A088580 -id:A088580 - cp's OEIS frontend

A063531 Numbers k such that sigma(k) + 1 is a square.

2, 7, 8, 14, 15, 23, 32, 33, 35, 47, 54, 56, 57, 60, 72, 78, 79, 84, 87, 92, 95, 120, 123, 124, 128, 138, 143, 154, 165, 167, 174, 184, 190, 196, 213, 223, 235, 242, 252, 253, 258, 267, 295, 312, 315, 319, 323, 327, 348, 359, 375, 378, 380, 393, 412, 423, 439

Offset: 1

Labos Elemer, Aug 02 2001

nonn

Numbers k such that A000203(k) = -1 + m^2 for some m.

			If k = p(p+2) is a product of twin primes (from A037074), then sigma(k) + 1 = 1 + (p+1)(p+3) = (p+2)^2, square of the larger twin. Other solutions can be either special primes = m^2 - 2 or composites like 120: sigma(120) = 120 + 60 + ... + 1 = 360 = 19^2 - 1. Square number solution is, e.g., 196: sigma(196) = 399 = 20^2 - 1.

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

Cf. A000203, A028871, A037074, A063530, A088580.

Select[Range[500],IntegerQ[Sqrt[DivisorSigma[1,#]+1]]&] (* Harvey P. Dale, Jul 02 2021 *)

{ n=0; for (a=1, 10^9, if (issquare(sigma(a) + 1), write("b063531.txt", n++, " ", a); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 25 2009

Minor edits from Franklin T. Adams-Watters, Aug 29 2009

A336924 a(n) = spf(1+sigma(n)), where spf is the smallest prime factor and sigma is the sum of divisors function.

Original entry on oeis.org

2, 2, 5, 2, 7, 13, 3, 2, 2, 19, 13, 29, 3, 5, 5, 2, 19, 2, 3, 43, 3, 37, 5, 61, 2, 43, 41, 3, 31, 73, 3, 2, 7, 5, 7, 2, 3, 61, 3, 7, 43, 97, 3, 5, 79, 73, 7, 5, 2, 2, 73, 3, 5, 11, 73, 11, 3, 7, 61, 13, 3, 97, 3, 2, 5, 5, 3, 127, 97, 5, 73, 2, 3, 5, 5, 3, 97, 13, 3, 11, 2, 127, 5, 3, 109, 7, 11, 181, 7, 5, 113, 13

Offset: 1

Antti Karttunen, Aug 09 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A020639, A071189, A088580.

Cf. also A336691, A336692, A336693, A336694, A336695, A336696.

A336924(n) = (factor((1+sigma(n)))[1, 1]);

a(n) = A020639(1+A000203(n)) = A020639(A088580(n)).

A336925 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336147(1+sigma(i)) = A336147(1+sigma(j)), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 1, 6, 7, 4, 8, 9, 2, 2, 1, 7, 10, 11, 12, 13, 14, 2, 15, 1, 12, 16, 17, 18, 19, 13, 1, 3, 20, 3, 21, 22, 15, 17, 23, 12, 24, 9, 25, 26, 19, 3, 2, 27, 28, 19, 13, 20, 29, 19, 29, 5, 23, 15, 4, 11, 24, 30, 1, 25, 31, 32, 33, 24, 31, 19, 6, 9, 34, 2, 35, 24, 4, 5, 36, 37, 33, 25, 9, 38, 39, 29, 40, 23, 41, 42, 4, 43, 31, 29, 44, 13, 45, 46, 47, 48, 49, 30

Offset: 1

Antti Karttunen, Aug 10 2020

nonn

Restricted growth sequence transform of the function f(n) = A336147(A088580(n)).
For all i, j:
  A324400(i) = A324400(j) => a(i) = a(j),
  a(i) = a(j) => A336691(i) = A336691(j),
  a(i) = a(j) => A336924(i) = A336924(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. also A336926.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A020639(n) = if(1==n, n, factor(n)[1, 1]);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A278221(n) = A046523(A122111(n));
Aux336147(n) = [A020639(n),A278221(n)];
v336925 = rgs_transform(vector(up_to, n, Aux336147(1+sigma(n))));
A336925(n) = v336925[n];

A368582 a(n) = floor((sigma(n) + 1) / 2).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 4, 8, 7, 9, 6, 14, 7, 12, 12, 16, 9, 20, 10, 21, 16, 18, 12, 30, 16, 21, 20, 28, 15, 36, 16, 32, 24, 27, 24, 46, 19, 30, 28, 45, 21, 48, 22, 42, 39, 36, 24, 62, 29, 47, 36, 49, 27, 60, 36, 60, 40, 45, 30, 84, 31, 48, 52, 64, 42, 72, 34, 63

Offset: 1

Peter Luschny, Dec 31 2023

nonn

Cf. A000203, A000079 (2^n), A000396 (perfect), A088580, A317306, A368207 (Bacher).

using Nemo
A368582(n::Int) = div(divisor_sigma(n, 1) + 1, 2)
println([A368582(n) for n in 1:68])

Array[Floor[(DivisorSigma[1, #] + 1)/2] &, 120] (* Michael De Vlieger, Dec 31 2023 *)

a(n) = (sigma(n)+1)\2; \\ Michel Marcus, Jan 03 2024

a(p) = (p + 1) / 2 for all odd prime p.
a(n) = n <=> n term of union of A000079 and A000396. (If there are no odd perfect numbers also of A317306).
a(n) = floor(A088580(n)/2). - Omar E. Pol, Dec 31 2023

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A063531 Numbers k such that sigma(k) + 1 is a square.

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A336924 a(n) = spf(1+sigma(n)), where spf is the smallest prime factor and sigma is the sum of divisors function.

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A336925 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336147(1+sigma(i)) = A336147(1+sigma(j)), for all i, j >= 1.

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A368582 a(n) = floor((sigma(n) + 1) / 2).

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