A329603 -id:A329603 - cp's OEIS frontend

A347117 Möbius transform of A329603.

2, 3, 6, 10, 16, 0, 48, 30, 12, 104, 96, 12, 240, 192, 8, 90, 336, 54, 576, 240, 16, 504, 720, 36, 24, 600, 40, 480, 1056, -122, 1680, 270, 96, 1104, 96, 132, 1920, 1224, 144, 720, 2736, 1064, 3360, 1200, 0, 1920, 3696, 108, 60, 126, 624, 1680, 4416, 422, 336, 1440, 768, 3144, 5616, -228, 6960, 3120, 416, 810, 624

Offset: 1

Antti Karttunen, Aug 21 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A008683, A156552, A329603.

Cf. also A332463, A347115, A347118.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, 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 };
A329603(n) = A005940(2+(3*A156552(n)));
A347117(n) = sumdiv(n,d,moebius(n/d)*A329603(d));

a(n) = Sum_{d|n} A008683(n/d) * A329603(d).

A353268 The least number with the same prime factorization pattern (A348717) as A329603(n) = A005940(1+(1+(3*A156552(n)))).

Original entry on oeis.org

2, 2, 8, 6, 18, 2, 50, 12, 20, 8, 98, 14, 242, 18, 32, 24, 338, 6, 578, 54, 72, 50, 722, 28, 42, 98, 60, 150, 1058, 2, 1682, 48, 200, 242, 162, 70, 1922, 338, 392, 108, 2738, 8, 3362, 294, 44, 578, 3698, 56, 110, 20, 968, 726, 4418, 12, 450, 300, 1352, 722, 5618, 26, 6962, 1058, 500, 96, 882, 18, 7442, 1014, 2312

Offset: 1

Antti Karttunen, Apr 09 2022

nonn

Index entries for sequences computed from indices in prime factorization

Cf. A005940, A156552, A348717, A353267.

Coincides with A352892 on even n, and with A329603 on odd n.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, 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 };
A329603(n) = A005940(2+(3*A156552(n)));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A353268(n) = A348717(A329603(n));

a(n) = A348717(A329603(n)).

For all n >= 1, a(2n) = A352892(2n), a(2n-1) = A329603(2n-1).

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

Original entry on oeis.org

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)

A329609 Numbers k such that A156552(k) is a multiple of 3.

Original entry on oeis.org

1, 4, 9, 10, 16, 21, 22, 25, 30, 34, 36, 39, 40, 46, 49, 55, 57, 62, 64, 66, 81, 82, 84, 85, 87, 88, 90, 91, 94, 100, 102, 105, 111, 115, 118, 120, 121, 129, 133, 134, 136, 138, 144, 146, 154, 155, 156, 159, 160, 166, 169, 183, 184, 186, 187, 189, 194, 195, 196, 198, 203, 205, 206, 213, 218, 220, 225, 228, 235, 237, 238, 246

Offset: 1

Antti Karttunen, Nov 21 2019

nonn

Not a multiplicative semigroup. For example, although 10 and 21 are present, 210 is missing. Compare to A332820. - Antti Karttunen, Jan 17 2023

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000290 (subsequence), A156552, A329603, A329604, A332812.
Positions of zeros in A329903, of nonzeros in A341353, of ones in A353269 (characteristic function), A353418 (Dirichlet inverse of char.fun), A359836.
Sequence A332449 sorted into ascending order.
Cf. also A353303, A353304 and A332820.

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
isA329609(n) = !(A156552(n)%3);

A341515 The Collatz or 3x+1 map (A006370) conjugated by unary-binary-encoding (A156552).

Original entry on oeis.org

1, 5, 2, 15, 3, 11, 5, 45, 4, 125, 7, 33, 11, 245, 6, 135, 13, 77, 17, 375, 10, 605, 19, 99, 9, 845, 8, 735, 23, 17, 29, 405, 14, 1445, 15, 231, 31, 1805, 22, 1125, 37, 1331, 41, 1815, 12, 2645, 43, 297, 25, 275, 26, 2535, 47, 539, 21, 2205, 34, 4205, 53, 51, 59, 4805, 20, 1215, 33, 1859, 61, 4335, 38, 3125, 67, 693

Offset: 1

Antti Karttunen, Feb 14 2021

nonn

Collatz-conjecture can be formulated via this sequence by postulating that all iterations of a(n), starting from any n > 1, will eventually reach the cycle [2, 5, 3].

Cf. A005940, A006370, A064989, A156552, A329603, A341510, A347115 (Möbius transform),
Sequences related to iterations of this sequence: A352890, A352891, A352892, A352893, A352894, A352896, A352897, A352898, A352899.
Cf. A341516 (a variant).

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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)};
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 };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));

If n is odd, then a(n) = A064989(n), otherwise a(n) = A329603(n) = A341510(n,2*n).

a(n) = A005940(1+A006370(A156552(n))).

A329604 Numbers k such that A156552(k) == 1 (mod 3); numbers k for which A156552(2*k) is a multiple of 3.

Original entry on oeis.org

2, 5, 8, 11, 15, 17, 18, 20, 23, 31, 32, 33, 41, 42, 44, 45, 47, 50, 51, 59, 60, 67, 68, 69, 72, 73, 77, 78, 80, 83, 92, 93, 97, 98, 99, 103, 109, 110, 114, 119, 123, 124, 125, 127, 128, 132, 135, 137, 141, 149, 153, 157, 161, 162, 164, 167, 168, 170, 174, 176, 177, 179, 180, 182, 188, 191, 197, 200, 201, 204, 207, 210, 211, 217, 219, 221, 222

Offset: 1

Antti Karttunen, Nov 21 2019

nonn

Even terms of A329609, divided by two.

Numbers k for which A156552(k) == 1 (mod 3). - Antti Karttunen, Feb 27 2020

Antti Karttunen, Table of n, a(n) for n = 1..20000

Sequence A329603 sorted into ascending order.
Positions of 1's in A329903 and in A332814.
Cf. A156552, A329609.
Cf. A001105 (subsequence apart from the initial 0).
Cf. A031368 (a subsequence of prime terms).
Cf. also A332812, A324814, A332821.

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
isA329604(n) = !(A156552(2*n)%3);

New primary definition added by Antti Karttunen, Mar 01 2020

A352892 Next even term in the trajectory of map x -> A341515(x), when starting from x=n; a(1) = 1. Here A341515 is the Collatz or 3x+1 map (A006370) conjugated by unary-binary-encoding (A156552).

Original entry on oeis.org

1, 2, 2, 6, 2, 2, 2, 12, 4, 8, 2, 14, 2, 18, 6, 24, 2, 6, 2, 54, 10, 50, 2, 28, 4, 98, 8, 150, 2, 2, 2, 48, 14, 242, 6, 70, 2, 338, 22, 108, 2, 8, 2, 294, 12, 578, 2, 56, 4, 20, 26, 726, 2, 12, 10, 300, 34, 722, 2, 26, 2, 1058, 20, 96, 14, 18, 2, 1014, 38, 32, 2, 140, 2, 1682, 18, 1734, 6, 50, 2, 216, 16, 1922, 2, 686

Offset: 1

Antti Karttunen, Apr 08 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A005940, A064989, A156552, A329603, A341515, A348717.
Cf. also A139391, A352893, A352894, A352896, A352897, A352898, A352899, A353267.
Coincides with A353268 on even n, and with A348717 on odd n.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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)};
A156552(n) = { my(f = factor(n), p, 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 };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A352892(n) = A348717(A341515(n));

A352892(n) = if(1==n, n, n = A341515(n); while(n%2, n = A341515(n)); (n)); \\ A slower alternative.

a(n) = A348717(A341515(n)).
For all n >= 1, a(2n) = A353268(2n), a(2n-1) = A348717(2n-1).
a(p) = 2 for all primes p.
For n > 1, a(n) = A005940(1+A139391(A156552(n))).

A347115 Möbius transform of A341515.

Original entry on oeis.org

1, 4, 1, 10, 2, 5, 4, 30, 2, 118, 6, 12, 10, 236, 2, 90, 12, 64, 16, 240, 4, 594, 18, 36, 6, 830, 4, 480, 22, -116, 28, 270, 6, 1428, 8, 132, 30, 1784, 10, 720, 36, 1076, 40, 1200, 4, 2622, 42, 108, 20, 144, 12, 1680, 46, 458, 12, 1440, 16, 4178, 52, -228, 58, 4772, 8, 810, 20, 1242, 60, 2880, 18, 2752, 66, 396

Offset: 1

Antti Karttunen, Aug 20 2021

sign

Cf. A008683, A064989, A329603, A341515.
Cf. A285702 (odd bisection), A347116 (even bisection).
Cf. also A332463, A347117, A347118, A348045.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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)};
A156552(n) = { my(f = factor(n), p, 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 };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A347115(n) = sumdiv(n,d,moebius(n/d)*A341515(d));

a(n) = A008683(n/d) * A341515(d).

A341510 Symmetric square array A(n,k) = A005940(1+A156552(n)+A156552(k)), read by antidiagonals starting with A(1,1).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 9, 9, 9, 9, 9, 7, 8, 10, 8, 8, 8, 8, 10, 8, 9, 7, 15, 7, 7, 7, 15, 7, 9, 10, 8, 10, 12, 10, 10, 12, 10, 8, 10, 11, 15, 7, 15, 25, 15, 25, 15, 7, 15, 11, 12, 14, 12, 10, 12, 18, 18, 12, 10, 12, 14, 12, 13, 25, 21, 25, 15, 25, 11, 25, 15, 25, 21, 25, 13

Offset: 1

Antti Karttunen, Feb 13 2021

Considered as a binary operation on the positive integers, A(x, y) returns the term of the Doudna-sequence from the position that is the sum of the positions of x and y in the same sequence. (This is based on giving the Doudna-sequence an offset of 0, rather than 1 as used in A005940.) - Peter Munn, Feb 14 2021

			The top left 16x16 corner of the array:
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10,  11,  12,  13,  14,  15, 16,
   2,  3,  4,  5,  6,  9, 10,  7,  8, 15,  14,  25,  22,  21,  12, 11,
   3,  4,  5,  6,  9,  8, 15, 10,  7, 12,  21,  18,  33,  20,  25, 14,
   4,  5,  6,  9,  8,  7, 12, 15, 10, 25,  20,  27,  28,  35,  18, 21,
   5,  6,  9,  8,  7, 10, 25, 12, 15, 18,  35,  16,  55,  30,  27, 20,
   6,  9,  8,  7, 10, 15, 18, 25, 12, 27,  30,  11,  42,  45,  16, 35,
   7, 10, 15, 12, 25, 18, 11, 16, 27, 14,  49,  20,  77,  50,  21, 24,
   8,  7, 10, 15, 12, 25, 16, 27, 18, 11,  24,  21,  40,  49,  14, 45,
   9,  8,  7, 10, 15, 12, 27, 18, 25, 16,  45,  14,  63,  24,  11, 30,
  10, 15, 12, 25, 18, 27, 14, 11, 16, 21,  50,  35,  70,  75,  20, 49,
  11, 14, 21, 20, 35, 30, 49, 24, 45, 50,  13,  36, 121,  22,  75, 32,
  12, 25, 18, 27, 16, 11, 20, 21, 14, 35,  36,  45,  60, 125,  30, 75,
  13, 22, 33, 28, 55, 42, 77, 40, 63, 70, 121,  60,  17,  98, 105, 48,
  14, 21, 20, 35, 30, 45, 50, 49, 24, 75,  22, 125,  98,  33,  36, 13,
  15, 12, 25, 18, 27, 16, 21, 14, 11, 20,  75,  30, 105,  36,  35, 50,
  16, 11, 14, 21, 20, 35, 24, 45, 30, 49,  32,  75,  48,  13,  50, 81,

Cf. A341511 (the lower triangular section).
Cf. A003961 (main diagonal), A329603 (skewed diagonal).
Cf. A297165 (row 2 and column 2, when started from its term a(1)).
Cf. also A003056, A268715, A341515, A341520, A348041.

up_to = 105;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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 };
A341510sq(n,k) = A005940(1+A156552(n)+A156552(k));
A341510list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A341510sq(col,(a-(col-1))))); (v); };
v341510 = A341510list(up_to);
A341510(n) = v341510[n];

A(n, k) = A(k, n) = A005940(1 + A156552(n) + A156552(k)).
A(n, n) = A003961(n).
A(n, 2*n) = A(2*n, n) = A329603(n).
A(n, 2) = A(2, n) = A297165(n).

A352893 Number of iterations of map x -> A352892(x) needed to reach x <= 2 when starting from x=n, or -1 if such number is never reached. Here A352892 is the next odd term in the Collatz or 3x+1 map (A139391) conjugated by unary-binary-encoding (A156552).

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 1, 5, 3, 6, 1, 4, 1, 3, 2, 5, 1, 2, 1, 6, 7, 8, 1, 4, 3, 8, 6, 3, 1, 1, 1, 39, 4, 44, 2, 41, 1, 44, 9, 11, 1, 6, 1, 8, 5, 10, 1, 38, 3, 7, 9, 8, 1, 5, 7, 37, 45, 10, 1, 9, 1, 56, 7, 39, 4, 3, 1, 44, 45, 40, 1, 41, 1, 39, 3, 44, 2, 8, 1, 11, 6, 15, 1, 3, 9, 15, 11, 13, 1, 4, 7, 10, 11, 32, 9, 38

Offset: 1

Antti Karttunen, Apr 08 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A005940, A064989, A139391, A156552, A286380, A329603, A341515, A352890, A352892, A352894.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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)};
A156552(n) = { my(f = factor(n), p, 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 };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A352892(n) = A348717(A341515(n));
A352893(n) = { my(k=0); while(n>2, n = A352892(n); k++); (k); };

\\ Much faster than above program:
A139391(n) = my(x = if(n%2, 3*n+1, n/2)); x/2^valuation(x, 2); \\ From A139391
A286380(n) = { my(k=0); while(n>1, n = A139391(n); k++); (k); };
A352893(n) = if(1==n,0,A286380(A156552(n)));

If n <= 2, a(n) = 0, otherwise a(n) = 1 + a(A352892(n)).
For n > 1, a(n) = A286380(A156552(n)).
a(p) = 1 for all odd primes p.
For n >= 1, A352894(n) <= a(n) <= A352890(n).

cp's OEIS Frontend

A347117 Möbius transform of A329603.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A353268 The least number with the same prime factorization pattern (A348717) as A329603(n) = A005940(1+(1+(3*A156552(n)))).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A329609 Numbers k such that A156552(k) is a multiple of 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A341515 The Collatz or 3x+1 map (A006370) conjugated by unary-binary-encoding (A156552).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A329604 Numbers k such that A156552(k) == 1 (mod 3); numbers k for which A156552(2*k) is a multiple of 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A352892 Next even term in the trajectory of map x -> A341515(x), when starting from x=n; a(1) = 1. Here A341515 is the Collatz or 3x+1 map (A006370) conjugated by unary-binary-encoding (A156552).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A347115 Möbius transform of A341515.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A341510 Symmetric square array A(n,k) = A005940(1+A156552(n)+A156552(k)), read by antidiagonals starting with A(1,1).

Views

Author

Keywords