cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A377879 Deficiency of squares: a(n) = 2n^2 - sigma(n^2).

Original entry on oeis.org

1, 1, 5, 1, 19, -19, 41, 1, 41, -17, 109, -115, 155, -7, 47, 1, 271, -199, 341, -161, 141, 37, 505, -499, 469, 71, 365, -199, 811, -1021, 929, 1, 449, 163, 683, -1159, 1331, 221, 663, -737, 1639, -1659, 1805, -251, 299, 361, 2161, -2035, 2001, -467, 1211, -265, 2755, -1819, 1927, -967, 1545, 631, 3421, -5293, 3659, 737
Offset: 1

Views

Author

Antti Karttunen, Nov 23 2024

Keywords

Comments

It is conjectured that 1's occur only when n is two's power (A000079), and that there are no -1's in this sequence. See comments in A033879 and in A337339.

Links

Crossrefs

Cf. A000290, A000079 (conjectured to give positions of all 1's), A033879, A378231 [= a(A003961(n))].
Cf. A000012, A083884.
Cf. also square array A083064.

Programs

Formula

a(n) = A033879(A000290(n)).

A379501 a(n) = (3/2)*A019565((2n-1)^2) - A019565(A001065((2n-1)^2)), where A019565 is the base-2 exp-function, and A001065 is the sum of proper divisors of n.

Original entry on oeis.org

2, 16, 216, 422, 470, 51016, 5082, 4446, 864, 106688, 1301846, 880, 204182, 1985872, 236964, 646310, 1030, 176778, 2799756, 96178962, 563400, 62092576, 1566805968, 27274, 559406, -16252236, 1040774592, 263042394, 7794826, 115781204, 13256922, -16386856, -1230440, 376172, -67188814, 222905278, 13547232, 28352541646
Offset: 1

Views

Author

Antti Karttunen, Jan 05 2025

Keywords

Links

Crossrefs

Cf. A001065, A016754, A019565, A379495, A379496.
Cf. also A378231, A337339.

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A379501(n) = { my(osq=((2*n)-1)^2); ((3/2)*A019565(osq) - A019565(sigma(osq)-osq)); };

Formula

a(n) = A379496(A016754(n)) = A019565(1+A016754(n)) - A379495(A016754(n)).
a(n) = (3/2)*A019565(A016754(n)) - A379495(A016754(n)).

A379490 Odd squares s such that 2*s is equal to bitwise-AND of 2*s and sigma(s).

Original entry on oeis.org

399736269009, 1013616036225, 1393148751631700625, 2998748839068013955625, 3547850289210724050225
Offset: 1

Views

Author

Antti Karttunen, Jan 13 2025

Keywords

Comments

If there are any quasiperfect numbers, i.e., numbers x for which sigma(x) = 2*x+1, then they should occur also in this sequence.
Square roots of these terms are: 632247, 1006785, 1180317225, 54760833075, 59563833735.
Question: Are there any solutions to similar equations "Odd squares s such that 2*s is equal to bitwise-AND of 2*s and A001065(s)" and "Odd squares s such that 3*s is equal to bitwise-AND of 3*s and sigma(s)"? Such sequences would contain odd triperfect numbers, if they exist (cf. A005820, A347391, A347884). - Antti Karttunen, Aug 19 2025
a(6) > 4*10^21. - Giovanni Resta, Aug 19 2025

Links

Crossrefs

Odd squares in A324647.
Intersection of A016754 and A324647.
Subsequence of A325311, which is a subsequence of A005231.
Cf. A000203, A001065, A004198, A318468.
Cf. also A336700, A336701, A337339, A337342, A348742, A379474, A379503, A379505, A379949 for other conditions that quasiperfect numbers should satisfy.

Programs

  • PARI
    k=0; forstep(n=1,oo,2, if(!((n-1)%(2^27)),print1("("n")")); if(!isprime(n) && omega(n)>=3, f = factor(n); sq=n^2; sig=prod(i=1,#f~,((f[i,1]^(1+(2*f[i,2])))-1) / (f[i,1]-1)); if(((2*sq)==bitand(2*sq, sig)), k++; print1(sq,", "))));

Extensions

a(4) and a(5) from Giovanni Resta, Aug 19 2025
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