 0 mod m, arXiv:2102.09941 [math.NT], 2021.

Cf. A019295 (ratio sigma^m(n)/n), A019276 (indices of records), A019277 (records), A000396.

a019294 n = snd $ until ((== 0) . (`mod` n) . fst)
                        (\(x, i) -> (a000203 x, i + 1)) (a000203 n, 1)
-- Reinhard Zumkeller, Aug 02 2012

a:=[]; f:=func; for n in [1..81] do k:=n; s:=1; while f(k) mod n ne 0 do k:=f(k); s:=s+1; end while; Append(~a,s); end for; a; // Marius A. Burtea, Jan 11 2020

A019294 := proc(n)
    local a,nitr ;
    a := 1 ;
    nitr := numtheory[sigma](n);
    while modp(nitr,n) <> 0 do
        nitr := numtheory[sigma](nitr) ;
        a := a+1 ;
    end do:
    return a;
end proc: # R. J. Mathar, Aug 22 2016

f[n_, m_] := Block[{d = DivisorSigma[1, n]}, If[ Mod[d, m] == 0, 0, d]]; Table[ Length[ NestWhileList[ f[ #, n] &, n, # != 0 &]] - 1, {n, 84}] (* Robert G. Wilson v, Jun 24 2005 *)
Table[Length[NestWhileList[DivisorSigma[1,#]&,DivisorSigma[1,n], !Divisible[ #,n]&]],{n,90}] (* Harvey P. Dale, Mar 04 2015 *)

a(n)=if(n<0,0,c=1; s=n; while(sigma(s)%n>0,s=sigma(s); c++); c)

apply( A019294(n,s=n)=for(k=1,oo,(s=sigma(s))%n||return(k)), [1..99]) \\ M. F. Hasler, Jan 07 2020

Conjecture: lim_{n -> oo} log(Sum_{k=1..n} a(k))/log(n) = C = 1.6... - Benoit Cloitre, Aug 24 2002
From Michel Marcus, Jan 02 2017: (Start)
a(n) = 1 for n in A007691.
a(n) = 2 for n in A019278 unless it belongs to A007691.
a(n) = 3 for n in A019292 unless it belongs to A007691 or A019278. (End)

Additional comments from Labos Elemer, Jun 20 2001

Edited by M. F. Hasler, Jan 07 2020

A175200 Numbers k such that rad(k) divides sigma(k).

Original entry on oeis.org

1, 6, 24, 28, 40, 54, 96, 120, 135, 216, 224, 234, 270, 360, 384, 486, 496, 540, 588, 600, 640, 672, 864, 891, 936, 1000, 1080, 1350, 1372, 1521, 1536, 1638, 1782, 1792, 1920, 1944, 2016, 2160, 2176, 3000, 3240, 3375, 3402, 3456, 3564, 3724, 3744, 3780, 4320

Offset: 1

Michel Lagneau, Mar 03 2010

nonn

rad(k) is the product of the distinct primes dividing k (A007947). sigma(k) is the sum of divisors of k (A000203). The odd numbers in this sequence (A336554) are rare: 1, 135, 891, 1521, 3375, 5733, 10935, 11907, 41067, 43875, ...
Also numbers k such that k divides sigma(k)^tau(k). - Arkadiusz Wesolowski, Nov 09 2013
This sequence is infinite. It contains an infinite number of even elements and an infinite number of odd ones. This is due to the fact that for every odd prime p and every prime q dividing p+1, p*q^r is prime-perfect when r = -1 + the multiplicative order of q modulo p. - Emmanuel Vantieghem, Oct 13 2014
For each term, it is possible to find an exponent k such that sigma(n)^k is divisible by n. A007691 (multi-perfect numbers) is a subsequence of terms that have k=1. A263928 is the subsequence of terms that have k=2. - Michel Marcus, Nov 03 2015
Pollack and Pomerance call these numbers "prime-abundant numbers". - Amiram Eldar, Jun 02 2020

			rad(6) = 6, sigma(6) = 12 = 6*2.
rad(24) = 6, sigma(24) = 60 = 6*10.
rad(43875) = 195, sigma(43875) = 87360 = 195*448.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 827.

Donovan Johnson, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12A, Paper A14, 2012.

Cf. A000203, A007947, A027598, A069235, A105402, A173615, A336554 (odd terms).

[n: n in [1..5000] | IsZero(DivisorSigma(1, n)^n mod n)];// Vincenzo Librandi, Aug 07 2018

for n from 1 to 5000 do : p1:= ifactors(n)[2] :p2 :=mul(p1[i][1], i=1..nops(p1)): if irem(sigma(n),p2) =0 then print (n): else fi: od :

Select[Range@5000, Divisible[DivisorSigma[1, #]^#, # ]&] (* Vincenzo Librandi, Aug 07 2018 *)

isok(n) = {fs = Set(factor(sigma(n))[,1]); fn = Set(factor(n)[,1]); fn == setintersect(fn, fs);} \\ Michel Marcus, Nov 03 2015

A011775 Numbers k such that k divides phi(k) * sigma(k).

Original entry on oeis.org

1, 6, 18, 24, 28, 40, 54, 72, 84, 96, 117, 120, 135, 162, 196, 200, 216, 224, 234, 252, 270, 288, 360, 384, 468, 486, 496, 540, 588, 600, 640, 648, 672, 756, 775, 819, 864, 891, 936, 1000, 1080, 1152, 1350, 1372, 1458, 1488, 1521, 1536, 1550, 1568, 1638, 1701, 1764

Offset: 1

R. K. Guy

nonn

Comments from Farideh Firoozbakht, Dec 01 2005: (Start)
I. All numbers of the form 2^(4m-1)*5^n where m & n are natural numbers are in the sequence. Because if s=2^(4m-1)*5^n then phi(s)=2^(4m-2)*4*5^(n-1); sigma(s)=(2^(4m)-1)*(5^(n+1)-1)/4 so phi(s)*sigma(s)=6*((16^m-1)/15)*((5^(n+1)-1)/4)*(2^(4m-1)*5^n)= 6*((16^m-1)/15)*((5^(n+1)-1)/4)*s, note that (16^m-1)/15 and (5^(n+1)-1)/4 are integers, hence s divides phi(s)*sigma(s).
II. All numbers of the form 2^(2m-1)*3^n where m & n are natural numbers (A228104) are in the sequence. Because if s=2^(2m-1)*3^n then phi(s)=2^(2m-2)*2*3^(n-1); sigma(s)=(2^(2m)-1)*(3^(n+1)-1)/2 so phi(s)*sigma(s)=((3^(n+1)-1)/2)*((4^m-1)/3)*(2^(2m-1)*3^n) =((3^(n+1)-1)/2)*((4^m-1)/3)*s, note that (3^(n+1)-1)/2 and (4^m-1)/3 are integers, hence s divides phi(s)*sigma(s).
So this sequence is infinite. Also it is obvious that perfect numbers (A000396) and multiply-perfect numbers(A007691) are subsequences of this sequence. (End)

Donovan Johnson, Table of n, a(n) for n = 1..10000
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Cf. A062354, A000396, A007691.

Select[Range[1770], IntegerQ[DivisorSigma[1, # ]*EulerPhi[ # ]/# ] &] (* Farideh Firoozbakht, Dec 01 2005 *)

is(n)=sigma(n)*eulerphi(n)%n==0 \\ Charles R Greathouse IV, Nov 27 2013

Corrected and extended by David W. Wilson

A046762 Numbers k such that the sum of the squares of the divisors of k is divisible by k.

Original entry on oeis.org

1, 10, 60, 65, 84, 130, 140, 150, 175, 260, 350, 420, 525, 780, 1050, 1105, 1820, 2100, 2210, 4420, 4650, 5425, 5460, 8840, 10500, 10850, 13260, 16275, 19720, 20150, 20737, 21700, 30225, 30940, 32045, 32550, 41474, 45500, 55250, 57350, 60450

Offset: 1

Labos Elemer

nonn

Compare with multiply perfect numbers A007691. Here Sum(divisors) is replaced by Sum(square of divisors).
Problem 11090 proves that this sequence is infinite. - T. D. Noe, Apr 18 2006
Cai, Chen, & Zhang prove that sigma_2(n)/n = b has only finitely many solutions for a given b, and hence (since this sequence is infinite) sigma_2(a(n))/a(n) is unbounded. - Charles R Greathouse IV, Jul 21 2016

			k = 65 = a(4), sigma(2,65) = 4420 = 65*68 = 68*k;
k = 1820 = a(17), the divisor-square sum is 4641000 = 2550*1820 = 2550*k.

Amiram Eldar, Table of n, a(n) for n = 1..3200 (terms 1..1000 from T. D. Noe)
Tianxin Cai, Deyi Chen, and Yong Zhang, Perfect numbers and Fibonacci primes (I), Int. J. Number Theory 11, 159 (2015).
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.

Cf. A007691.

Select[Range[70000],Divisible[DivisorSigma[2,#],#]&] (* Harvey P. Dale, Dec 15 2010 *)

is(n)=sigma(n,2)%n==0 \\ Charles R Greathouse IV, Feb 04 2013

A065997 Numbers n such that sigma(n) / n is prime.

Original entry on oeis.org

6, 28, 120, 496, 672, 8128, 523776, 33550336, 459818240, 1476304896, 8589869056, 14182439040, 31998395520, 51001180160, 137438691328, 518666803200, 13661860101120, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 2305843008139952128

Offset: 1

Joseph L. Pe, Dec 10 2001

This is a subsequence of the sequence of multiply perfect numbers A007691.
The prime values of sigma(n) / n are A219545.
Numbers whose abundancy index is a prime. There are two visible bends (sudden changes in the growth rate) in the scatter plot. Compare also to the scatter plot of A336702. - Antti Karttunen, Feb 25 2022

Antti Karttunen, Table of n, a(n) for n = 1..597 (computed from the b-file of A007691 prepared by T. D. Noe, using Flammenkamp's data)

Subsequence of A007691 and of A342924.
Cf. A000396, A005820, A046060 (subsequences).
Cf. A219545, A336702.

isA065997(n) = { my(p=sigma(n)/n); (1==denominator(p) && isprime(p)); }; \\ Antti Karttunen, Feb 25 2022

Terms a(10) to a(14) from Jonathan Sondow, Nov 22 2012

Extended by T. D. Noe, Nov 26 2012

cp's OEIS Frontend

A341606 Square array A(n,k) = A017666(A246278(n,k)), read by falling antidiagonals; denominator of abundancy index as applied onto prime shift array A246278.

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A054030 Sigma(n)/n for n such that sigma(n) is divisible by n.

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Maple

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A336849 a(n) = A003961(n) / gcd(A003961(n), sigma(A003961(n))), where A003961 is the prime shift towards larger primes.

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A127724 k-imperfect numbers for some k >= 1.

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A351458 Numbers k for which k * gcd(sigma(k), A276086(k)) is equal to sigma(k) * gcd(k, A276086(k)), where A276086 is the primorial base exp-function, and sigma gives the sum of divisors of its argument.

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A019294 Number (> 0) of iterations of sigma (A000203) required to reach a multiple of n when starting with n.

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Magma

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A175200 Numbers k such that rad(k) divides sigma(k).

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