cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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

Views

Author

Antti Karttunen, Nov 03 2022

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • PARI
    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); };

Formula

a(n) = A356309(n) - n.
If a(n) > 0, then A000035(a(n)) = A000035(n) and A329041(n, a(n)) > 1.

A356309 The least j >= n such that n and A276086(j) are relatively prime, where A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 2, 6, 4, 5, 6, 7, 8, 12, 30, 11, 12, 13, 14, 30, 16, 17, 18, 19, 30, 24, 22, 23, 24, 30, 26, 30, 28, 29, 30, 31, 32, 36, 34, 210, 36, 37, 38, 42, 60, 41, 210, 43, 44, 60, 46, 47, 48, 210, 60, 54, 52, 53, 54, 60, 210, 60, 58, 59, 60, 61, 62, 210, 64, 65, 66, 67, 68, 72, 210, 71, 72, 73, 74, 90, 76, 2310, 78
Offset: 0

Views

Author

Antti Karttunen, Nov 04 2022

Keywords

Links

Crossrefs

Cf. A276086, A356302, A356313.
Cf. A324583 (positions of the fixed points), A356314 (positions of the terms that are primorial numbers), A356316 (where a(n) is a multiple of n), A356318 (where a nontrivial multiple), A356319 (where n < a(n) < 2*n).

Programs

  • Mathematica
    f[nn_] := Block[{m = 1, i = 1, n = nn, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m]; Array[Block[{k = #}, While[! CoprimeQ[#, f[k]], k++]; k] &, 79, 0] (* Michael De Vlieger, Nov 06 2022, after Jean-François Alcover at A276086 *)
  • PARI
    A356309(n) = (n+A356302(n)); \\ See code in the latter sequence.

Formula

a(n) = n + A356302(n).

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

Views

Author

Antti Karttunen, Mar 10 2019

Keywords

Comments

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

Links

Crossrefs

Cf. A276086, A324198.
Cf. A324583 (complement).

Programs

  • PARI
    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, ", ")));

A324655 a(n) = A000005(A276086(n)).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 2, 4, 4, 8, 6, 12, 3, 6, 6, 12, 9, 18, 4, 8, 8, 16, 12, 24, 5, 10, 10, 20, 15, 30, 2, 4, 4, 8, 6, 12, 4, 8, 8, 16, 12, 24, 6, 12, 12, 24, 18, 36, 8, 16, 16, 32, 24, 48, 10, 20, 20, 40, 30, 60, 3, 6, 6, 12, 9, 18, 6, 12, 12, 24, 18, 36, 9, 18, 18, 36, 27, 54, 12, 24, 24, 48, 36, 72, 15, 30, 30, 60, 45, 90, 4, 8, 8
Offset: 0

Views

Author

Antti Karttunen, Mar 10 2019

Keywords

Comments

Alternative construction: write n down in primorial base (as in A049345, taking care of not mangling digits larger than 9), increment all the digits by one, and multiply together to get a(n). a(0) = 1 either as an empty product, or as a product of any number of 1's. See examples.

Examples

			For n = 11, its primorial base representation is "121" as 11 = 1*A002110(2) + 2*A002110(1) + 1*A002110(0) = 1*6 + 2*2 + 1*1, thus a(11) = (1+1)*(2+1)*(1+1) = 12.
For n = 13, its primorial base representation is "201" as 13 = 2*6 + 0*2 + 1*1, thus a(13) = (2+1)*(0+1)*(1+1) = 6.

Links

Crossrefs

Cf. A000005, A002110 (positions of 2's), A049345, A276086.
Cf. also A267263, A276150, A324650, A324653 for omega, bigomega, phi and sigma analogs.

Programs

  • PARI
    A324655(n) = { my(t=1,m); forprime(p=2, , if(!n, return(t)); m = n%p; t *= (1+m); n = (n-m)/p); };
  • PARI
    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; };
A324655(n) = numdiv(A276086(n));

Formula

a(n) = A000005(A276086(n)).
a(A002110(n)) = 2.

A328233 Numbers n such that the arithmetic derivative of A276086(n) is prime.

Original entry on oeis.org

3, 7, 9, 33, 37, 38, 211, 213, 218, 241, 242, 246, 247, 249, 2313, 2317, 2319, 2341, 2342, 2346, 2521, 2523, 2526, 2529, 2550, 2553, 2559, 30031, 30038, 30039, 30061, 30062, 30063, 30066, 30069, 30242, 30243, 30249, 30270, 30278, 30279, 32341, 32342, 32347, 32370, 32373, 32377, 32379, 32551, 32553, 510513, 510518, 510519
Offset: 1

Views

Author

Antti Karttunen, Oct 09 2019

Keywords

Comments

Numbers n for which A327860(n) = A003415(A276086(n)) is a prime.
Numbers n such that A276086(n) is in A157037.
Terms come in distinct "batches", where in each batch they are "slightly more" than the nearest primorial (A002110) below. This is explained by the fact that for A276086(n) to be a squarefree (which is the necessary condition for A157037), n's primorial base expansion (A049345) must not contain digits larger than 1. Thus this is a subsequence of A276156.
Numbers n such that A327860(A276086(n)) = A003415(A276087(n)) is a prime [A276087(n) is in A157037] are much rarer: 2, 4, 30, 212, 421, 30045, 510511, 512820, 9729723, ...
For all terms k in this sequence, A327969(k) <= 4, and particularly A327969(k) = 2 when k is a prime. Otherwise, when k is not a prime, but A003415(k) is, A327969(k) = 3, while for other cases (when k is neither prime nor in A157037), we have A327969(k) = 4.

Links

Crossrefs

Cf. A002110, A003415, A051674, A157037, A276086, A327860, A327969, A327978, A328232, A328240.
Subsequence of A276156, of A328116, and of A328242.

Programs

  • PARI
    A327860(n) = { my(m=1, i=0, s=0, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), my(e=((n%nextpr)/pr)); m *= (prime(i)^e); s += (e / prime(i)); n-=(n%nextpr)); pr=nextpr); (s*m); };
isA328233(n) = isprime(A327860(n));

A329029 a(n) = A069359(A276086(n)), where A276086 is the primorial base exp-function and A069359(n) = n * Sum_{p|n} 1/p.

Original entry on oeis.org

0, 1, 1, 5, 3, 15, 1, 7, 8, 31, 24, 93, 5, 35, 40, 155, 120, 465, 25, 175, 200, 775, 600, 2325, 125, 875, 1000, 3875, 3000, 11625, 1, 9, 10, 41, 30, 123, 12, 59, 71, 247, 213, 741, 60, 295, 355, 1235, 1065, 3705, 300, 1475, 1775, 6175, 5325, 18525, 1500, 7375, 8875, 30875, 26625, 92625, 7, 63, 70, 287, 210, 861, 84, 413, 497, 1729
Offset: 0

Views

Author

Antti Karttunen, Nov 07 2019

Keywords

Comments

A380535 gives the indices n where a(n) is a multiple of A053669(n). This can be seen from the formula a(n) = A003557(A276086(n)) * A069359(A328571(n)). The left hand side of the product is a multiple of A053669(n) if and only if A276088(n) > 1, while the right hand side is never a multiple of A053669(n), as it is equal to A329031(n) = A003415(A007947(A276086(n))). - Antti Karttunen, Feb 11 2025

Links

Crossrefs

Cf. A003415, A003557, A053669, A069359, A276086, A276088, A328571, A328572, A329031, A380535.
Coincides with A327860 on the positions given by A276156.

Programs

  • PARI
    A329029(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); if(e, m *= (p^e); s += (1/p)); n = n\p; p = nextprime(1+p)); (s*m); };
  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A069359(n) = (n*sumdiv(n, d, isprime(d)/d));
A329029(n) = A069359(A276086(n));

Formula

a(n) = A069359(A276086(n)).
a(n) = A328572(n) * A329031(n) = A003557(A276086(n)) * A069359(A328571(n)). - Antti Karttunen, Feb 11 2025

A328110 Fixed points of A327860: numbers k for which A003415(A276086(k)) = k, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 7, 8, 2556
Offset: 1

Views

Author

Antti Karttunen, Oct 08 2019

Keywords

Comments

Applying A276086 to these terms gives the fixed points of A327859: 1, 2, 10, 15, 5005, ..., i.e., A369650 without any of the terms of A100716.
No more terms below <= 2550136832.
From Antti Karttunen, Feb 09 2024: (Start)
The known five terms are all members of A276156, which is equal to the claim that the intersection of A048103 and A369650 is squarefree. See the example, and also comments in A351088 and in A380527.
Even terms here must be multiples of 4, see comment in A327860.
No terms of A047257 may occur in this sequence, which is equal to the claim that A276086(a(n)) is never a multiple of 9. See comment in A327859.
(End)

Examples

			Computing A327860(2556) is easy, because it is a member of A276156, as 2556 = 6 + 30 + 210 + 2310. Therefore A327860(2556) = A003415(5*7*11*13) = (5*7*11) + (5*7*13) + (5*11*13) + (7*11*13) = 2556, and 2556 is included in this sequence. - _Antti Karttunen_, Feb 04 2024

Crossrefs

Cf. A002110, A002620, A003415, A047257, A048103, A100716, A276085, A276086, A327859, A327860, A369650.
After 0, the intersection of A351087 and A380527, thus like the latter, also this is conjectured to be a subsequence of A276156.
After two initial terms (0 & 1), a subsequence of A328118. Subsequence of A351088.

Programs

  • PARI
    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); };
isA328110(n) = (A327860(n) == 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

Views

Author

Antti Karttunen, Oct 15 2019

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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);

Formula

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

A344592 a(n) = A003557(A276086(A108951(n))).

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 1, 1, 7, 1, 125, 1, 11, 16807, 15, 1, 35, 1, 343, 161051, 13, 1, 25, 9317, 17, 1, 1331, 1, 2401, 1, 1, 371293, 19, 253333223, 42875, 1, 23, 1419857, 1, 1, 1, 1, 2197, 14641, 29, 1, 49, 371293, 6684099653, 2476099, 4913, 1, 55, 37349, 19487171, 6436343, 31, 1, 5929, 1, 37, 20449, 21, 582622237229761, 1792160394037
Offset: 1

Views

Author

Antti Karttunen, May 26 2021

Keywords

Links

Crossrefs

Cf. A003557, A085731, A108951, A276086, A324886, A328572, A329047, A342002, A342920, A346091, A346092 [= A276085(a(n))].
Cf. A344591 (positions of ones), A344593 (rgs-transform).

Programs

  • Mathematica
    Block[{b = MixedRadix[Reverse@ Prime@ Range@ 20]}, Array[#/(Times @@ FactorInteger[#][[All, 1]]) &@ Apply[Times, Power @@@ #] &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[#, b] &@ Apply[Times, Map[(Times @@ Prime@ Range@ PrimePi@ #1)^#2 & @@ # &, FactorInteger[#]]] &, 66]] (* Michael De Vlieger, Jul 14 2021 *)
  • PARI
    A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) }; \\ From A108951
A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };
A344592(n) = A328572(A108951(n));
  • PARI
    A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A344592(n) = A003557(A276086(A108951(n)));

Formula

a(n) = A328572(A108951(n)) = A003557(A324886(n)) = A003557(A276086(A108951(n))).
a(n) = A329047(n) / A342920(n).
a(n) = A085731(A324886(n)) = gcd(A324886(n), A329047(n)) = A324886(n) / A346091(n). - Antti Karttunen, Jul 09 2021

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

Views

Author

Antti Karttunen, Feb 03 2022

Keywords

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • PARI
    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));

Formula

a(n) = gcd(n, A327860(n)) = gcd(n, A003415(A276086(n))).
a(n) = A373145(A276086(n)). - Antti Karttunen, Jun 18 2024
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