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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A326042 a(n) = A064989(sigma(A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, and A064989 shifts it back towards smaller primes.

Original entry on oeis.org

1, 1, 2, 11, 1, 2, 2, 3, 29, 1, 5, 22, 4, 2, 2, 49, 3, 29, 2, 11, 4, 5, 6, 6, 34, 4, 22, 22, 1, 2, 17, 55, 10, 3, 2, 319, 10, 2, 8, 3, 7, 4, 2, 55, 29, 6, 8, 98, 85, 34, 6, 44, 6, 22, 5, 6, 4, 1, 29, 22, 13, 17, 58, 1091, 4, 10, 4, 33, 12, 2, 31, 87, 3, 10, 68, 22, 10, 8, 10, 49, 469, 7, 12, 44, 3, 2, 2, 15, 25, 29, 8, 66, 34, 8
Offset: 1

Views

Author

Antti Karttunen, Jun 16 2019

Keywords

Comments

For any other number n than those in A326182 we have a(n) < A003961(n).
Fixed points k (for which a(k) = k) satisfy A003973(k) = 2^e * A003961(k) for some exponent e >= 0. Applying A003961 to such numbers gives the odd terms in A336702, of which there are likely to be just a single instance, its initial 1. (Clarified Nov 07 2021).
Conjecture: There are no other fixed points than a(1) = 1. If true, then there are no odd perfect numbers. This condition is equivalent to the condition that if A161942 has no fixed points larger than one, then there are no odd perfect numbers. This follows as whenever k is a fixed point, that is, a(k) = k, then we should also have A003961(a(k)) = A003961(A064989(sigma(A003961(k)))) = A161942(A003961(k)) = A003961(k). Note that A003961 is an injective and surjective mapping from natural numbers to odd numbers, A064989 is its (left) inverse, and composition A003961(A064989(n)) is equivalent to A000265(n).
From Antti Karttunen, Aug 05 2020: (Start)
For any hypothetical odd perfect number x, we would have A003973(k) = 2 * A003961(k), with k = A064989(x) and x = A003961(k). Thus we would have a(k) = A064989(sigma(A003961(k))) = A064989(sigma(x)) = A064989(2*x) = A064989(x) = k. On the other hand, A003973(k) = sigma(A003961(k)) < A003961(A003961(k)) [see A286385 for the reason why], so a necessary condition for this is that x should be one of the terms of A246282. (Clarified Dec 01 2020).
(End)

Links

Crossrefs

Cf. A000037, A000203, A000265, A000593, A003961, A003973, A064989, A161942, A162284, A246282, A286385, A326041, A326182, A336702 (numbers whose abundancy index is a power of 2).
Cf. A348736 [n - a(n)], A348738 [a(n) < n], A348739 [a(n) > n], A348750 [= A064989(a(A003961(n)))], A348940 [gcd(n,a(n))], A348941, A348942, A351456, A353767, A353790, A353794.
Cf. also A332223 for another conjugation of sigma.

Programs

  • Mathematica
    f1[p_, e_] := NextPrime[p]^e; a1[1] = 1; a1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[2, e_] := 1; f2[p_, e_] := NextPrime[p, -1]^e; a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := a2[DivisorSigma[1, a1[n]]]; Array[a, 100] (* Amiram Eldar, Nov 07 2021 *)
  • PARI
    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)));

Formula

a(n) = A064989(A003973(n)) = A064989(sigma(A003961(n))).
For k in A000037, a(k) = A064989(A003973(k)/2) = A064989((1/2)*sigma(A003961(k))).
Multiplicative with a(p^e) = A064989((q^(e+1)-1)/(q-1)), where q = nextPrime(p). - Antti Karttunen, Nov 05 2021
a(n) = A353790(n) / A353767(n) = A353794(n) / A351456(n). - Antti Karttunen, May 13 2022

Extensions

Keyword:mult added by Antti Karttunen, Nov 05 2021

A348739 Numbers k for which A326042(k) > k.

Original entry on oeis.org

4, 9, 12, 16, 18, 25, 32, 36, 44, 48, 49, 64, 72, 81, 96, 99, 100, 108, 124, 144, 147, 162, 169, 176, 180, 192, 196, 225, 236, 243, 252, 256, 279, 284, 288, 300, 320, 324, 361, 372, 396, 400, 405, 432, 441, 448, 450, 468, 484, 486, 496, 507, 512, 529, 531, 567, 576, 588, 604, 612, 625, 639, 648, 675, 676, 700, 704
Offset: 1

Views

Author

Antti Karttunen, Nov 02 2021

Keywords

Comments

Terms that occur also in A337386 are: 180, 300, 720, 900, 960, 1008, 1200, 1440, 1620, 1800, 2016, 2400, ...

Links

Crossrefs

Positions of negative terms in A348736.
Cf. A326182 (subsequence after its initial 1), A348738.
Cf. A000203, A003961, A064989, A161942, A191218, A326042, A337386, A348742, A348749 (corresponding odd numbers), A348942.

Programs

  • Mathematica
    f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := NextPrime[p]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; Select[Range[700], s1[DivisorSigma[1, s2[#]]] > # &] (* Amiram Eldar, Nov 04 2021 *)
  • PARI
    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)));
isA348739(n) = (A326042(n)>n);

A326183 Numbers n for which A003961(n) <= A064989(sigma(A003961(n))), but which are not squares.

Original entry on oeis.org

97560000, 140040000, 161640000, 188280000, 392760000, 414360000, 429480000, 432360000, 443160000, 474840000, 494280000, 522360000, 582840000, 589320000, 673560000, 715320000, 767160000, 842040000, 888120000, 903960000, 918360000, 932760000, 1026360000, 1047240000
Offset: 1

Views

Author

Antti Karttunen, Jun 17 2019

Keywords

Comments

At least for 1 <= n <= 24 it holds that A052126(a(n)) is square, and moreover, apart from a(20), that square is 360000, while A052126(a(20)) = A052126(903960000) = 29160000.
The largest prime factors (that are also unitary in case of 24 initial terms) are A006530(a(n)): 271, 389, 449, 523, 1091, 1151, 1193, 1201, 1231, 1319, 1373, 1451, 1619, 1637, 1871, 1987, 2131, 2339, 2467, 31, 2551, 2591, 2851, 2909.

Links

Crossrefs

Setwise difference A326182 \ A000290.
Cf. A000203, A003961, A006530, A052126, A064989.

A337344 Odd numbers k such that A064989(sigma(k)) >= k.

Original entry on oeis.org

1, 9, 25, 225, 289, 729, 1681, 2401, 2601, 3481, 5041, 6561, 7225, 7921, 10201, 15129, 15625, 17161, 18225, 19881, 21609, 27889, 28561, 29929, 31329, 35721, 42025, 45369, 59049, 60025, 62001, 65025, 71289, 83521, 85849, 87025, 88209, 91809, 114921, 123201, 126025, 130321, 140625, 146689, 154449, 164025, 172225
Offset: 1

Views

Author

Antti Karttunen, Aug 26 2020

Keywords

Comments

Applying A064989 to these terms and sorting into ascending order gives A326182.
Conversely, this sequence is obtained when the sequence b(n) = A003961(A326182(n)) is sorted into ascending order.
Not all terms are squares. For example, 12121028325 = A003961(A326183(1)) = 3^6 * 5^2 * 7^4 * 277 is also term, and this is true for all terms of A326183 similarly prime shifted. Interestingly, for n = 1..24, A003961(A326183(n)) is a term of A228058.

Links

Crossrefs

Odd terms of A337343.
Cf. A000203, A003961, A064989, A228058, A326042, A326182, A326183.

Programs

  • PARI
    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)};
isA337344(n) = ((n%2)&&(A064989(sigma(n))>=n));
Showing 1-4 of 4 results.

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