A337339 -id:A337339 - cp's OEIS frontend

A337340 a(n) = (A337339(n)-A003961(n)) / 2.

0, 1, 4, 16, 9, 49, 25, 169, 144, 100, 36, 484, 64, 38, 289, 1600, 81, 1369, 121, 961, 729, 361, 196, 4489, 576, 625, 3844, 139, 225, 2704, 324, 14641, 1024, 784, 1444, 12544, 400, 1156, 1764, 8836, 441, 6724, 529, 3364, 7569, 1849, 676, 40804, 3600, 5329, 2209, 5776, 841, 34969, 2025, 4262, 3249, 386, 900, 24649, 1089

Offset: 1

Antti Karttunen, Aug 24 2020

sign

Not all terms are squares. For example, a(14) = 38, a(28) = 139 and a(56) = 4262 are not squares.

Not all terms are nonnegative. The first three negative terms are a(1354) = -290, a(7078) = -1363 and a(15722) = -2386.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A003961, A337337, A337339.

Cf. also A337341.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337340(n) = { my(h=A003961(n),s=(h^2),u=(s+1)/2); ((u/gcd(1+sigma(s), u)) - h)/2; };

A337341 a(n) = A337339(n) - n.

Original entry on oeis.org

0, 3, 10, 37, 20, 107, 54, 357, 304, 211, 74, 1001, 132, 95, 598, 3265, 164, 2795, 246, 1965, 1492, 739, 398, 9089, 1176, 1275, 7786, 349, 452, 5483, 654, 29493, 2080, 1591, 2930, 25277, 804, 2343, 3574, 17821, 884, 13571, 1062, 6801, 15268, 3739, 1358, 81965, 7272, 10755, 4462, 11653, 1688, 70259, 4086, 8765, 6556, 807

Offset: 1

Antti Karttunen, Aug 24 2020

nonn

Superficially, it seems that A000027 offers an lower bound for A337339 as there seems to be no negative terms in this sequence, but of course it is not guaranteed.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000027, A337339, A337340.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337341(n) = { my(h=A003961(n),s=(h^2),u=(s+1)/2); ((u/gcd(1+sigma(s), u)) - n); };

a(n) = A337339(n) - n.

A336700 Numbers k such that the odd part of (1+k) divides (1 + odd part of sigma(k)).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 2431, 2943, 3775, 4095, 8191, 13311, 14335, 16383, 17407, 21951, 22527, 32767, 34335, 44031, 57855, 65535, 85375, 131071, 204799, 262143, 376831, 524287, 923647, 1048575, 1562623, 1632255, 2056191, 2097151, 2744319, 4194303, 6815743, 8388607, 8781823, 10059775, 16777215

Offset: 1

Antti Karttunen, Aug 02 2020

nonn

Numbers k for which A337194(k) = 1+A161942(k) is a multiple of A000265(1+k).

Conjecture: After 1, all terms are of the form 4u+3 (in A004767). If this could be proved, then it would refute at once the existence of both the odd perfect numbers and the quasiperfect numbers. Concentrating on the latter is probably easier, as it is known that all quasiperfect numbers must be odd squares, thus k is of the form 4u+1, in which case the condition given in A336701 that A000265(1+A000265(sigma(k))) must be equal to A000265(1+k) reduces to a simpler form, A000265(1+sigma(k)) = (1+k)/2, and as k = s^2, with s odd, so (s^2 + 1)/2 should divide 1+sigma(s^2). Does that condition allow any other solutions than s=1 ? See A337339.

Antti Karttunen, Table of n, a(n) for n = 1..77; all terms < 2^32
Paolo Cattaneo, Sui numeri quasiperfetti, Bollettino dell’Unione Matematica Italiana, Serie 3, Vol.6(1951), n.1, p. 59-62.
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 where odd perfect numbers must occur, if they exist at all
Index entries for sequences related to sigma(n)

Cf. A000203, A000265, A004767, A016754, A161942, A228058, A336697, A336698, A337339, A337342.

Subsequences: A000225, A336701 (terms where the quotient is a power of 2).

Block[{f}, f[n_] := n/2^IntegerExponent[n, 2]; Select[Range[2^20], Mod[f[1 + f[DivisorSigma[1, #]]], f[1 + #]] == 0 &] ] (* Michael De Vlieger, Aug 22 2020 *)

A000265(n) = (n>>valuation(n,2));
isA336700(n) = !((1+A000265(sigma(n)))%A000265(1+n));

A337337 a(n) = gcd(1+sigma(s), (s+1)/2), where s is the square of n once prime-shifted (s = A003961(n)^2 = A003961(n^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 24 2020

nonn

All terms are in A007310, because all terms of A337336 are.

Cf. A003961, A003973, A007310, A048673, A336697, A336844, A337194, A337335, A337336, A337338, A337339.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337337(n) = { my(s=(A003961(n)^2)); gcd((s+1)/2, 1+sigma(s)); };

A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A336697(n) = { my(s=((n+n-1)^2)); gcd((s+1)/2, 1+sigma(s)); };
A337337(n) = A336697(A048673(n));

a(n) = gcd((s+1)/2, 1+sigma(s)), where s = A003961(n)^2 = A003961(n^2).
a(n) = gcd(A048673(n^2), 1+A003973(n^2)).
a(n) = gcd(A048673(n^2), A337194(A003961(n)^2)) = gcd(A337336(n), A336844(n^2)).
a(n) = A336697(A048673(n)).
a(n) = A337335(n^2).

A378231 Deficiency of prime-shifted squares: a(n) = 2*A003961(n^2) - sigma(A003961(n^2)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, 5, 19, 41, 41, 47, 109, 365, 469, 141, 155, 299, 271, 449, 683, 3281, 341, 1097, 505, 1041, 1927, 663, 811, 2567, 2001, 1211, 11719, 3509, 929, -921, 1331, 29525, 2777, 1545, 4277, 6749, 1639, 2333, 4933, 9141, 1805, 851, 2161, 5235, 16733, 3815, 2755, 22979, 13177, 6805, 6239, 9671, 3421, 27347, 6131, 31049

Offset: 1

Antti Karttunen, Nov 23 2024

sign

See comments in A377879 and in A337339.

Cf. A000203, A003961, A003973, A033879, A344587, A377879.

Cf. also A337339.

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A344587(n) = { my(u=A003961(n)); (u+u - sigma(u)); };
A378231(n) = A344587(n^2);

a(n) = A344587(n^2) = A377879(A003961(n)).

A336697 a(n) = gcd((s+1)/2, 1+sigma(s)), where s is the n-th odd square, (2n-1)^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 18 2020

nonn

See comments in A336700, A337337, and A337339.

Antti Karttunen, Table of n, a(n) for n = 1..16396
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A016754, A048673, A336700, A337337, A337339.

Array[GCD[(# + 1)/2, 1 + DivisorSigma[1, #]] &[(2 # - 1)^2] &, 120] (* Michael De Vlieger, Aug 24 2020 *)

A336697(n) = { my(s=((n+n-1)^2)); gcd((s+1)/2,1+sigma(s)); };

For all n >= 1, a(A048673(n)) = A337337(n).

A337336 a(n) = A048673(n^2).

Original entry on oeis.org

1, 5, 13, 41, 25, 113, 61, 365, 313, 221, 85, 1013, 145, 545, 613, 3281, 181, 2813, 265, 1985, 1513, 761, 421, 9113, 1201, 1301, 7813, 4901, 481, 5513, 685, 29525, 2113, 1625, 2965, 25313, 841, 2381, 3613, 17861, 925, 13613, 1105, 6845, 15313, 3785, 1405, 82013, 7321, 10805, 4513, 11705, 1741, 70313, 4141, 44105

Offset: 1

Antti Karttunen, Aug 24 2020

nonn

All terms are odd and neither there are multiples of 3, thus only terms of A007310 occur here.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000290, A003961, A007310, A040001, A048673, A337337, A337338, A337339.

A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A337336(n) = A048673(n^2);

a(n) = A048673(A000290(n)) = (1+(A003961(n)^2))/2.

For all n>= 1, A010872(a(n)) = A040001(n).

A337338 Numerator of (1+sigma(s)) / ((s+1)/2), where s is the square of n prime-shifted once (s = A003961(n)^2 = A003961(n^2)).

Original entry on oeis.org

2, 14, 32, 122, 58, 404, 134, 1094, 782, 742, 184, 3752, 308, 346, 1768, 9842, 382, 10154, 554, 6898, 4124, 2380, 872, 33884, 2802, 3992, 19532, 1238, 994, 22972, 1408, 88574, 5674, 4954, 7582, 94502, 1724, 7190, 9518, 62302, 1894, 53600, 2258, 22144, 44518, 11324, 2864, 305072, 16106, 36414, 11812, 37148, 3542, 253904

Offset: 1

Antti Karttunen, Aug 24 2020

Cf. A003961, A003973, A048673, A336697, A336844, A337194, A337337.

Cf. A337339 (denominators).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337338(n) = { my(s=(A003961(n)^2),t=1+sigma(s)); (t/gcd(t, (s+1)/2)); };
\\ Or as:
A337338(n) = { my(s=A003961(n^2)); numerator((1+sigma(s))/((s+1)/2)); };

a(n) = A337194(A003961(n)^2) / A337337(n).

A337342 Numbers k such that A048673(k) divides 1+A003973(k).

Original entry on oeis.org

1, 10, 584, 3824, 23008, 5033216

Offset: 1

Antti Karttunen, Aug 24 2020

Numbers k such that A048673(k) = A337335(k). Equivalently, numbers k such that (A003961(k)+1)/2 divides 1+A003973(k).
No squares larger than one in this sequence => No quasiperfect numbers. See also A337339. For any x corresponding to a quasiperfect number qp = A003961(x), the quotient (1+A003973(x)) / A048673(x) should be 4. Thus that A003961(x) should also be a member of A325311.
At least for the terms x = a(2) .. a(6) here, the quotient (1+A003973(x)) / A048673(x) = 3. The terms for which the quotient is 3 are precisely those which by prime shifting become the terms of A007593 (that are all odd), thus the terms y = A064989(A007593(n)), for n >= 1, form a subsequence of this sequence.
a(7) > 2^28.
Terms 65810851904356352, 30943274395471606363637940224, 40102483616531202199118491418624 are also in the sequence, but their positions are unknown. (Adapted from Jud McCranie's Dec 16 1999 comment in A007593).

Cf. A003961, A003973, A007593, A048673, A064989, A209922, A324899, A325311, A336700, A337335, A337339.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337342(n) = { my(s=A003961(n)); !((1+sigma(s))%((1+s)/2)); };

A348941 a(n) = n / gcd(n, A326042(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 7, 15, 16, 17, 18, 19, 20, 21, 22, 23, 4, 25, 13, 27, 14, 29, 15, 31, 32, 33, 34, 35, 36, 37, 19, 39, 40, 41, 21, 43, 4, 45, 23, 47, 24, 49, 25, 17, 13, 53, 27, 11, 28, 57, 58, 59, 30, 61, 62, 63, 64, 65, 33, 67, 68, 23, 35, 71, 24, 73, 37, 75, 38, 77, 39, 79, 80, 81, 82

Offset: 1

Antti Karttunen, Nov 04 2021

Denominator of ratio A326042(n) / n.

If there are no more 1's in this sequence after the initial one, then there are no odd terms of A336702 (numbers whose abundancy index is a power of 2) larger than one, and neither there are odd terms in A005820 or in A046060. Compare to similar conditions given in A336848, A336849 and A337339.

Cf. A005820, A046060, A326042, A336702, A336848, A336849, A337339, A348736, A348940, A348942 (numerators).

f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s[n_] := Times @@ f1 @@@ FactorInteger[n]; f[p_, e_] := s[((q = NextPrime[p])^(e + 1) - 1)/(q - 1)]; s2[1] = 1; s2[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := n/GCD[n, s2[n]]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
A348941(n) = (n / gcd(n, A326042(n)));

a(n) = n / A348940(n) = n / gcd(n, A326042(n)).

cp's OEIS Frontend

A337340 a(n) = (A337339(n)-A003961(n)) / 2.

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A337341 a(n) = A337339(n) - n.

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A336700 Numbers k such that the odd part of (1+k) divides (1 + odd part of sigma(k)).

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A337337 a(n) = gcd(1+sigma(s), (s+1)/2), where s is the square of n once prime-shifted (s = A003961(n)^2 = A003961(n^2)).

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A378231 Deficiency of prime-shifted squares: a(n) = 2*A003961(n^2) - sigma(A003961(n^2)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).

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A336697 a(n) = gcd((s+1)/2, 1+sigma(s)), where s is the n-th odd square, (2n-1)^2.

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A337336 a(n) = A048673(n^2).

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A337338 Numerator of (1+sigma(s)) / ((s+1)/2), where s is the square of n prime-shifted once (s = A003961(n)^2 = A003961(n^2)).

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