A332225 -id:A332225 - cp's OEIS frontend

A332223 a(1) = 1, and for n > 1, a(n) = A005940(1+sigma(A156552(n))).

1, 2, 4, 5, 8, 9, 16, 7, 25, 18, 32, 25, 64, 21, 21, 49, 128, 27, 256, 35, 40, 121, 512, 49, 125, 385, 49, 121, 1024, 13, 2048, 13, 225, 1573, 105, 77, 4096, 57, 187, 343, 8192, 63, 16384, 65, 55, 4693, 32768, 121, 625, 32, 15625, 85, 65536, 81, 180, 91, 253, 9945, 131072, 175, 262144, 508079, 625, 847, 729, 169, 524288, 2057, 2601, 105, 1048576

Offset: 1

Antti Karttunen, Feb 12 2020

nonn

From Antti Karttunen, Jul 31 - Aug 06 2020: (Start)
As a curiosity, like with sigma, also here a(14) = a(15). [Cf. also A003973 and A341512]
Question: is it possible that a(k) = 2*k for any k? If not, then the deficiency (A033879) cannot be -1, and there are no quasiperfect numbers. If there were such cases, then A156552(k) = q would be an instance of quasiperfect number, which should also be an odd square, thus k would need to be of the form 4u+2.
In range n <= 10000, a(n) is a nontrivial multiple of n only at n = [25, 35, 343, 539, 847, 3315] with a(n) = [125, 105, 2401, 2695, 2541, 9945]. The quotients are thus also odd: 5, 3, 7, 5, 3, 3.
This rather meager empirical evidence motivates a conjecture that no quotient a(n)/n may be an even integer, and particularly, never a power of 2 larger than one, which (when translated back to the ordinary, unconjugated sigma) claims that it is not possible that sigma(n) = 2^k * n + 2^k - 1, for any n > 1, k > 0. See also A336700 and A336701, where this leads to a rather surprising empirical observation.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000 (computed using Hans Havermann's factorization of A156552)
P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
V. Siva Rama Prasad and C. Sunitha, On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 3, 73-78.
Eric Weisstein's World of Mathematics, Quasiperfect Number
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A005940, A027687, A156552, A246277, A323243, A324201, A324899, A324909, A332224, A332225, A332463 (Möbius transform).
Cf. A003961, A332449, A332450, A332451, A332460 (for other functions similarly conjugated).
Cf. also A003973, A033879, A326042, A332222, A336700, A336701, A341512.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A332223(n) = if(1==n,n,A005940(1+sigma(A156552(n))));

A332223(n) = if(1==n,n,A005940(1+sumdiv(A156552(n),d,d))); \\ Antti Karttunen, Aug 04 2020

For n > 1, a(n) = A005940(1+A000203(A156552(n))) = A005940(1+A323243(n)).
a(A324201(n)) = A003961(A324201(n)). [It's an open problem whether A324201 gives all such solutions]
For n > 1, a(n) = A005940(1 + (Sum_{d|A156552(n)} d)). - Antti Karttunen, Aug 04 2020

A332224 a(n) = A087808(sigma(n)).

Original entry on oeis.org

1, 2, 4, 3, 4, 8, 8, 4, 5, 10, 8, 12, 6, 16, 16, 5, 10, 7, 12, 14, 32, 20, 16, 16, 5, 14, 24, 24, 8, 40, 32, 6, 32, 12, 32, 10, 12, 16, 24, 18, 14, 64, 16, 28, 14, 40, 32, 20, 13, 11, 40, 34, 12, 32, 40, 32, 48, 18, 16, 56, 10, 64, 40, 7, 28, 80, 36, 12, 64, 80, 40, 34, 22, 26, 20, 40, 64, 56, 48, 22, 17, 12, 28, 96, 24, 68, 32, 36, 18

Offset: 1

Antti Karttunen, Feb 12 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16383
Index entries for sequences related to sigma(n)

Cf. A000203, A087808, A332225.

Cf. also A324293.

Block[{s = DivisorSigma[1, Range[90]], t}, t = Nest[Append[#1, If[EvenQ[#2], 2 #1[[#2/2 + 1]], #1[[(#2 - 1)/2 + 1]] + 1]] & @@ {#, Length@ #} &, {0}, Max@ s]; t[[Most@ s + 1]] ] (* Michael De Vlieger, Feb 12 2020 *)

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
A332224(n) = A087808(sigma(n));

a(n) = A087808(A000203(n)).

A332445 Numbers k of the form 4m+1 for which A087808(sigma(k)) is equal to 2*A087808(k).

Original entry on oeis.org

2009, 19377, 37809, 59373, 74673, 115677, 270041, 310329, 354609, 357309, 720425, 732321, 841437, 2071737, 2612269, 3131149, 3866461, 3930929, 5172093, 5593981, 7118753, 7903961, 8224173, 9327393, 9438129, 11452321, 12708025, 18857209, 18861889, 18875313, 19110321, 20278269, 20709225, 20950061, 23963597, 24895153

Offset: 1

Antti Karttunen, Feb 14 2020

nonn

Numbers k such that A332224(k) is equal to A087808(2*k) and k == 1 mod 4.
Notably, the only square among the first 299 terms is a(248) = 5808421369 = 76213^2. sigma(5808421369) = 5808497583 == 3 (mod 4) == 7 (mod 8). Of the remaining 298 terms < 2^33, 92 are such that sigma(k) == 6 (mod 8) and 206 are such that sigma(k) == 2 (mod 8), that is, are terms of A332227.
Question: Why the terms come in clusters? Compare also the scatterplots of A087808 and A332224, and a similar sequence A332465.

Intersection of A016813 and A332446.
Cf. also A228058, A332227, A332465.
Cf. A000203, A087808, A286357, A332224, A332225, A332458.

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA332445(n) = ((1==(n%4))&&(2*A087808(n)==A087808(sigma(n))));

A332443 Numbers k such that A332224(k) = A087808(sigma(k)) is odd.

Original entry on oeis.org

1, 4, 9, 16, 18, 25, 49, 50, 64, 81, 100, 121, 169, 200, 256, 338, 361, 392, 400, 441, 450, 529, 578, 625, 648, 676, 722, 729, 784, 800, 841, 900, 961, 1024, 1089, 1156, 1225, 1250, 1296, 1352, 1369, 1568, 1600, 1682, 1800, 1849, 2025, 2116, 2209, 2312, 2401, 2450, 2592, 2601, 2704, 2738, 3042, 3136, 3200, 3249, 3362, 3364, 3481, 3600

Offset: 1

Antti Karttunen, Feb 13 2020

nonn

Numbers k such that A332224(k) is odd, or equally, that A332448(k) is zero.

Antti Karttunen, Table of n, a(n) for n = 1..26213
Index entries for sequences related to sigma(n)

Cf. A000203, A087808, A332224, A332225, A332444.

Subsequence of A028982. Positions of zeros in A332448.

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA332443(n) = (A087808(sigma(n))%2);
A332443list(u) = { my(v1=vector(2*u,n,2*(n^2)), v2=vector(sqrtint(v1[#v1]),n,n^2)); select(isA332443,Vec(setunion(v1,v2))); };
v332443 = A332443list(8192);
A332443(n) = v332443[n];

A332229 Even numbers k such that A156552(k) is not a power of prime, and for which A323243(k) = sigma(A156552(k)) is congruent to 2 modulo 8.

Original entry on oeis.org

290, 434, 550, 826, 858, 1394, 1798, 2254, 2418, 2546, 2950, 3094, 3910, 4150, 4382, 4930, 5590, 6138, 6358, 6390, 6710, 6966, 7514, 7546, 7622, 7658, 7990, 8550, 8798, 8906, 9230

Offset: 1

Antti Karttunen, Feb 13 2020

nonn

Numbers k for which A156552(k) is in A332228.

Sequence A005940(1+A332228(n)), n >= 1, sorted into ascending order.

Cf. A000203, A005940, A156552, A323243, A332225, A332228.

Subsequence of A324814.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
isA332228(n) = ((n%2)&&!isprimepower(n)&&2==(sigma(n)%8));
isA332229(n) = isA332228(A156552(n));

v156552sigs = readvec("a156552.txt"); \\ Factorization file for A156552 prepared by Hans Havermann, available at https://oeis.org/A156552/a156552.txt
isA156552not_a_primepower(n) = if(n<=2,0,my(prsig=v156552sigs[n]); length(prsig[1])>1);
A323243(n) = if(n<=2,n-1,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,((ps[i]^(1+es[i]))-1)/(ps[i]-1)));
isA332229(n) = (!(n%2)&&isA156552not_a_primepower(n)&&(2==(A323243(n)%8)));
k=0; for(n=1,10000,if(isA332229(n),k++; print1(n,", ")));

A332444 Numbers k that are squares or twice-squares, but for which A087808(sigma(k)) is even.

Original entry on oeis.org

2, 8, 32, 36, 72, 98, 128, 144, 162, 196, 225, 242, 288, 289, 324, 484, 512, 576, 882, 968, 1058, 1152, 1444, 1458, 1521, 1681, 1764, 1922, 1936, 2048, 2178, 2304, 2500, 2809, 2888, 2916, 3025, 3528, 3698, 3872, 4418, 4608, 4802, 5000, 5329, 5776, 5832, 6498, 7056, 7396, 7569, 7688, 7744, 7921, 7938, 8192, 8281

Offset: 1

Antti Karttunen, Feb 13 2020

nonn

Numbers k for which A000203(k) is odd and A332224(k) is even.

Antti Karttunen, Table of n, a(n) for n = 1..19515
Index entries for sequences related to sigma(n)

Setwise difference of A028982 \ A332443.

Cf. A000203, A087808, A332224, A332225.

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA332444(n) = (((!(n%2)&&issquare(n/2))||issquare(n)) && !(A087808(sigma(n))%2));
A332444list(u) = { my(v1=vector(2*u,n,2*(n^2)), v2=vector(sqrtint(v1[#v1]),n,n^2)); select(isA332444,Vec(setunion(v1,v2))); };
v332444 = A332444list(12000);
A332444(n) = v332444[n];

cp's OEIS Frontend

A332223 a(1) = 1, and for n > 1, a(n) = A005940(1+sigma(A156552(n))).

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A332224 a(n) = A087808(sigma(n)).

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A332445 Numbers k of the form 4m+1 for which A087808(sigma(k)) is equal to 2*A087808(k).

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A332443 Numbers k such that A332224(k) = A087808(sigma(k)) is odd.

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A332229 Even numbers k such that A156552(k) is not a power of prime, and for which A323243(k) = sigma(A156552(k)) is congruent to 2 modulo 8.

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A332444 Numbers k that are squares or twice-squares, but for which A087808(sigma(k)) is even.

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