A324198 -id:A324198 - cp's OEIS frontend

A355442 a(n) = gcd(A003961(n), A276086(n)), where A003961 is fully multiplicative with a(p) = nextprime(p), and A276086 is primorial base exp-function.

1, 3, 1, 9, 1, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5, 9, 1, 25, 1, 3, 5, 3, 1, 5, 1, 3, 125, 9, 1, 7, 1, 3, 1, 3, 7, 5, 1, 3, 5, 63, 1, 5, 1, 3, 175, 3, 1, 5, 1, 21, 5, 9, 1, 125, 7, 3, 5, 3, 1, 7, 1, 3, 1, 9, 7, 5, 1, 3, 5, 21, 1, 25, 1, 3, 245, 9, 1, 5, 1, 21, 125, 3, 1, 5, 7, 3, 5, 9, 1, 7, 1, 3, 1, 3, 7, 5, 1, 3, 5, 441

Offset: 1

Antti Karttunen, Jul 13 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..11550
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial base

Cf. A003961, A020639, A276086, A355001 [smallest prime factor of a(n)], A355456 [= gcd(sigma(n), a(n))], A355692 (Dirichlet inverse), A355820, A355821 (positions of 1's).

Cf. also A322361, A324198, A351459.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355442(n) = gcd(A003961(n), A276086(n));

a(n) = gcd(A003961(n), A276086(n)).

A356302 The least k >= 0 such that n and A276086(n+k) are relatively prime, where A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 20, 0, 0, 0, 0, 15, 0, 0, 0, 0, 10, 3, 0, 0, 0, 5, 0, 3, 0, 0, 0, 0, 0, 3, 0, 175, 0, 0, 0, 3, 20, 0, 168, 0, 0, 15, 0, 0, 0, 161, 10, 3, 0, 0, 0, 5, 154, 3, 0, 0, 0, 0, 0, 147, 0, 0, 0, 0, 0, 3, 140, 0, 0, 0, 0, 15, 0, 2233, 0, 0, 10, 3, 0, 0, 126, 5, 0, 3, 0, 0, 0, 119, 0, 3, 0, 0, 0, 0, 112

Offset: 0

Antti Karttunen, Nov 03 2022

For all nonzero terms, adding a(n) to n in primorial base generates at least one carry. See the formula involving A329041.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A276086, A324198, A356309, A329041.
Cf. A324583 (positions of zeros), A324584 (of nonzeros), A356318 (positions where a(n) > 0 and a multiple of n), A356319 (where 0 < a(n) < n).
Cf. A358213, A358214 (conjectured positions of records and their values).
Cf. also A356303, A356304.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A356302(n) = { my(k=0); while(gcd(A276086(n+k),n)!=1,k++); (k); };

a(n) = A356309(n) - n.

If a(n) > 0, then A000035(a(n)) = A000035(n) and A329041(n, a(n)) > 1.

A324584 Numbers n that share a prime factor with A276086(n).

Original entry on oeis.org

3, 9, 10, 15, 20, 21, 25, 27, 33, 35, 39, 40, 42, 45, 49, 50, 51, 55, 56, 57, 63, 69, 70, 75, 77, 80, 81, 84, 85, 87, 91, 93, 98, 99, 100, 105, 110, 111, 112, 115, 117, 119, 123, 126, 129, 130, 133, 135, 140, 141, 145, 147, 153, 154, 159, 160, 161, 165, 168, 170, 171, 175, 177, 182, 183, 189, 190, 195, 196, 200, 201, 203, 205

Offset: 1

Antti Karttunen, Mar 10 2019

nonn

Numbers n for which A324198(n) <> 1.

Cf. A276086, A324198.

Cf. A324583 (complement).

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324198(n) = gcd(n,A276086(n));
for(n=1, oo, if(1!=A324198(n), print1(n, ", ")));

A328386 a(n) = A276086(n) mod n.

Original entry on oeis.org

0, 1, 0, 1, 3, 5, 3, 7, 3, 5, 2, 1, 11, 5, 0, 1, 8, 17, 3, 15, 15, 3, 19, 1, 0, 3, 24, 25, 27, 7, 14, 21, 9, 29, 21, 35, 33, 29, 15, 35, 15, 7, 6, 41, 15, 11, 1, 11, 35, 25, 48, 23, 9, 1, 5, 21, 30, 51, 44, 49, 37, 23, 42, 57, 37, 47, 21, 55, 21, 35, 8, 1, 41, 49, 0, 5, 28, 41, 5, 55, 57, 21, 26, 49, 50, 27, 6, 9, 73, 73, 49

Offset: 1

Antti Karttunen, Oct 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..30030

Cf. A276086, A324198, A328382, A328387 (positions of zeros).

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Array[Mod[Apply[Times, Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #}] &@ IntegerDigits[#, b], #] &, 91]] (* Michael De Vlieger, Oct 15 2019 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328386(n) = (A276086(n)%n);

a(n) = A276086(n) mod n.

A351083 a(n) = gcd(n, A003415(A276086(n))), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 7, 8, 1, 1, 1, 2, 1, 1, 5, 16, 1, 3, 1, 10, 1, 1, 1, 4, 25, 1, 1, 2, 1, 1, 1, 2, 1, 17, 5, 12, 1, 1, 13, 2, 1, 1, 1, 4, 5, 1, 1, 2, 1, 25, 1, 4, 1, 3, 5, 2, 1, 1, 1, 2, 1, 1, 7, 4, 1, 1, 1, 2, 1, 7, 1, 24, 1, 1, 5, 2, 7, 1, 1, 80, 1, 1, 1, 14, 5, 1, 1, 8, 1, 3, 91, 4, 1, 1, 1, 2, 1, 49, 1, 4

Offset: 0

Antti Karttunen, Feb 03 2022

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003415, A276086, A324198, A327860, A328572, A351080, A351084, A351087 (fixed points), A354823 (Dirichlet inverse), A373145, A373599 (indices of multiples of 3 in this sequence).
Cf. A166486 (a(n) mod 2, parity of terms, see comment in A327860).
Cf. also A345000.

Array[Block[{i, m, n = #, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; GCD[#, If[m < 2, 0, m Total[#2/#1 & @@@ FactorInteger[m]]]]] &, 101, 0] (* Michael De Vlieger, Feb 04 2022 *)

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A351083(n) = gcd(n, A327860(n));

a(n) = gcd(n, A327860(n)) = gcd(n, A003415(A276086(n))).

a(n) = A373145(A276086(n)). - Antti Karttunen, Jun 18 2024

A351251 Denominator of n / A276086(n).

Original entry on oeis.org

1, 2, 3, 2, 9, 18, 5, 10, 15, 10, 9, 90, 25, 50, 75, 10, 225, 450, 125, 250, 75, 250, 1125, 2250, 625, 50, 1875, 1250, 5625, 11250, 7, 14, 21, 14, 63, 18, 35, 70, 105, 70, 63, 630, 25, 350, 525, 70, 1575, 3150, 875, 250, 105, 1750, 7875, 15750, 4375, 1750, 1875, 8750, 39375, 78750, 49, 98, 147, 14, 441, 882, 245, 490

Offset: 0

Antti Karttunen, Feb 05 2022

Antti Karttunen, Table of n, a(n) for n = 0..11550

Cf. A276086, A324198, A351250 (numerators), A351253.

Cf. also A351231.

Array[Block[{i, m, n = #, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; Denominator[#/m]] &, 68, 0] (* Michael De Vlieger, Feb 06 2022 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351251(n) = denominator(n/A276086(n));

a(n) = A276086(n) / gcd(n,A276086(n)) = A276086(n) / A324198(n).

a(n) = A276086(A351253(n)).

A328387 Numbers k such that A276086(k) is a multiple of k.

Original entry on oeis.org

1, 3, 15, 25, 75, 105, 147, 175, 343, 385, 525, 539, 625, 735, 825, 1029, 1155, 1225, 1331, 1375, 1617, 1815, 2695, 3003, 3025, 3675, 3773, 3993, 4375, 5005, 5145, 5577, 5775, 6655, 6825, 8085, 8125, 8281, 8575, 9075, 9555, 9625, 10725, 11011, 11319, 12675, 12705, 13013, 13377, 15015, 15379, 15925, 17303, 17745, 17787, 17875

Offset: 1

Antti Karttunen, Oct 15 2019

nonn

All terms are odd. Of the first 3003 terms, 1709 are multiples of five.

Antti Karttunen, Table of n, a(n) for n = 1..3003

Cf. A276086, A324198.
Indices of 0's in A328386. Indices of 1's in A351250.
Subsequence of A048103 and of A358226.
Cf. also A370114, A358231.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA328387(n) = (0==(A276086(n)%n));

A364500 a(n) = gcd(n, A005940(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 8, 1, 10, 1, 12, 1, 2, 3, 16, 1, 2, 1, 20, 7, 2, 1, 24, 1, 2, 3, 4, 1, 6, 1, 32, 1, 2, 1, 4, 1, 2, 3, 40, 1, 14, 1, 4, 5, 2, 1, 48, 1, 2, 3, 4, 1, 6, 5, 8, 1, 2, 1, 12, 1, 2, 9, 64, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 15, 4, 11, 6, 1, 80, 1, 2, 1, 28, 5, 2, 3, 8, 1, 10, 7, 4, 1, 2, 5, 96, 1, 2, 33, 4

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A005940, A364499, A364501, A364502.

Cf. also A324198, A339969, A364255.

nn = 100; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Array[GCD[a[#], #] &, nn] (* Michael De Vlieger, Jul 28 2023 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A364500(n) = gcd(n, A005940(n));

A364500(n) = { my(orgn=n,p=2,rl=0,z=1); n--; while(n, if(!(n%2), p=nextprime(1+p), rl++; if(1==(n%4), z *= p^min(rl,valuation(orgn,p)); rl=0)); n>>=1); (z); };

a(n) = gcd(n, A364499(n)) = gcd(A005940(n), A364499(n)).

a(n) = n / A364501(n) = A005940(n) / A364502(n).

A379486 Numbers k for which gcd(k,A003961(k))gcd(sigma(k),A276086(k)) is equal to gcd(k,A276086(k))gcd(sigma(k),A003961(k)), where A003961(n) is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 2, 4, 6, 14, 16, 18, 24, 26, 28, 40, 54, 62, 64, 66, 74, 86, 102, 114, 122, 134, 138, 146, 152, 162, 169, 174, 176, 182, 184, 186, 206, 222, 234, 254, 270, 280, 282, 289, 290, 302, 304, 306, 308, 314, 318, 326, 338, 342, 354, 360, 361, 366, 368, 380, 384, 386, 402, 414, 422, 426, 434, 438, 441, 446, 448, 456, 474, 496

Offset: 1

Antti Karttunen, Jan 01 2025

nonn

Cf. A000203, A003961, A276086, A322361, A324198, A324644, A342671, A379485 (characteristic function), A379487, A379488.
Positions of 0's in A379489.
Cf. A379491 (subsequence, terms that are multiperfect numbers, A007691).

PARI
```
is_A379486 = A379485;
```

{Numbers k such that A379487(k) = A379488(k)}.

{Numbers k such that A322361(k)/A324198(k) = A324644(k)/A342671(k)}.

A324577 a(n) = A025487(n) * A324576(n) = A025487(n) * A276086(A025487(n)).

Original entry on oeis.org

2, 6, 36, 30, 120, 300, 3600, 15000, 210, 672, 1260, 42000, 2940, 28224, 88200, 164640, 288120, 4609920, 216090000, 21176820, 564715200, 2310, 11880, 18480, 4435200, 19404000, 66555720, 44370480000, 50820, 1306800, 2845920, 63748608, 5856903360, 328703760, 306790176000, 12298440, 7906140000, 645668100, 33746919360, 15874550866944

Offset: 1

Antti Karttunen, Mar 09 2019

nonn

Note that A324198(A025487(n)) = gcd(A025487(n), A324576(n)) = 1 for all n, because each term of A025487 is a product of primorials.

Cf. A025487, A276086, A324198, A324576, A324579, A324580.

Cf. also A324582 (a subsequence).

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324577(n) = A025487(n)*A276086(A025487(n));

a(n) = A025487(n) * A324576(n) = A025487(n) * A276086(A025487(n)).

a(n) = A324580(A025487(n)).

cp's OEIS Frontend

A355442 a(n) = gcd(A003961(n), A276086(n)), where A003961 is fully multiplicative with a(p) = nextprime(p), and A276086 is primorial base exp-function.

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A356302 The least k >= 0 such that n and A276086(n+k) are relatively prime, where A276086 is the primorial base exp-function.

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A324584 Numbers n that share a prime factor with A276086(n).

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A328386 a(n) = A276086(n) mod n.

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A351083 a(n) = gcd(n, A003415(A276086(n))), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

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A351251 Denominator of n / A276086(n).

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A328387 Numbers k such that A276086(k) is a multiple of k.

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A364500 a(n) = gcd(n, A005940(n)).

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A379486 Numbers k for which gcd(k,A003961(k))*gcd(sigma(k),A276086(k)) is equal to gcd(k,A276086(k))*gcd(sigma(k),A003961(k)), where A003961(n) is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.

A379486 Numbers k for which gcd(k,A003961(k))gcd(sigma(k),A276086(k)) is equal to gcd(k,A276086(k))gcd(sigma(k),A003961(k)), where A003961(n) is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.