A332450 -id:A332450 - cp's OEIS frontend

A332449 a(n) = A005940(1+(3*A156552(n))).

1, 4, 9, 10, 25, 16, 49, 30, 21, 36, 121, 22, 169, 100, 81, 90, 289, 40, 361, 250, 225, 196, 529, 66, 55, 484, 105, 490, 841, 64, 961, 270, 441, 676, 625, 154, 1369, 1156, 1089, 750, 1681, 144, 1849, 1210, 39, 1444, 2209, 198, 91, 84, 1521, 1690, 2809, 120, 1225, 1470, 2601, 2116, 3481, 34, 3721, 3364, 1029, 810, 3025, 400

Offset: 1

Antti Karttunen, Feb 14 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A329609 (terms sorted into ascending order).
Cf. A000290, A003961, A005117 (positions of squares), A005940, A010052, A156552, A277010, A329603, A332450, A332451, A347119, A347120, A353267 [= A348717(a(n))], A353269, A353270 [= gcd(n, a(n))], A353271, A353272, A353273.
Cf. also A332223.

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
A332449(n) = A005940(1+(3*A156552(n)));

a(n) = A005940(1+(3*A156552(n))).
a(p) = p^2 for all primes p.
a(u) = A332451(u) and A010052(a(u)) = 1 for all squarefree numbers (A005117).
a(A003961(n)) = A003961(a(n)) = A005940(1+(6*A156552(n))).
From Antti Karttunen, Apr 10 2022: (Start)
a(n) = A347119(n) * A000290(A347120(n)) = A353270(n) * A353272(n).
a(A353269(n)) = 1 for all n.
(End)

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

Original entry on oeis.org

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

A366263 Doudna sequence permuted by Blue code: a(n) = A005940(1+A193231(n)).

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 9, 8, 16, 27, 25, 18, 15, 12, 10, 7, 14, 11, 21, 20, 35, 30, 24, 45, 81, 32, 54, 125, 36, 75, 49, 50, 100, 147, 121, 98, 225, 72, 150, 245, 625, 162, 64, 243, 250, 343, 375, 108, 33, 28, 22, 13, 40, 63, 55, 42, 90, 175, 135, 48, 77, 70, 60, 105, 210, 385, 315, 120, 143, 154, 140, 231, 525, 180

Offset: 0

Antti Karttunen, Oct 06 2023

Cf. A005940, A193231, A234022, A268389, A277818, A286601, A286602, A332450, A366264 (inverse map).

Cf. also A365810, A366260, A366275.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) };
A366263(n) = A005940(1+A193231(n));

a(n) = A332450(A005940(1+n)).
For all n >= 0, A001222(a(n)) = A234022(n) and A046523(a(n)) = A286601(n).
For all n >= 1, A055396(a(n)) = A277818(n) = 1+A268389(n).

A332451 a(n) = A005940(1+A048724(A156552(n))).

Original entry on oeis.org

1, 4, 9, 6, 25, 16, 49, 10, 15, 36, 121, 54, 169, 100, 81, 14, 289, 24, 361, 150, 225, 196, 529, 250, 35, 484, 21, 294, 841, 64, 961, 22, 441, 676, 625, 90, 1369, 1156, 1089, 490, 1681, 144, 1849, 726, 375, 1444, 2209, 686, 77, 60, 1521, 1014, 2809, 40, 1225, 1210, 2601, 2116, 3481, 486, 3721, 3364, 735, 26, 3025, 400

Offset: 1

Antti Karttunen, Feb 15 2020

nonn

Cf. A000290, A003961, A005117 (gives the positions of squares), A005940, A008836, A010052, A048724, A156552, A277010, A293448, A332449, A332450.
Permutation of A028260.
Cf. A332460 for complementary sequence (after its initial 1).

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A048724(n) = bitxor(n, 2*n); \\ From A048724
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
A332451(n) = A005940(1+A048724(A156552(n)));

a(n) = A005940(1+A048724(A156552(n))).
a(p) = p^2 for all primes p.
For all squarefree numbers u, a(u) = A332449(u) and A010052(a(u)) = 1.
a(A003961(n)) = A003961(a(n)).
a(A293448(n)) = A293448(a(n)).
a(A332450(n)) = A332450(A003961(n)); A332450(a(n)) = A003961(A332450(n)).
A008836(a(n)) = +1 for all n.

A332460 a(n) = A005940(1+A065621(A156552(n))), with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 8, 5, 18, 7, 12, 27, 50, 11, 32, 13, 98, 75, 20, 17, 30, 19, 72, 147, 242, 23, 108, 125, 338, 45, 200, 29, 162, 31, 28, 363, 578, 245, 48, 37, 722, 507, 300, 41, 450, 43, 392, 243, 1058, 47, 500, 343, 70, 867, 968, 53, 42, 605, 588, 1083, 1682, 59, 128, 61, 1922, 675, 44, 845, 882, 67, 1352, 1587, 1250, 71, 180, 73, 2738, 105

Offset: 1

Antti Karttunen, Feb 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A005940, A008836, A010052, A065621, A156552, A332449, A332450.
Cf. A332451 for a complementary sequence (after its initial 1).
Permutation of A026424 after the initial 1.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A065621(n) = bitxor(n-1,n+n-1);
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
A332460(n) = A005940(1+A065621(A156552(n)));

a(1) = 1, and for n > 1, a(n) = A005940(1+A065621(A156552(n))).
a(p) = p for all primes p.
a(A003961(n)) = A003961(a(n)).
A008836(a(n)) = -1 for all n >= 2.

cp's OEIS Frontend

A332449 a(n) = A005940(1+(3*A156552(n))).

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

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A366263 Doudna sequence permuted by Blue code: a(n) = A005940(1+A193231(n)).

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A332451 a(n) = A005940(1+A048724(A156552(n))).

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A332460 a(n) = A005940(1+A065621(A156552(n))), with a(1) = 1.

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