A019565 -id:A019565 - cp's OEIS frontend

A319990 a(n) = Product_{d|n, dA019565(d)^[0 == d mod 3].

1, 1, 1, 1, 1, 6, 1, 1, 6, 1, 1, 90, 1, 1, 6, 1, 1, 1260, 1, 1, 6, 1, 1, 3150, 1, 1, 84, 1, 1, 18900, 1, 1, 6, 1, 1, 1455300, 1, 1, 6, 1, 1, 9900, 1, 1, 17640, 1, 1, 242550, 1, 1, 6, 1, 1, 19209960, 1, 1, 6, 1, 1, 764032500, 1, 1, 9240, 1, 1, 2340, 1, 1, 6, 1, 1, 7283776500, 1, 1, 1260, 1, 1, 35100, 1, 1, 38808, 1, 1, 94594500, 1, 1, 6, 1, 1

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A293214, A319991, A319992, A320003, A320010 (rgs-transform).

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319990(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };

a(n) = Product_{d|n, dA019565(d)^[0 == d mod 3].
a(n) = A293214(n) / (A319991(n)*A319992(n)).
For all n >= 1:
A007814(a(n)) = A320003(n).
A195017(a(n)) = 0 mod 3.

A332824 a(n) = Product_{d|n} A019565(phi(d)), where phi is Euler totient function A000010.

Original entry on oeis.org

2, 4, 6, 12, 10, 36, 30, 60, 90, 100, 42, 540, 70, 900, 210, 420, 22, 8100, 66, 2100, 3150, 1764, 330, 18900, 550, 4900, 2970, 94500, 770, 44100, 2310, 4620, 6930, 484, 11550, 4252500, 130, 4356, 16170, 115500, 182, 9922500, 546, 291060, 242550, 108900, 2730, 1455300, 8190, 302500, 858, 1131900, 1430, 8820900, 19110

Offset: 1

Antti Karttunen, Feb 25 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000010, A019565, A097248, A318834, A332825, A332826, A332827.

Cf. A048675 (a left inverse).

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A332824(n) = { my(m=1); fordiv(n,d,m *= A019565(eulerphi(d))); (m); };

a(n) = Product_{d|n} A332825(d).
a(n) = A318834(n) * A332825(n).
A048675(a(n)) = n.
A097248(a(n)) = A019565(n).

A351549 Numbers k for which k * gcd(sigma(k), A019565(k)) is equal to sigma(k) * gcd(k, A019565(k)).

Original entry on oeis.org

1, 1456, 15480, 114660, 2244600, 3894768, 25108200, 27052704, 65021040, 112402080, 1973921400

Offset: 1

Antti Karttunen, Feb 19 2022

Numbers k such that their abundancy index [sigma(k)/k] is equal to A351557(k)/A351556(k).
Question: If the above ratio is neither 1 nor 2, must it then be > 2? Are all even terms abundant?
a(12) > 2281701376 if it exists.

Cf. A000203, A005101, A007947, A019565, A080398, A351556, A351557, A351558, A351559.
Subsequence of A386424.
Cf. also A351458, A351548.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
isA351549(n) = { my(s=sigma(n), z=A019565(n)); (n*gcd(s,z))==(s*gcd(n,z)); };

A376406 a(0) = 1, and for n > 0, a(n) = A019565(Sum_{i=0..n-1} a(i)), where A019565 is the base-2 exp-function.

Original entry on oeis.org

1, 2, 6, 14, 330, 10166, 12075690, 1174153011328084322, 73582975079922326904310062621361286633125176265747127754

Offset: 0

Antti Karttunen, Nov 04 2024

nonn

a(9) has 272 digits and a(10) has 1523 digits.
The lexicographically earliest infinite sequence x for which A048675(x(n)) gives the partial sums of x (shifted right once). This follows because the "least k" condition in the alternative formula also ensures that each k is squarefree, as we have A097248(n) = A019565(A048675(n)) <= n for all n, with equivalence only when n is squarefree.
Compare also to A376408.

			Starting with a(0) = 1, we take partial sums of previous terms, and apply A019565 to get the next term, and in the rightmost column, we "unbox" that term by applying A048675 to get A376407(n), which thus gives the partial sums of a(0)..a(n-1):
a(0)                               = 1          -> 0
a(1) = A019565(1)                  = 2,         -> 1     = 1
a(2) = A019565(1+2)                = 6,         -> 3     = 1+2
a(3) = A019565(1+2+6)              = 14,        -> 9     = 1+2+6
a(4) = A019565(1+2+6+14)           = 330,       -> 23    = 1+2+6+14
a(5) = A019565(1+2+6+14+330)       = 10166,     -> 353   = 1+2+6+14+330
a(6) = A019565(1+2+6+14+330+10166) = 12075690,  -> 10519 = 1+2+6+14+330+10166
etc.

Antti Karttunen, Table of n, a(n) for n = 0..9
Index entries for sequences related to binary expansion of n

Cf. A019565, A048675, A097248, A376408.
Cf. A376407 (= A048675(a(n)), also gives the partial sums from its second term onward).
Subsequence of A005117.
Cf. also analogous sequences A002110 (for A276085), A093502 (for A056239), A376399 (for A276075).

up_to = 12;
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A376406list(up_to) = { my(v=vector(up_to), s=1); v[1]=1; for(n=2,up_to,v[n] = A019565(s); s += v[n]); (v); };
v376406 = A376406list(1+up_to);
A376406(n) = v376406[1+n];

a(n) = A019565(A376407(n)) = A019565(Sum_{i=0..n-1} a(i)).
a(0) = 1, and for n > 0, a(n) is the least k such that A048675(k) = a(n-1) + A048675(a(n-1)), where A048675 is the base-2 log-function.
For n > 0, a(n) <= a(n-1) * A019565(a(n-1)).

A277810 Square array A(r,c) = A019565(A277820(r,c)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

2, 6, 3, 10, 15, 30, 210, 21, 14, 5, 22, 1155, 462, 35, 70, 858, 39, 910, 55, 330, 105, 1870, 3315, 72930, 5005, 2002, 33, 42, 9699690, 5187, 2926, 85, 714, 2145, 770, 7, 46, 111546435, 238602, 11305, 248710, 3927, 390, 77, 154, 4002, 87, 93763670, 21505, 152490, 440895, 3094, 91, 546, 231, 7130, 13485, 620310, 1078282205, 2306486, 9867, 114114, 17017, 170170, 1365, 2310

Offset: 1

Antti Karttunen, Nov 01 2016

Permutation of squarefree numbers (A005117) after their initial term 1.

			The top left corner of the array:
    2,    6,     10,   210,      22,    858,      1870,    9699690
    3,   15,     21,  1155,      39,   3315,      5187,  111546435
   30,   14,    462,   910,   72930,   2926,    238602,   93763670
    5,   35,     55,  5005,      85,  11305,     21505, 1078282205
   70,  330,   2002,   714,  248710, 152490,   2306486,   60138078
  105,   33,   2145,  3927,  440895,   9867,   1870935,  691587897
   42,  770,    390,  3094,  114114, 520030,    162690,  581334754
    7,   77,     91, 17017,     133,  33649,     50141, 6685349671
  154,  546, 170170,   570,    6118, 254562, 357505330,   51269790
  231, 1365,   7293,  3135, 1312311, 983535,  11599797,  589602585

Antti Karttunen, Table of n, a(n) for n = 1..1540; the first 55 antidiagonals of the array

Transpose: A277809.
The topmost row: A123098, the leftmost column: A277811.
Cf. A005117, A019565, A277820.

(define (A277810 n) (A277810bi (A002260 n) (A004736 n)))
(define (A277810bi row col) (A019565 (A277820bi row col)))

A(r,c) = A019565(A277820(r,c)).

A277811 Column 1 of A277810: a(n) = A019565(A065621(n)).

Original entry on oeis.org

2, 3, 30, 5, 70, 105, 42, 7, 154, 231, 2310, 385, 110, 165, 66, 11, 286, 429, 4290, 715, 10010, 15015, 6006, 1001, 182, 273, 2730, 455, 130, 195, 78, 13, 442, 663, 6630, 1105, 15470, 23205, 9282, 1547, 34034, 51051, 510510, 85085, 24310, 36465, 14586, 2431, 374, 561, 5610, 935, 13090, 19635, 7854, 1309, 238, 357, 3570, 595, 170, 255, 102, 17

Offset: 1

Antti Karttunen, Nov 01 2016

Antti Karttunen, Table of n, a(n) for n = 1..8192

Column 1 of A277810.
Permutation of A030059.
Cf. A019565, A065621.

(define (A277811 n) (A019565 (A065621 n)))

a(n) = A019565(A065621(n)).

A286611 Numbers k for which A019565(k) <= A087207(k).

Original entry on oeis.org

17, 19, 34, 37, 41, 43, 47, 53, 59, 61, 65, 67, 69, 71, 73, 74, 79, 82, 83, 86, 89, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 129, 131, 134, 137, 139, 141, 142, 145, 146, 148, 149, 151, 157, 158, 163, 164, 166, 167, 173, 177, 178, 179, 181, 183, 191, 193, 194, 197, 199, 201, 202, 206, 211, 212

Offset: 1

Antti Karttunen, Jun 20 2017

Any 2-cycle of A087207 and also any 2-cycle of A019565 (in which case A019565(x) = A087207(x) for both members of the cycle), if such cycles exist at all, must have the both of its members included in this sequence.

Cf. A019565, A087207, A285315, A285316, A286609.

Intersection with A286608 gives A286612.

A007947(n) = factorback(factorint(n)[, 1]);
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ After Michel Marcus
A087207(n) = A048675(A007947(n));
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
isA286611(n) = (A019565(n) <= A087207(n));
n=0; k=1; while(k <= 10000, n=n+1; if(isA286611(n),write("b286611.txt", k, " ", n);k=k+1));

;; With Antti Karttunen's IntSeq-library.
(define A286611 (MATCHING-POS 1 1 (lambda (n) (<= (A019565 n) (A087207 n)))))

A300830 a(n) = Product_{d|n} A019565(d)^(1-A008966(n/d)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 2, 1, 1, 12, 1, 1, 1, 30, 1, 6, 1, 20, 1, 1, 1, 540, 2, 1, 12, 60, 1, 1, 1, 210, 1, 1, 1, 2520, 1, 1, 1, 1260, 1, 1, 1, 84, 20, 1, 1, 94500, 2, 6, 1, 140, 1, 540, 1, 18900, 1, 1, 1, 25200, 1, 1, 60, 2310, 1, 1, 1, 44, 1, 1, 1, 8731800, 1, 1, 12, 132, 1, 1, 1, 346500, 168, 1, 1, 39600, 1, 1, 1, 41580, 1, 1260

Offset: 1

Antti Karttunen, Mar 16 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A008683, A008966, A019565.

Cf. also A293214, A300831, A300832, A300833.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300830(n) = { my(m=1); fordiv(n,d,if(!moebius(n/d),m *= A019565(d))); m; };

a(n) = Product_{d|n} A019565(d)^(1-abs(A008683(n/d))).

a(n) = A293214(n) / (A300831(n)*A300832(n)).

A318469 Multiplicative with a(p^e) = A019565(A003714(e)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 10, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 7, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 20, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 14, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 20, 10, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 14, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Antti Karttunen, Aug 30 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A003714, A019565, A072649, A318464, A318465.

Cf. also A293442, A293443, A300834, A318470.

A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318469(n) = factorback(apply(e -> A019565(A003714(e)),factor(n)[,2]));

For all n >= 1, A001222(a(n)) = A318464(n).

A318835 Restricted growth sequence transform of A318834, product_{d|n, dA019565(A000010(d)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 5, 6, 2, 7, 2, 8, 9, 8, 2, 10, 2, 11, 12, 13, 2, 14, 15, 16, 12, 14, 2, 17, 2, 18, 19, 20, 21, 22, 2, 23, 24, 25, 2, 26, 2, 27, 28, 29, 2, 30, 9, 31, 32, 33, 2, 34, 24, 35, 36, 37, 2, 38, 2, 39, 40, 39, 41, 42, 2, 43, 44, 45, 2, 46, 2, 47, 48, 49, 50, 51, 2, 52, 53, 54, 2, 55, 56, 57, 58, 59, 2, 60, 61, 62, 63, 64, 65, 66, 2

Offset: 1

Antti Karttunen, Sep 04 2018

nonn

For all i, j: a(i) = a(j) => A051953(i) = A051953(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A019565, A318831, A318834,

Cf. also A293215, A293217, A293223, A293224, A293232, A300833, A300835.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(eulerphi(d)))); m; };
v318835 = rgs_transform(vector(up_to,n,A318834(n)));
A318835(n) = v318835[n];

cp's OEIS Frontend

A319990 a(n) = Product_{d|n, dA019565(d)^[0 == d mod 3].

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A332824 a(n) = Product_{d|n} A019565(phi(d)), where phi is Euler totient function A000010.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A351549 Numbers k for which k * gcd(sigma(k), A019565(k)) is equal to sigma(k) * gcd(k, A019565(k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A376406 a(0) = 1, and for n > 0, a(n) = A019565(Sum_{i=0..n-1} a(i)), where A019565 is the base-2 exp-function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A277810 Square array A(r,c) = A019565(A277820(r,c)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

A277811 Column 1 of A277810: a(n) = A019565(A065621(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A286611 Numbers k for which A019565(k) <= A087207(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Scheme

A300830 a(n) = Product_{d|n} A019565(d)^(1-A008966(n/d)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A318469 Multiplicative with a(p^e) = A019565(A003714(e)).

Views