A019565 -id:A019565 - cp's OEIS frontend

A351082 a(n) = Product_{d|n} A019565(A289814(d)); a product obtained from the 2-digits present in ternary expansions of the divisors of n.

1, 2, 1, 2, 2, 6, 3, 12, 1, 4, 2, 6, 1, 12, 6, 36, 6, 30, 5, 40, 15, 20, 10, 540, 30, 60, 1, 12, 2, 36, 1, 72, 6, 36, 36, 30, 1, 20, 1, 240, 2, 540, 3, 120, 30, 100, 10, 8100, 15, 600, 90, 900, 30, 210, 28, 1008, 35, 28, 14, 7560, 21, 84, 105, 504, 28, 1260, 7, 504, 210, 3024, 42, 94500, 35, 140, 3150, 700, 420, 18900

Offset: 1

Antti Karttunen, Jan 31 2022

Cf. A019565, A289814, A293221, A351081, A351092.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A289814(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); };
A351082(n) = { my(m=1); fordiv(n,d,m *= A019565(A289814(d))); (m); };

a(n) = A019565(A289814(n)) * A293222(n).

A379495 a(n) = A019565(A001065(n)), where A019565 is the base-2 exp-function, and A001065 is the sum of proper divisors of n.

Original entry on oeis.org

1, 2, 2, 6, 2, 15, 2, 30, 5, 7, 2, 11, 2, 21, 14, 210, 2, 110, 2, 165, 42, 105, 2, 65, 15, 11, 70, 385, 2, 273, 2, 2310, 210, 55, 70, 4290, 2, 165, 22, 429, 2, 2145, 2, 91, 26, 231, 2, 595, 7, 546, 110, 1365, 2, 51, 22, 17, 330, 13, 2, 7735, 2, 39, 182, 30030, 66, 1785, 2, 3003, 462, 357, 2, 102102, 2, 91, 286, 17, 66

Offset: 1

Antti Karttunen, Jan 05 2025

nonn

Cf. A001065, A019565, A379496, A379501.

Cf. also A379493.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A379495(n) = A019565(sigma(n)-n);

A379501 a(n) = (3/2)*A019565((2n-1)^2) - A019565(A001065((2n-1)^2)), where A019565 is the base-2 exp-function, and A001065 is the sum of proper divisors of n.

Original entry on oeis.org

2, 16, 216, 422, 470, 51016, 5082, 4446, 864, 106688, 1301846, 880, 204182, 1985872, 236964, 646310, 1030, 176778, 2799756, 96178962, 563400, 62092576, 1566805968, 27274, 559406, -16252236, 1040774592, 263042394, 7794826, 115781204, 13256922, -16386856, -1230440, 376172, -67188814, 222905278, 13547232, 28352541646

Offset: 1

Antti Karttunen, Jan 05 2025

sign

Cf. A001065, A016754, A019565, A379495, A379496.

Cf. also A378231, A337339.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A379501(n) = { my(osq=((2*n)-1)^2); ((3/2)*A019565(osq) - A019565(sigma(osq)-osq)); };

a(n) = A379496(A016754(n)) = A019565(1+A016754(n)) - A379495(A016754(n)).

a(n) = (3/2)*A019565(A016754(n)) - A379495(A016754(n)).

A103790 a(n) = the minimum k that makes prime(n)+A019565(k) prime.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 5, 3, 1, 3, 3, 1, 5, 3, 3, 1, 3, 3, 1, 3, 5, 3, 9, 3, 1, 3, 1, 7, 9, 5, 3, 1, 5, 1, 3, 3, 5, 3, 3, 1, 5, 1, 3, 1, 7, 7, 3, 1, 5, 3, 1, 5, 3, 3, 3, 1, 3, 3, 1, 5, 9, 3, 1, 13, 7, 3, 5, 1, 5, 3, 7, 3, 3, 5, 3, 7, 11, 7, 5, 1, 5, 1, 3, 5, 3, 7, 3, 1, 15, 11, 7, 13, 7, 5, 3, 9, 1, 13, 3, 5, 3, 3

Offset: 1

Lei Zhou, Feb 16 2005

All elements except the first one are odd. This suggests a new way looking for large primes candidates.

			Prime(1)+A019565(0)=2+1=3 is prime, so a(1)=0;
Prime(4)+A019565(3)=7+6=13 is prime, so a(4)=3;

Cf. A019565.

A019565 = Function[tn, k1 = tn; o = 1; tt = 1; While[k1 > 0, k2 = Mod[k1, 2]; If[k2 == 1, tt = tt*Prime[o]]; k1 = (k1 - k2)/2; o = o + 1]; tt]; Do[npd = Prime[n]; ts = 1; tt = ts; cp = npd + A019565[tt]; While[ ! (PrimeQ[cp]), ts = ts + 1; tt = ts; cp = npd + A019565[ tt]]; Print[ts], {n, 3, 200} ]

A103797 Indices of n such that A019565(n)-1 is prime.

Original entry on oeis.org

2, 3, 7, 9, 11, 21, 27, 29, 31, 39, 41, 47, 51, 53, 55, 61, 63, 67, 71, 81, 89, 91, 107, 113, 121, 123, 129, 131, 135, 139, 143, 149, 151, 157, 159, 163, 169, 175, 179, 183, 187, 191, 197, 199, 207, 211, 217, 223, 235, 241, 251, 259, 261, 269, 279, 281, 287, 295

Offset: 0

Lei Zhou, Feb 22 2005

nonn

			A019565(0)=1, 1-1=0 is not prime;
A019565(1)=2, 2-1=1 is not prime;
A019565(2)=3, 3-1=2 is prime, so a(1)=2.

Cf. A019565, A103796.

A019565 = Function[tn, k1 = tn; o = 1; tt = 1; While[k1 > 0, k2 = Mod[k1, 2]; If[k2 == 1, tt = tt*Prime[o]]; k1 = (k1 - k2)/2; o = o + 1]; tt]; Do[cp = A019565[n] - 1; If[PrimeQ[cp], Print[n]], {n, 0, 1000}]

A103799 Indices n such that A019565(n)+2 is prime.

Original entry on oeis.org

0, 2, 4, 6, 10, 12, 14, 16, 22, 24, 26, 34, 36, 38, 44, 46, 50, 62, 64, 66, 70, 74, 78, 82, 84, 90, 92, 96, 104, 106, 116, 118, 124, 130, 132, 138, 142, 144, 150, 154, 162, 164, 170, 172, 176, 186, 190, 194, 224, 230, 242, 252, 258, 262, 264, 270, 274, 278, 302, 308

Offset: 1

Lei Zhou, Feb 22 2005

			A019565(0)=1, 1+2=3 is prime, so a(1)=0;
A019565(1)=2, 2+2=4 is not prime,
A019565(2)=3, 3+2=5 is prime, so a(2)=2;

Cf. A019565, A103796, A103797.

A019565 = Function[tn, k1 = tn; o = 1; tt = 1; While[k1 > 0, k2 = Mod[k1, 2]; If[k2 == 1, tt = tt*Prime[o]]; k1 = (k1 - k2)/2; o = o + 1]; tt]; Do[cp = A019565[n] + 2; If[PrimeQ[cp], Print[n]], {n, 0, 1000}]

A288570 Partial sums of A019565.

Original entry on oeis.org

1, 3, 6, 12, 17, 27, 42, 72, 79, 93, 114, 156, 191, 261, 366, 576, 587, 609, 642, 708, 763, 873, 1038, 1368, 1445, 1599, 1830, 2292, 2677, 3447, 4602, 6912, 6925, 6951, 6990, 7068, 7133, 7263, 7458, 7848, 7939, 8121, 8394, 8940, 9395, 10305, 11670, 14400, 14543

Offset: 0

N. J. A. Sloane, Jun 26 2017

N. J. A. Sloane, Table of n, a(n) for n = 0..8191

Cf. A019565.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A288570(n) = sum(i=0,n,A019565(i)); \\ Antti Karttunen, Aug 19 2021

G.f.: (1/(1 - x)) * Product_{k>=0} (1 + prime(k+1)*x^(2^k)). - Ilya Gutkovskiy, Aug 18 2021

A293444 a(n) = A293442(A293442(n)), where A293442 is multiplicative with a(p^e) = A019565(e).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 4, 2, 3, 2, 4, 2, 3, 3, 2, 2, 4, 2, 4, 3, 3, 2, 6, 2, 3, 4, 4, 2, 6, 2, 4, 3, 3, 3, 3, 2, 3, 3, 6, 2, 6, 2, 4, 4, 3, 2, 4, 2, 4, 3, 4, 2, 6, 3, 6, 3, 3, 2, 6, 2, 3, 4, 4, 3, 6, 2, 4, 3, 6, 2, 6, 2, 3, 4, 4, 3, 6, 2, 4, 2, 3, 2, 6, 3, 3, 3, 6, 2, 6, 3, 4, 3, 3, 3, 6, 2, 4, 4, 3, 2, 6, 2, 6, 6

Offset: 1

Antti Karttunen, Oct 31 2017

nonn

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

Cf. A019565, A048675, A064547, A293442.

(define (A293444 n) (A293442 (A293442 n)))

a(n) = A293442(A293442(n)).

A048675(a(n)) = A064547(n) = A001222(A293442(n)).

A303770 Divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A303773(n)).

Original entry on oeis.org

1, 2, 6, 3, 15, 5, 10, 30, 210, 42, 21, 7, 14, 70, 35, 105, 1155, 165, 55, 11, 22, 66, 33, 231, 77, 154, 462, 2310, 330, 110, 770, 385, 5005, 455, 91, 13, 26, 78, 39, 195, 65, 130, 390, 2730, 182, 546, 273, 1365, 15015, 3003, 429, 143, 286, 858, 4290, 1430, 715, 2145, 36465, 561, 51, 17, 34, 102, 510, 170, 85, 255, 1785, 357, 119, 238, 714

Offset: 0

Antti Karttunen, May 05 2018

nonn

Each a(n+1) is either a divisor or a multiple of a(n).

Antti Karttunen, Table of n, a(n) for n = 0..10092

Cf. A005117, A019565, A303773.

Cf. A303760 (a simpler variant).

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A303770(n) = A019565(A303773(n)); \\ Uses also code from A303773.

a(n) = A019565(A303773(n)).

A319680 a(n) = Product_{d|n} A019565(d)^[moebius(n/d) = +1].

Original entry on oeis.org

2, 3, 6, 5, 10, 30, 30, 7, 14, 42, 42, 105, 70, 210, 420, 11, 22, 198, 66, 165, 220, 330, 330, 385, 154, 462, 462, 1155, 770, 207900, 2310, 13, 52, 78, 156, 975, 130, 390, 780, 455, 182, 147420, 546, 1365, 5460, 2730, 2730, 1001, 286, 4290, 1716, 2145, 1430, 30030, 8580, 5005, 4004, 6006, 6006, 7882875, 10010, 30030, 180180, 17, 68

Offset: 1

Antti Karttunen, Sep 29 2018

nonn

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

Cf. A019565, A300831, A319681 (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
A319680(n) = { my(m=1); fordiv(n,d,if((1==moebius(n/d)), m *= A019565(d))); m; };

a(n) = A019565(n) * A300831(n).

cp's OEIS Frontend

A351082 a(n) = Product_{d|n} A019565(A289814(d)); a product obtained from the 2-digits present in ternary expansions of the divisors of n.

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A379495 a(n) = A019565(A001065(n)), where A019565 is the base-2 exp-function, and A001065 is the sum of proper divisors of n.

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A379501 a(n) = (3/2)*A019565((2n-1)^2) - A019565(A001065((2n-1)^2)), where A019565 is the base-2 exp-function, and A001065 is the sum of proper divisors of n.

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A103790 a(n) = the minimum k that makes prime(n)+A019565(k) prime.

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Mathematica

A103797 Indices of n such that A019565(n)-1 is prime.

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Mathematica

A103799 Indices n such that A019565(n)+2 is prime.

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Mathematica

A288570 Partial sums of A019565.

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A293444 a(n) = A293442(A293442(n)), where A293442 is multiplicative with a(p^e) = A019565(e).

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A303770 Divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A303773(n)).

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Formula

A319680 a(n) = Product_{d|n} A019565(d)^[moebius(n/d) = +1].