A341528 -id:A341528 - cp's OEIS frontend

A341512 a(n) = A341529(n) - A341528(n) = (sigma(n)A003961(n)) - (nsigma(A003961(n))).

0, 1, 2, 11, 2, 36, 4, 85, 46, 58, 2, 324, 4, 120, 120, 575, 2, 693, 4, 566, 248, 172, 6, 2340, 94, 270, 788, 1176, 2, 1800, 6, 3661, 348, 358, 336, 5967, 4, 492, 548, 4210, 2, 3744, 4, 1820, 2490, 744, 6, 15372, 380, 2271, 720, 2826, 6, 11304, 392, 8760, 992, 946, 2, 15480, 6, 1232, 5164, 22631, 636, 5904, 4, 3866

Offset: 1

Antti Karttunen, Feb 22 2021

Cf. A000203, A000720, A001223, A003961, A003973, A286385, A341528, A341529, A341530.

Cf. Sequences A001359, A029710, A031924 give the positions of 2's, 4's and 6's in this sequence, or at least subsets of such positions.

Array[#2 DivisorSigma[1, #1] - #1 DivisorSigma[1, #2] & @@ {#, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &, 68] (* Michael De Vlieger, Feb 22 2021 *)

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341528(n) = (n*sigma(A003961(n)));
A341529(n) = (sigma(n)*A003961(n));
A341512(n) = (A341529(n)-A341528(n));

a(n) = A341529(n) - A341528(n) = (sigma(n)*A003961(n)) - (n*sigma(A003961(n))).
For all primes p, a(p) = (q*(p+1)) - (p*(q+1)) = (pq + q) - (pq + p) = q - p = A001223(A000720(p)), where q = nextprime(p) = A003961(p).
And in general, a(p^e) = (q^e * (p^(e+1)-1)/(p-1)) - ((p^e) * (q^(e+1)-1)/(q-1)), where q = A003961(p).
Thus, a(p^2) = (p + 1)*q^2 - p^2*q - p^2,
      a(p^3) = (p^2 + p + 1)*q^3 - p^3*q^2 - p^3*q - p^3,
      a(p^4) = (p^3 + p^2 + p + 1)*q^4 - p^4*q^3 - p^4*q^2 - p^4*q - p^4,
      etc.

A342667 Largest prime in the numerator of ratio A341528(n)/A341529(n) (when presented in its lowest terms).

Original entry on oeis.org

1, 2, 3, 13, 5, 2, 7, 2, 31, 5, 11, 13, 13, 7, 3, 11, 17, 31, 19, 13, 7, 11, 23, 2, 19, 13, 13, 13, 29, 2, 31, 13, 11, 17, 5, 31, 37, 19, 13, 5, 41, 7, 43, 11, 31, 23, 47, 11, 7, 19, 17, 13, 53, 13, 11, 7, 19, 29, 59, 13, 61, 31, 31, 1093, 13, 11, 67, 17, 23, 5, 71, 31, 73, 37, 19, 19, 7, 13, 79, 11, 71, 41, 83, 13

Offset: 1

Antti Karttunen, Mar 23 2021

nonn

Equally, largest prime in the denominator of ratio A342661(A003961(n)) / A342662(A003961(n)).

Cf. A000203, A003961, A006530, A064989, A341526, A341528, A341529, A326041, A342661, A342662, A342664, A342668.

Cf. also A341607.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A341526(n) = { my(s=A003961(n)); numerator((sigma(s)*n)/(sigma(n)*s)); };
A342667(n) = A006530(A341526(n));

a(n) = A006530(A341526(n)).

a(n) = A006530(A342664(A003961(n))).

A347124 Möbius transform of A341528, n * sigma(A003961(n)).

Original entry on oeis.org

1, 7, 17, 44, 39, 119, 83, 268, 261, 273, 153, 748, 233, 581, 663, 1616, 339, 1827, 455, 1716, 1411, 1071, 689, 4556, 1385, 1631, 3933, 3652, 927, 4641, 1177, 9712, 2601, 2373, 3237, 11484, 1553, 3185, 3961, 10452, 1803, 9877, 2063, 6732, 10179, 4823, 2537, 27472, 6433, 9695, 5763, 10252, 3179, 27531, 5967, 22244

Offset: 1

Antti Karttunen, Aug 24 2021

Multiplicative because A341528 is.

Cf. A002117, A003961, A003973, A008683, A151800, A341528, A341512, A346239, A347125.

f[p_, e_] := Module[{q = NextPrime[p]}, p^(e-1) * (q^e * (p*q-1) - p + 1)/(q-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2023 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A003973(n) = sigma(A003961(n));
A341528(n) = (n*A003973(n));
A347124(n) = sumdiv(n,d,moebius(n/d)*A341528(d));

a(n) = Sum_{d|n} A008683(n/d) * A341528(d).
a(n) = A347125(n) - A346239(n).
Multiplicative with a(p^e) = p^(e-1)*(q^e*(p*q-1)-p+1)/(q-1), where q = A151800(p). - Sebastian Karlsson, Sep 02 2021
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = (1/zeta(3)) / Product_{p prime} (1 - (q(p)-p)/p^2 - q(p)/p^3) = 5.6488805... , and q(p) = A151800(p). - Amiram Eldar, Dec 24 2023

A342668 Largest prime in the denominator of ratio A341528(n)/A341529(n) (when presented in its lowest terms).

Original entry on oeis.org

1, 3, 5, 7, 7, 5, 11, 3, 13, 7, 13, 7, 17, 11, 7, 31, 19, 13, 23, 7, 11, 13, 29, 5, 31, 17, 5, 11, 31, 7, 37, 3, 13, 19, 11, 7, 41, 23, 17, 7, 43, 11, 47, 3, 13, 29, 53, 31, 11, 31, 19, 17, 59, 5, 13, 11, 23, 31, 61, 7, 67, 37, 13, 127, 17, 13, 71, 19, 29, 11, 73, 13, 79, 41, 31, 23, 13, 17, 83, 31, 11, 43, 89, 11

Offset: 1

Antti Karttunen, Mar 24 2021

nonn

Cf. A000203, A003961, A006530, A341527, A341528, A341529, A342667.

Cf. A341628 (same sequence applied onto prime shift array A246278).

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A341527(n) = { my(s=A003961(n)); denominator((sigma(s)*n)/(sigma(n)*s)); };
A342668(n) = A006530(A341527(n));

a(n) = A006530(A341527(n)).

A378228 Dirichlet inverse of A341528, where A341528(n) = n * sigma(A003961(n)), and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, -8, -18, 12, -40, 144, -84, 0, 45, 320, -154, -216, -234, 672, 720, 0, -340, -360, -456, -480, 1512, 1232, -690, 0, 175, 1872, 0, -1008, -928, -5760, -1178, 0, 2772, 2720, 3360, 540, -1554, 3648, 4212, 0, -1804, -12096, -2064, -1848, -1800, 5520, -2538, 0, 539, -1400, 6120, -2808, -3180, 0, 6160, 0, 8208, 7424

Offset: 1

Antti Karttunen, Nov 23 2024

Multiplicative because A341528 is.

Dirichlet inverse of A341528.
Cf. A003973, A378221.
Cf. also A378229.

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341528(n) = (n*sigma(A003961(n)));
memoA378228 = Map();
A378228(n) = if(1==n,1,my(v); if(mapisdefined(memoA378228,n,&v), v, v = -sumdiv(n,d,if(dA341528(n/d)*A378228(d),0)); mapput(memoA378228,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA341528(n/d) * a(d).

A016754 Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.

Original entry on oeis.org

1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569, 7921, 8281, 8649, 9025

Offset: 0

N. J. A. Sloane

The brown rat (rattus norwegicus) breeds very quickly. It can give birth to other rats 7 times a year, starting at the age of three months. The average number of pups is 8. The present sequence gives the total number of rats, when the intervals are 12/7 of a year and a young rat starts having offspring at 24/7 of a year. - Hans Isdahl, Jan 26 2008
Numbers n such that tau(n) is odd where tau(x) denotes the Ramanujan tau function (A000594). - Benoit Cloitre, May 01 2003
If Y is a fixed 2-subset of a (2n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007
Binomial transform of [1, 8, 8, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 8, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
All terms of this sequence are of the form 8k+1. For numbers 8k+1 which aren't squares see A138393. Numbers 8k+1 are squares iff k is a triangular number from A000217. And squares have form 4n(n+1)+1. - Artur Jasinski, Mar 27 2008
Sequence arises from reading the line from 1, in the direction 1, 25, ... and the line from 9, in the direction 9, 49, ..., in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008
Equals the triangular numbers convolved with [1, 6, 1, 0, 0, 0, ...]. - Gary W. Adamson & Alexander R. Povolotsky, May 29 2009
First differences: A008590(n) = a(n) - a(n-1) for n>0. - Reinhard Zumkeller, Nov 08 2009
Central terms of the triangle in A176271; cf. A000466, A053755. - Reinhard Zumkeller, Apr 13 2010
Odd numbers with odd abundance. Odd numbers with even abundance are in A088828. Even numbers with odd abundance are in A088827. Even numbers with even abundance are in A088829. - Jaroslav Krizek, May 07 2011
Appear as numerators in the non-simple continued fraction expansion of Pi-3: Pi-3 = K_{k>=1} (1-2*k)^2/6 = 1/(6+9/(6+25/(6+49/(6+...)))), see also the comment in A007509. - Alexander R. Povolotsky, Oct 12 2011
Ulam's spiral (SE spoke). - Robert G. Wilson v, Oct 31 2011
All terms end in 1, 5 or 9. Modulo 100, all terms are among { 1, 9, 21, 25, 29, 41, 49, 61, 69, 81, 89 }. - M. F. Hasler, Mar 19 2012
Right edge of both triangles A214604 and A214661: a(n) = A214604(n+1,n+1) = A214661(n+1,n+1). - Reinhard Zumkeller, Jul 25 2012
Also: Odd numbers which have an odd sum of divisors (= sigma = A000203). - M. F. Hasler, Feb 23 2013
Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1) with hypotenuse c (A020882) and respective even leg b (A231100); sequence gives values c-b, sorted with duplicates removed. - K. G. Stier, Nov 04 2013
For n>1 a(n) is twice the area of the irregular quadrilateral created by the points ((n-2)*(n-1),(n-1)*n/2), ((n-1)*n/2,n*(n+1)/2), ((n+1)*(n+2)/2,n*(n+1)/2), and ((n+2)*(n+3)/2,(n+1)*(n+2)/2). - J. M. Bergot, May 27 2014
Number of pairs (x, y) of Z^2, such that max(abs(x), abs(y)) <= n. - Michel Marcus, Nov 28 2014
Except for a(1)=4, the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016
a(n) is the sum of 2n+1 consecutive numbers, the first of which is n+1. - Ivan N. Ianakiev, Dec 21 2016
a(n) is the number of 2 X 2 matrices with all elements in {0..n} with determinant = 2*permanent. - Indranil Ghosh, Dec 25 2016
Engel expansion of Pi*StruveL_0(1)/2 where StruveL_0(1) is A197037. - Benedict W. J. Irwin, Jun 21 2018
Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; the segments on the hypotenuse {p = a(n)/A001844(n), q = A060300(n)/A001844(n) = A001844(n) - p} and their ratio p/q = a(n)/A060300(n) are irreducible fractions in Q\Z. X values are A005408, Y values are A046092, Z values are A001844. - Ralf Steiner, Feb 25 2020
a(n) is the number of large or small squares that are used to tile primitive squares of type 2 (A344332). - Bernard Schott, Jun 03 2021
Also, positive odd integers with an odd number of odd divisors (for similar sequence with 'even', see A348005). - Bernard Schott, Nov 21 2021
a(n) is the least odd number k = x + y, with 0 < x < y, such that there are n distinct pairs (x,y) for which x*y/k is an integer; for example, a(2) = 25 and the two corresponding pairs are (5,20) and (10,15). The similar sequence with 'even' is A016742 (see Comment of Jan 26 2018). - Bernard Schott, Feb 24 2023
From Peter Bala, Jan 03 2024: (Start)
The sequence terms are the exponents of q in the series expansions of the following infinite products:
1) q*Product_{n >= 1} (1 - q^(16*n))*(1 + q^(8*n)) = q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + ....
2) q*Product_{n >= 1} (1 + q^(16*n))*(1 - q^(8*n)) = q - q^9 - q^25 + q^49 + q^81 - q^121 - q^169 + + - - ....
3) q*Product_{n >= 1} (1 - q^(8*n))^3 = q - 3*q^9 + 5*q^25 - 7*q^49 + 9*q^81 - 11*q^121 + 13*q^169 - + ....
4) q*Product_{n >= 1} ( (1 + q^(8*n))*(1 - q^(16*n))/(1 + q^(16*n)) )^3 = q + 3*q^9 - 5*q^25 - 7*q^49 + 9*q^81 + 11*q^121 - 13*q^169 - 15*q^225 + + - - .... (End)

L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 586.

Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe).
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.
Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary, Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.
John Elias, Illustration: 8-fold Square Progression of Ulam's Spiral.
Milan Janjic, Two Enumerative Functions.
Scientific American, Cover of the March 1964 issue.
Amelia Carolina Sparavigna, Groupoids of OEIS A002378 and A016754 Numbers (oblong and odd square numbers), Politecnico di Torino (Italy, 2019).
Leo Tavares, Illustration: Diamond Triangles.
Leo Tavares, Illustration: Diamond Stars.
Eric Weisstein's World of Mathematics, Moore Neighborhood.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A000384, A001263, A001539, A001844, A003881, A005408, A006752, A014105, A016742, A016802, A016814, A016826, A016838, A033996, A046092, A060300, A138393, A167661, A167700.
Cf. A000447 (partial sums).
Cf. A005917, A344330, A344332.
Cf. A348005, A379481 [= a(A048673(n)-1)].
Partial sums of A022144.
Positions of odd terms in A341528.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A014634, A003154.

a016754 n = a016754_list !! n
a016754_list = scanl (+) 1 $ tail a008590_list
-- Reinhard Zumkeller, Apr 02 2012

[n^2: n in [1..100 by 2]]; // Vincenzo Librandi, Jan 03 2017

Range[1, 100, 2]^2 (* Paolo Xausa, Feb 13 2025 *)

A016754(n):=(n+n+1)^2$
makelist(A016754(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

A016754(n)=(n<<1+1)^2 \\ Charles R Greathouse IV, Jun 16 2011, corrected and edited by M. F. Hasler, Apr 11 2023

def A016754(n): return ((n<<1)|1)**2 # Chai Wah Wu, Jul 06 2023

a(n) = 1 + Sum_{i=1..n} 8*i = 1 + 8*A000217(n). - Xavier Acloque, Jan 21 2003; Zak Seidov, May 07 2006; Robert G. Wilson v, Dec 29 2010
O.g.f.: (1+6*x+x^2)/(1-x)^3. - R. J. Mathar, Jan 11 2008
a(n) = 4*n*(n + 1) + 1 = 4*n^2 + 4*n + 1. - Artur Jasinski, Mar 27 2008
a(n) = A061038(2+4n). - Paul Curtz, Oct 26 2008
Sum_{n>=0} 1/a(n) = Pi^2/8 = A111003. - Jaume Oliver Lafont, Mar 07 2009
a(n) = A000290(A005408(n)). - Reinhard Zumkeller, Nov 08 2009
a(n) = a(n-1) + 8*n with n>0, a(0)=1. - Vincenzo Librandi, Aug 01 2010
a(n) = A033951(n) + n. - Reinhard Zumkeller, May 17 2009
a(n) = A033996(n) + 1. - Omar E. Pol, Oct 03 2011
a(n) = (A005408(n))^2. - Zak Seidov, Nov 29 2011
From George F. Johnson, Sep 05 2012: (Start)
a(n+1) = a(n) + 4 + 4*sqrt(a(n)).
a(n-1) = a(n) + 4 - 4*sqrt(a(n)).
a(n+1) = 2*a(n) - a(n-1) + 8.
a(n+1) = 3*a(n) - 3*a(n-1) + a(n-2).
(a(n+1) - a(n-1))/8 = sqrt(a(n)).
a(n+1)*a(n-1) = (a(n)-4)^2.
a(n) = 2*A046092(n) + 1 = 2*A001844(n) - 1 = A046092(n) + A001844(n).
Limit_{n -> oo} a(n)/a(n-1) = 1. (End)
a(n) = binomial(2*n+2,2) + binomial(2*n+1,2). - John Molokach, Jul 12 2013
E.g.f.: (1 + 8*x + 4*x^2)*exp(x). - Ilya Gutkovskiy, May 23 2016
a(n) = A101321(8,n). - R. J. Mathar, Jul 28 2016
Product_{n>=1} A033996(n)/a(n) = Pi/4. - Daniel Suteu, Dec 25 2016
a(n) = A014105(n) + A000384(n+1). - Bruce J. Nicholson, Nov 11 2017
a(n) = A003215(n) + A002378(n). - Klaus Purath, Jun 09 2020
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=0} a(n)/n! = 13*e.
Sum_{n>=0} (-1)^(n+1)*a(n)/n! = 3/e. (End)
Sum_{n>=0} (-1)^n/a(n) = A006752. - Amiram Eldar, Oct 10 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = cosh(Pi/2).
Product_{n>=1} (1 - 1/a(n)) = Pi/4 (A003881). (End)
From Leo Tavares, Nov 24 2021: (Start)
a(n) = A014634(n) - A002943(n). See Diamond Triangles illustration.
a(n) = A003154(n+1) - A046092(n). See Diamond Stars illustration. (End)
From Peter Bala, Mar 11 2024: (Start)
Sum_{k = 1..n+1} 1/(k*a(k)*a(k-1)) = 1/(9 - 3/(17 - 60/(33 - 315/(57 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*2^2 ))))).
3/2 - 2*log(2) = Sum_{k >= 1} 1/(k*a(k)*a(k-1)) = 1/(9 - 3/(17 - 60/(33 - 315/(57 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*2^2 - ... ))))).
Row 2 of A142992. (End)
From Peter Bala, Mar 26 2024: (Start)
8*a(n) = (2*n + 1)*(a(n+1) - a(n-1)).
Sum_{n >= 0} (-1)^n/(a(n)*a(n+1)) = 1/2 - Pi/8 = 1/(9 + (1*3)/(8 + (3*5)/(8 + ... + (4*n^2 - 1)/(8 + ... )))). For the continued fraction use Lorentzen and Waadeland, p. 586, equation 4.7.9 with n = 1. Cf. A057813. (End)

Additional description from Terrel Trotter, Jr., Apr 06 2002

A341529 a(n) = sigma(n) * A003961(n), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of the divisors of n.

Original entry on oeis.org

1, 9, 20, 63, 42, 180, 88, 405, 325, 378, 156, 1260, 238, 792, 840, 2511, 342, 2925, 460, 2646, 1760, 1404, 696, 8100, 1519, 2142, 5000, 5544, 930, 7560, 1184, 15309, 3120, 3078, 3696, 20475, 1558, 4140, 4760, 17010, 1806, 15840, 2068, 9828, 13650, 6264, 2544, 50220, 6897, 13671, 6840, 14994, 3186, 45000, 6552, 35640

Offset: 1

Antti Karttunen, Feb 16 2021

Question: Does the maximum value of ratio A341529(n)/A341528(n) stay below 2?
From Amiram Eldar and Antti Karttunen, Jan 28 2023: (Start)
Answer to the above question is yes: Sup_{n>=1} A341529(n)/A341528(n) = 2.
Proof:
  f(n) = A341529(n)/A341528(n) is a multiplicative function with f(p^e) = (1 + 1/p + ... + 1/p^e)/(1 + 1/q + ... + 1/q^e), where q = nextprime(p).
  First we prove a lemma which states that f(p^(1+e)) / f(p^e) > 1, for any prime p, and exponent e.
  We note that (sigma(p^(1+e))/(p^(1+e))) / (sigma(p^e)/(p^e)) = (sigma(p^(1+e))/(p*sigma(p^e))) = sigma(p^(1+e)) / (sigma(p^(1+e)) - 1), so setting q = nextprime(p), we can write the ratio f(p^(1+e)) / f(p^e) as (sigma(p^(1+e))/(sigma(p^(1+e))-1)) / (sigma(q^(1+e))/(sigma(q^(1+e))-1)), and to prove this to be > 1, we just note that the denominator is less than the numerator, because sigma(p^e) is monotonically growing with respect to the increasing prime p.
  Since q > p, we have f(p^e) > 1 for all p and all e>=1, and together with the above lemma this shows that f(n) <= f(n*m) for all m>=1.
  Suppose n = Product_i p_i^e_i, and let pmax = max(p_i), emax = max(e_i), so n is a divisor of m = (pmax#)^emax, and f(n) < f(m), where p# = 2 * 3 * ... * p is the primorial of p, A034386(p).
  Then f(m) = f(2^emax) * f(3^emax) * ... * f(pmax^emax) = (1 + 1/2 + ... + 1/2^emax)/(1 + 1/3 + ... + 1/3^emax)) * (1 + 1/3 + ... + 1/3^emax)/(1 + 1/5 + ... + 1/5^emax)) * ... * (1 + 1/p + ... + 1/p^emax)/(1 + 1/q + ... + 1/q^emax))[telescoping product] = (1 + 1/2 + ... + 1/2^emax)/(1 + 1/qmax + ... + 1/qmax^emax) <= (1 + 1/2 + ... + 1/2^emax) < 2, where qmax = nextprime(pmax).
  So we have f(n) < 2 for all n.
  To prove that 2 is the supremum, we have lim_{e,k -> oo) f(prime(k)#^e) = 2.
(End)

Cf. A000203, A003961, A003973, A028982 (positions of odd terms), A072861, A151800, A286385, A341512, A341526, A341527, A341528, A341530, A342661, A342667, A342671, A342672.

Cf. also arrays A341605/A341606.

Array[DivisorSigma[1, #]*Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &, 56] (* Michael De Vlieger, Feb 22 2021 *)

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341529(n) = (sigma(n)*A003961(n));

Multiplicative with a(p^e) = q^e * (p^(e+1)-1)/(p-1), where q = nextPrime(p).
a(n) = A000203(n) * A003961(n).
For all n > 1, a(n) > A341528(n).
For all n >= 1, A072861(n) <= a(n) <= A003961(n)^2. [See A286385].
a(n) = A341528(n) + A341512(n) = A342671(n) * A342672(n) = A342661(A003961(n)). - Antti Karttunen, Mar 22 2021
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} p^4*(p-1)/((p^3-nextprime(p))*(p^2-nextprime(p))) = 3.0664809..., where nextprime is A151800. - Amiram Eldar, Dec 08 2022

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

cp's OEIS Frontend

A341512 a(n) = A341529(n) - A341528(n) = (sigma(n)*A003961(n)) - (n*sigma(A003961(n))).

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A342667 Largest prime in the numerator of ratio A341528(n)/A341529(n) (when presented in its lowest terms).

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A347124 Möbius transform of A341528, n * sigma(A003961(n)).

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A342668 Largest prime in the denominator of ratio A341528(n)/A341529(n) (when presented in its lowest terms).

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A378228 Dirichlet inverse of A341528, where A341528(n) = n * sigma(A003961(n)), and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

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A016754 Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.

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A341529 a(n) = sigma(n) * A003961(n), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of the divisors of n.

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

A341512 a(n) = A341529(n) - A341528(n) = (sigma(n)A003961(n)) - (nsigma(A003961(n))).

A341530 a(n) = gcd(nsigma(A003961(n)), sigma(n)A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors of n.

A341527 Denominator of ratio nsigma(A003961(n)) / sigma(n)A003961(n), where sigma is the sum of divisors of n, and A003961 shifts the prime factorization of n one step towards larger primes.