A387721 -id:A387721 - cp's OEIS frontend

A107758 (+2)Sigma(n): If n = Product p_i^r_i then a(n) = Product (2 + Sum p_i^s_i, s_i=1 to r_i) = Product (1 + (p_i^(r_i+1)-1)/(p_i-1)), a(1) = 1.

1, 4, 5, 8, 7, 20, 9, 16, 14, 28, 13, 40, 15, 36, 35, 32, 19, 56, 21, 56, 45, 52, 25, 80, 32, 60, 41, 72, 31, 140, 33, 64, 65, 76, 63, 112, 39, 84, 75, 112, 43, 180, 45, 104, 98, 100, 49, 160, 58, 128, 95, 120, 55, 164, 91, 144, 105, 124, 61, 280, 63, 132, 126, 128, 105, 260, 69, 152, 125, 252, 73, 224, 75, 156, 160

Offset: 1

Yasutoshi Kohmoto, May 25 2005

			a(6) = (2+2)*(2+3) = 20.

Daniel Suteu, Table of n, a(n) for n = 1..10000
Sagar Mandal, Divisibility and Sequence Properties of sigma+ and phi+, arXiv:2508.11660 [math.GM], 2025.
József Sándor and Krassimir Atanassov, Some new arithmetic functions, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856. See sigma+ function.

Cf. A000203, A055653, A072691, A107759.

Cf. A052396 (k such that a(k) = 2k), A387720 (k such that a(k) < 2k), A387721 (k such that a(k) > 2k), A387725, A386390 (k such that k-1 | a(k)).

A107758 := proc(n) local pf,p ; if n = 1 then 1; else pf := ifactors(n)[2] ; mul( 1+(op(1,p)^(op(2,p)+1)-1)/(op(1,p)-1), p=pf) ; end if; end proc:
seq(A107758(n),n=1..60) ; # R. J. Mathar, Jan 07 2011

Table[DivisorSum[n, DivisorSigma[1, #] &, CoprimeQ[n/#, #] &], {n, 54}] (* Michael De Vlieger, Jun 27 2018 *)
f[p_, e_] := 1 + (p^(e + 1) - 1)/(p - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Aug 26 2022 *)

a(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, sigma(d))); \\ Daniel Suteu, Jun 27 2018

A107758(n) =  { my(f = factor(n)); prod(k=1, #f~, 1+sigma(f[k, 1]^f[k, 2])); }; \\ Antti Karttunen, Sep 06 2025

a(n) = Sum_{d|n, gcd(n/d, d) = 1} sigma(d), where sigma(d) is the sum of the divisors of d. - Daniel Suteu, Jun 27 2018

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.0741158... . - Amiram Eldar, Nov 01 2022

More terms from Antti Karttunen, Sep 06 2025

A052396 (+2)-sigma perfect numbers: numbers k such that (+2)sigma(k) = 2*k, where (+2)sigma(k) = A107758(k).

Original entry on oeis.org

2, 4, 8, 16, 32, 63, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 34587, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 170271801, 268435456, 536870912, 1073741824, 2147483648, 4294967296

Offset: 1

Yasutoshi Kohmoto Mar 13 2000

2^n is a term for all n>=1. - Amiram Eldar, Aug 26 2022

			Factorizations: even examples: 2, 2^2, 2^3, 2^4,...; odd examples: a(6) = 3^2*7, a(17) = 3^4*7*61, a(30) = 3^6*7*61*547.

Cf. A000079, A107758.

Complement of (A387720 U A387721).

f[p_, e_] := 1 + (p^(e + 1) - 1)/(p - 1); s[n_] := Times @@ f @@@ FactorInteger[n]; s[1] = 1; Select[Range[5*10^6], s[#] == 2*# &] (* Amiram Eldar, Aug 26 2022 *)

Corrected by Franklin T. Adams-Watters, Oct 25 2006

a(30) corrected and a(31)-a(35) added by Amiram Eldar, Aug 26 2022

A387720 Numbers k for which A107758(k) < 2*k, where A107758 is sigma+, multiplicative function with a(p^e) = 1+sigma(p^e).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159

Offset: 1

Antti Karttunen, Sep 06 2025

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A107758.
Complement of (A052396 U A387721).
Cf. also A005100, A246281, A387710.

A107758(n) =  { my(f = factor(n)); prod(k=1, #f~, 1+sigma(f[k, 1]^f[k, 2])); };
is_A387720(n) = (A107758(n)<(2*n));

A387722 Numbers k for which A107758(k) > 2k, but for all whose proper divisors d|k, dA107758(d) <= 2d.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 385, 386, 394, 398, 422, 429, 446, 454, 455, 458, 466, 478, 482, 502, 514, 526, 538, 542, 554, 561, 562, 566

Offset: 1

Antti Karttunen, Sep 06 2025

Not the same as the positions of 1's in A387725.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A107758, A387725.
Setwise difference A387721 \ A387723.
Cf. also A091191, A337372, A387712.

A107758(n) =  { my(f = factor(n)); prod(k=1, #f~, 1+sigma(f[k, 1]^f[k, 2])); };
is_A387722(n) = if((A107758(n)<=(2*n)), 0, fordiv(n, d, if(dA107758(d)>(2*d), return(0))); (1));

A387723 Numbers k for which A107758(k) > 2k, and also for some of the proper divisors d|k, dA107758(d) > 2d.

Original entry on oeis.org

12, 18, 20, 24, 28, 30, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 92, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 116, 120, 124, 126, 130, 132, 135, 136, 138, 140, 144, 148, 150, 152, 154, 156, 160, 162, 164, 165, 168, 170, 172, 174, 176, 180, 182, 184, 186, 188

Offset: 1

Antti Karttunen, Sep 06 2025

Not the same as positions of terms > 1 in A387725. For example, A387725(225) = 3, although 225 is not present in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A107758, A387725.
Setwise difference A387721 \ A387722.
Cf. also A341610, A387713.

A107758(n) =  { my(f = factor(n)); prod(k=1, #f~, 1+sigma(f[k, 1]^f[k, 2])); };
is_A387723(n) = if((A107758(n)<=(2*n)), 0, fordiv(n, d, if(dA107758(d)>(2*d), return(1))); (0));

A387725 Number of unitary divisors d of n for which A107758(d) > 2*d, where A107758 is sigma+, multiplicative function with a(p^e) = 1+sigma(p^e).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 4, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 4, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 4, 0, 1, 0, 0, 0, 3, 0, 1, 0, 3, 0, 1, 0, 1, 1, 1, 0, 3, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 3

Offset: 1

Antti Karttunen, Sep 06 2025

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A107758, A387721 (positions of positive terms).

Cf. also A387715, A337345.

A107758(n) =  { my(f = factor(n)); prod(k=1, #f~, 1+sigma(f[k, 1]^f[k, 2])); };
A387725(n) = sumdiv(n,d,(1==gcd(d,n/d)) && (A107758(d)>(2*d)));

a(n) = Sum_{d|n} [gcd(d,n/d)==1 and A107758(d) > 2*d], where [ ] is the Iverson bracket.

cp's OEIS Frontend

A107758 (+2)Sigma(n): If n = Product p_i^r_i then a(n) = Product (2 + Sum p_i^s_i, s_i=1 to r_i) = Product (1 + (p_i^(r_i+1)-1)/(p_i-1)), a(1) = 1.

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A052396 (+2)-sigma perfect numbers: numbers k such that (+2)sigma(k) = 2*k, where (+2)sigma(k) = A107758(k).

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A387720 Numbers k for which A107758(k) < 2*k, where A107758 is sigma+, multiplicative function with a(p^e) = 1+sigma(p^e).

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A387722 Numbers k for which A107758(k) > 2*k, but for all whose proper divisors d|k, dA107758(d) <= 2*d.

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A387723 Numbers k for which A107758(k) > 2*k, and also for some of the proper divisors d|k, dA107758(d) > 2*d.

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A387725 Number of unitary divisors d of n for which A107758(d) > 2*d, where A107758 is sigma+, multiplicative function with a(p^e) = 1+sigma(p^e).

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A387722 Numbers k for which A107758(k) > 2k, but for all whose proper divisors d|k, dA107758(d) <= 2d.

A387723 Numbers k for which A107758(k) > 2k, and also for some of the proper divisors d|k, dA107758(d) > 2d.