A019565 -id:A019565 - cp's OEIS frontend

A329352 a(n) = Product_{d|n} A019565(d)^A010051(n/d).

1, 2, 2, 3, 2, 18, 2, 5, 6, 30, 2, 75, 2, 90, 60, 7, 2, 210, 2, 105, 180, 126, 2, 245, 10, 210, 14, 525, 2, 66150, 2, 11, 252, 66, 300, 1155, 2, 198, 420, 385, 2, 173250, 2, 825, 2940, 990, 2, 847, 30, 3234, 132, 1155, 2, 15246, 420, 2695, 396, 2310, 2, 2223375, 2, 6930, 1540, 13, 700, 64350, 2, 195, 1980, 171990, 2, 5005, 2, 390, 32340, 975, 1260

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

			The divisors of 30 are [1, 2, 3, 5, 6, 10, 15, 30], of which only d = 6, 10 and 15 are such that 30/d is a prime, thus a(n) = A019565(6) * A019565(10) * A019565(15) = 15 * 21 * 210 = 66150.

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

Cf. A010051, A019565, A048675, A069359, A329353 (rgs-transform).
Cf. also A329350.
Differs from A300832 for the first time at n=30, where a(30) = 66150, while A300832(30) = 132300.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A329352(n) = { my(m=1); fordiv(n,d,if(isprime(n/d), m *= A019565(d))); (m); };

a(n) = Product_{d|n} A019565(d)^A010051(n/d).

For all n, A048675(a(n)) = A069359(n).

A339812 Number of prime divisors of (A019565(n) - 1), counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 2, 4, 2, 2, 2, 5, 2, 4, 1, 3, 2, 3, 3, 3, 1, 8, 1, 2, 1, 3, 2, 2, 2, 6, 2, 2, 1, 4, 1, 5, 2, 2, 3, 4, 1, 2, 3, 3, 1, 4, 1, 6, 1, 6, 3, 3, 2, 5, 1, 2, 1, 4, 2, 3, 1, 4, 2, 2, 1, 2, 2, 3, 2, 5, 2, 4, 2, 3, 1, 6, 2, 2, 3, 3, 2, 4, 1, 3, 1, 5, 2, 2, 2, 4, 4, 2, 3, 6, 2, 2, 2, 2, 2

Offset: 1

Antti Karttunen, Dec 18 2020

nonn

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

Cf. A001222, A019565, A339809, A339810.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339812(n) = bigomega(A019565(n)-1);

a(n) = A001222(A339809(n)) = A001222(A019565(n)-1).

a(n) = A001222(A339810(n)).

A339814 The exponent of the highest power of 2 dividing (A019565(2n) - 1).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 3, 1, 5, 1, 2, 2, 1, 7, 1, 2, 1, 6, 1, 1, 4, 1, 2, 1, 2, 1, 5, 3, 1, 2, 1, 4, 1, 2, 1, 1, 2, 1, 3, 1, 4, 1, 2, 2, 1, 4, 1, 2, 1, 4, 1, 1, 5, 1, 2, 1, 2, 1, 4, 3, 1, 2, 1, 1, 3, 1, 2, 2, 1, 3, 1, 4, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 3, 6, 1, 2, 1, 2, 1, 4, 1, 1, 5, 1, 2, 1, 3, 1, 2, 2, 1, 3, 1, 5, 1

Offset: 1

Antti Karttunen, Dec 18 2020

nonn

The 2-adic valuation of A339809(2n).

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

Bisection of A339813.

Cf. A003961, A007814, A019565, A339809, A339815, A339822.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339814(n) = valuation((A019565(2*n)-1),2);

a(n) = A007814(A339809(2*n)) = A007814(A019565(2*n)-1).

a(n) = A007814(A003961(A019565(n))-1).

A339898 a(n) = A019565(2n)-1 mod A000265(phi(A019565(2n))).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1, 2, 0, 2, 4, 4, 1, 5, 9, 14, 0, 2, 1, 2, 0, 2, 4, 5, 7, 8, 9, 14, 10, 32, 9, 29, 0, 0, 0, 0, 1, 2, 0, 2, 1, 0, 4, 4, 3, 11, 4, 14, 1, 2, 0, 2, 7, 5, 3, 2, 0, 2, 4, 14, 6, 20, 34, 14, 0, 2, 4, 5, 24, 20, 16, 23, 28, 41, 9, 29, 112, 68, 24, 74, 3, 11, 19, 5, 27, 2, 58, 14, 16, 50, 84, 119, 388, 356

Offset: 0

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A053575, A019565, A339809, A339821, A339971, A339899, A339901.

Cf. A339973 (positions of zeros).

A000265(n) = (n>>valuation(n,2));
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339898(n) = { my(x=A019565(2*n)); ((x-1)%A000265(eulerphi(x))); };

a(n) = A339809(2*n) modulo A339971(n), where A339971(n) = A053575(A019565(2n)).

A351091 a(n) = Product_{d|A000265(n)} A019565(A289813(d)); a product obtained from the 1-digits present in ternary expansions of the odd divisors of n.

Original entry on oeis.org

2, 2, 6, 2, 6, 6, 4, 2, 30, 6, 10, 6, 60, 4, 90, 2, 10, 30, 4, 6, 36, 10, 6, 6, 12, 60, 210, 4, 14, 90, 84, 2, 210, 10, 84, 30, 140, 4, 18900, 6, 210, 36, 140, 10, 3150, 6, 14, 6, 168, 12, 210, 60, 14, 210, 60, 4, 36, 14, 6, 90, 4, 84, 900, 2, 900, 210, 60, 10, 90, 84, 10, 30, 4, 140, 540, 4, 60, 18900, 4, 6, 2310

Offset: 1

Antti Karttunen, Jan 31 2022

Cf. A000265, A019565, A289813, A351081, A351092, A351093 (rgs-transform).

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); };
A351091(n) = { my(m=1); fordiv(n>>valuation(n,2),d,m *= A019565(A289813(d))); (m); };

a(n) = A351081(A000265(n)).

A351092 a(n) = Product_{d|A000265(n)} A019565(A289814(d)); a product obtained from the 2-digits present in ternary expansions of the odd divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 2, 1, 1, 3, 6, 1, 6, 1, 5, 2, 15, 2, 10, 1, 30, 1, 1, 3, 2, 6, 1, 1, 6, 6, 36, 1, 1, 5, 1, 2, 2, 15, 3, 2, 30, 10, 10, 1, 15, 30, 90, 1, 30, 1, 28, 3, 35, 2, 14, 6, 21, 1, 105, 1, 28, 6, 7, 6, 210, 36, 42, 1, 35, 1, 3150, 5, 420, 1, 105, 2, 1, 2, 2, 15, 12, 3, 6, 2, 6, 30, 3, 10, 1, 10

Offset: 1

Antti Karttunen, Jan 31 2022

Cf. A000265, A019565, A289814, A351082, A351091, A351094 (rgs-transform).

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); };
A351092(n) = { my(m=1); fordiv(n>>valuation(n,2),d,m *= A019565(A289814(d))); (m); };

a(n) = A351082(A000265(n)).

A351559 a(n) = A048675(gcd(sigma(n), A019565(n))).

Original entry on oeis.org

0, 2, 1, 0, 1, 2, 1, 0, 0, 2, 3, 8, 9, 2, 3, 0, 1, 2, 1, 0, 1, 2, 3, 0, 0, 10, 1, 8, 5, 2, 1, 0, 1, 2, 3, 32, 1, 6, 1, 0, 9, 2, 1, 8, 33, 2, 3, 0, 0, 2, 3, 0, 1, 6, 3, 0, 1, 2, 3, 8, 1, 2, 33, 0, 1, 2, 65, 0, 1, 2, 3, 0, 1, 2, 1, 12, 1, 10, 5, 0, 16, 2, 3, 0, 1, 18, 7, 0, 1, 2, 9, 8, 1, 2, 7, 0, 1, 2, 35, 0, 65, 2, 33

Offset: 1

Antti Karttunen, Feb 19 2022

Cf. A000203, A004198, A007947, A019565, A080398, A351557, A351560.

Cf. also A324644, A351558.

Table[If[# == 1, 0, Total[#2*2^PrimePi[#1] & @@@ FactorInteger[#]]/2] &@ GCD[DivisorSigma[1, n], Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[n, 2], 1]], {n, 103}] (* Michael De Vlieger, Feb 20 2022 *)

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A351559(n) = A048675(gcd(sigma(n), A019565(n)));

a(n) = A048675(A351557(n)) = A048675(gcd(sigma(n), A019565(n))).

a(n) = n AND A351560(n), where AND is bitwise-and, A004198.

A376408 a(0) = 1, and for n > 0, a(n) = a(n-1) * A019565(a(n-1)), where A019565 is the base-2 exp-function.

Original entry on oeis.org

1, 2, 6, 90, 353430, 274407373885179150, 2443417474326613595267894539584266773823049253134356678751627846400290750

Offset: 0

Antti Karttunen, Nov 04 2024

nonn

a(7) has 407 digits, and a(8) has 2804 digits.

Like A376406, this satisfies A048675(a(n)) = a(n-1) + A048675(a(n-1)), for all n >= 1, that is, applying A048675 to the terms gives the partial sums shifted right once, A376409. However, unlike A376406, this is not a subsequence of A005117: a(3) = 90 is the first term that is not squarefree. Neither can we say that this is the lexicographically largest of such sequences, as there are also infinite sequences that begin as 1, 2, 6, 120, 38, ... or as 1, 2, 6, 120, 2042040, ... that satisfy the same condition.

Antti Karttunen, Table of n, a(n) for n = 0..7
Index entries for sequences related to binary expansion of n

Cf. A019565, A048675, A376406.
Cf. A376409 (= A048675(a(n)), also partial sums from its second term onward).
Cf. also analogous sequences A002110 (for A276086) and A376400 (for A276076).

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A376408(n) = if(!n,1,my(x=A376408(n-1)); x*A019565(x));

A103785 Primes of the form A019565(2^n-1-k)+A019565(k) with minimum k.

Original entry on oeis.org

3, 7, 31, 211, 2311, 15017, 85091, 1616621, 22309297, 3234846617, 200560490131, 3710369067407, 20283350901829, 872184088778017, 307444891294245707, 775932344695001107, 961380175077106319537, 19548063559901161830551

Offset: 1

Lei Zhou, Feb 15 2005

This sequence can also be defined as: The Primes of the form primorial P(n)/A019565(k)+A019565(k) with minimum k. Conjecture: sequence is defined for any n>=1.

			for n=1, A019565(2^1-1-0)+A019565(0)=2+1=3 is prime, so a(1)=3;
for n=6, A019565(2^6-1-1)+A019565(1)=15015+2=15017 is prime, so a(6)=15017;

Cf. A019565, A002110.

nmax = 2^2048; npd = 1; n = 1; npd = npd*Prime[n]; While[npd < nmax, tn = 0; tt = 1; cp = npd/tt + tt; While[(IntegerQ[cp]) && (! (PrimeQ[cp])), tn = tn + 1; tt = 1; k1 = tn; o = 1; While[k1 > 0, k2 = Mod[k1, 2]; If[k2 == 1, tt = tt*Prime[o]]; k1 = (k1 - k2)/2; o = o + 1]; cp = npd/tt + tt]; Print[cp]; n = n + 1; npd = npd*Prime[n]]

A103787 a(n) = number of k's that make primorial P(n)/A019565(k)+A019565(k) prime, A019565(k)^2<=P(n).

Original entry on oeis.org

1, 2, 4, 8, 12, 21, 40, 70, 117, 263, 450, 703, 1385, 2423, 5501, 8617, 18249, 29352, 61970, 103568, 209309, 404977, 853279, 1609502, 3008915, 5342983, 10287184, 19087437, 38498011, 78520137, 145642314

Offset: 1

Lei Zhou, Feb 15 2005

If we remove the restriction A019565(k)^2<=P(n), every term gets doubled.

Number of distinct primes of the form d + P(n)/d, where P(n) is the n-th primorial A002110(n) and d is a divisor of P(n).

			P(1)=2, A019565(0)=1, 2/1+1=3 is prime, a(1)=1;
P(2)=6, A019565(0)=1, 6/1+1=7; A019565(1)=2, 6/2+2=5; so a(2)=2.

Cf. A002110, A019565, A103785, A103786.

npd = 1; Do[npd = npd*Prime[n]; tn = 0; tt = 1; cp = npd/tt + tt; ct = 0; While[IntegerQ[cp], If[(cp >= (tt*2)) && PrimeQ[cp], ct = ct + 1]; tn = tn + 1; tt = 1; k1 = tn; o = 1; While[k1 > 0, k2 = Mod[k1, 2]; If[k2 == 1, tt = tt*Prime[o]]; k1 = (k1 - k2)/2; o = o + 1]; cp = npd/tt + tt]; Print[ct], {n, 1, 22}]
Table[ps=Prime[Range[n]]; cnt=0; Do[b=IntegerDigits[i,2,n]; p=Times@@(ps^b) + Times@@(ps^(1-b)); If[PrimeQ[p], cnt++], {i,0,2^(n-1)-1}]; cnt, {n,22}]

a(n) = A088627(A002110(n)/2).

a(28)-a(31) from James G. Merickel, Aug 07 2015

cp's OEIS Frontend

A329352 a(n) = Product_{d|n} A019565(d)^A010051(n/d).

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A339812 Number of prime divisors of (A019565(n) - 1), counted with multiplicity.

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A339814 The exponent of the highest power of 2 dividing (A019565(2n) - 1).

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A339898 a(n) = A019565(2n)-1 mod A000265(phi(A019565(2n))).

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A351091 a(n) = Product_{d|A000265(n)} A019565(A289813(d)); a product obtained from the 1-digits present in ternary expansions of the odd divisors of n.

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A351092 a(n) = Product_{d|A000265(n)} A019565(A289814(d)); a product obtained from the 2-digits present in ternary expansions of the odd divisors of n.

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A351559 a(n) = A048675(gcd(sigma(n), A019565(n))).

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A376408 a(0) = 1, and for n > 0, a(n) = a(n-1) * A019565(a(n-1)), where A019565 is the base-2 exp-function.

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A103785 Primes of the form A019565(2^n-1-k)+A019565(k) with minimum k.

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