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.

Showing 1-4 of 4 results.

A048674 Fixed points of A048673 and A064216: Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) = (2*n)-1, where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

1, 2, 3, 25, 26, 33, 93, 1034, 970225, 8550146, 325422273, 414690595, 1864797542, 2438037206
Offset: 1

Views

Author

Antti Karttunen, Jul 14 1999

Keywords

Comments

Equally: after 1, numbers n such that, if the prime factorization of 2n-1 = Product_{k >= 1} (p_k)^(c_k) then Product_{k >= 1} (p_{k-1})^(c_k) = n.
Factorization of the initial terms: 1, 2, 3, 5^2, 2*13, 3*11, 3*31, 2*11*47, 5^2*197^2, 2*11*47*8269, 3*11*797*12373, 5*11^2*433*1583, 2*23*59*101*6803, 2*11*53*1201*1741.
The only 3-cycle of permutation A048673 in range 1 .. 402653184 is (2821 3460 5639).
For 2-cycles, take setwise difference of A245449 and this sequence.
Numbers k for which A336853(k) = k-1. - Antti Karttunen, Nov 26 2021

Examples

			25 is present, as 2*25 - 1 = 49 = p_4^2, and p_3^2 = 5*5 = 25.
26 is present, as 2*26 - 1 = 51 = 3*17 = p_2 * p_8, and p_1 * p_7 = 2*13 = 26.
Alternatively, as 26 = 2*13 = p_1 * p_7, and ((p_2 * p_8)+1)/2 = ((3*17)+1)/2 = 26 also, thus 26 is present.

Crossrefs

Fixed points of permutation pair A048673/A064216.
Positions of zeros in A349573.
Subsequence of the following sequences: A245449, A269860, A319630, A349622, A378980 (see also A379216).
This sequence is also obtained as a setwise difference of the following pairs of sequences: A246281 \ A246351, A246352 \ A246282, A246361 \ A246371, A246372 \ A246362.
Cf. A000040, A003961 (A045965), A064989, A285701, A336853.
Cf. also A348514 (fixed points of map A108228, similar to A048673).

Programs

  • Maple
    A048673 := n -> (A003961(n)+1)/2;
A048674list := proc(upto_n) local b,i; b := [ ]; for i from 1 to upto_n do if(A048673(i) = i) then b := [ op(b), i ]; fi; od: RETURN(b); end;
  • Mathematica
    Join[{1}, Reap[For[n = 1, n < 10^7, n++, ff = FactorInteger[n]; If[Times @@ Power @@@ (NextPrime[ff[[All, 1]]]^ff[[All, 2]]) == 2 n - 1, Print[n]; Sow[n]]]][[2, 1]]] (* Jean-François Alcover, Mar 04 2016 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA048674(n) = ((n+n)==(1+A003961(n))); \\ Antti Karttunen, Nov 26 2021

Extensions

Entry revised and the names in Maple-code cleaned by Antti Karttunen, Aug 25 2014
Terms a(11) - a(14) added by Antti Karttunen, Sep 11-13 2014

A270430 Numbers n such that A048673(n) and A064216(n) are of the same parity.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 12, 13, 16, 17, 20, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 39, 40, 41, 42, 48, 49, 50, 52, 53, 58, 62, 64, 65, 68, 69, 74, 75, 77, 80, 81, 82, 85, 90, 93, 97, 98, 99, 100, 101, 102, 104, 105, 106, 108, 109, 111, 113, 114, 116, 117, 120, 121, 124, 125, 126, 128, 130, 132, 133, 136, 137, 139, 141, 144
Offset: 1

Views

Author

Antti Karttunen, Mar 17 2016

Keywords

Comments

See A270434 for the possible bias favoring this sequence over the complement A270431.

Links

Crossrefs

Complement: A270431.
Left inverse: A270432.
Cf. A245449 (a subsequence).
Cf. A000035, A048673, A064216, A270434.
Cf. also A269860.

Programs

  • Mathematica
    f[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; g[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; Select[Range@ 144, Xor[EvenQ@ f@ #, OddQ@ g@ #] &] (* Michael De Vlieger, Mar 17 2016 *)

Formula

Other identities. For all n >= 1:
A270432(a(n)) = n.

A269861 Numbers n such that n and A048673(n) are of opposite parity.

Original entry on oeis.org

4, 5, 7, 10, 12, 14, 15, 16, 17, 19, 21, 29, 30, 34, 36, 38, 40, 41, 42, 43, 44, 45, 48, 51, 52, 53, 55, 56, 57, 58, 61, 63, 64, 65, 67, 73, 77, 79, 82, 86, 87, 90, 91, 92, 100, 101, 102, 103, 106, 108, 110, 113, 114, 115, 120, 122, 123, 124, 125, 126, 127, 129, 130, 132, 134, 135, 136, 137, 140, 144, 146, 148, 149
Offset: 1

Views

Author

Antti Karttunen, Mar 16 2016

Keywords

Comments

Union of even terms of A246261 and odd terms of A246263.

Links

Crossrefs

Complement: A269860.
Left inverse: A269862.
Cf. A000035, A048673, A246261, A246263.
Cf. also A270431.

Programs

  • Mathematica
    f[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; Select[Range@ 150, Xor[EvenQ@ f@ #, EvenQ@ #] &] (* Michael De Vlieger, Mar 17 2016 *)

Formula

Other identities. For all n >= 1:
A269862(a(n)) = n.

A349573 a(n) = A048673(n) - n, where A048673(n) = (A003961(n)+1) / 2, and A003961(n) shifts the prime factorization of n one step towards larger primes.

Original entry on oeis.org

0, 0, 0, 1, -1, 2, -1, 6, 4, 1, -4, 11, -4, 3, 3, 25, -7, 20, -7, 12, 7, -2, -8, 44, 0, 0, 36, 22, -13, 23, -12, 90, 0, -5, 4, 77, -16, -3, 4, 55, -19, 41, -19, 15, 43, -2, -20, 155, 12, 24, -3, 25, -23, 134, -9, 93, 1, -11, -28, 98, -27, -6, 75, 301, -5, 32, -31, 18, 4, 46, -34, 266, -33, -12, 48, 28, -5, 50, -37
Offset: 1

Views

Author

Antti Karttunen, Nov 23 2021

Keywords

Links

Crossrefs

Cf. A003961, A048673, A064216, A349571.
Cf. A048674 (positions of zeros), A246351 (negative terms), A246281 (nonpositive terms), A246352 (nonnegative terms), A246282 (positive terms), A269860 (even terms), A269861 (odd terms).

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p]^e; a[1] = 0; a[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2 - n; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *)
  • PARI
    A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A349573(n) = (A048673(n)-n);

Formula

a(n) = A048673(n) - n.
a(n) = Sum_{d|n, dA349571(n/d).
Showing 1-4 of 4 results.

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