A111333 -id:A111333 - cp's OEIS frontend

A062875 Records in A046112 (or A006339).

1, 5, 25, 125, 3125, 15625, 390625, 1953125, 48828125, 6103515625, 30517578125, 3814697265625, 95367431640625, 476837158203125, 11920928955078125, 1490116119384765625, 186264514923095703125, 931322574615478515625, 116415321826934814453125

Offset: 1

Eric W. Weisstein

nonn

A111333 gives where records occur in A046112; A102781 gives where records occur in A006339.

Ray Chandler, Table of n, a(n) for n = 1..416 (a(417) exceeds 1000 digits).
Eric Weisstein's World of Mathematics, Circle Lattice Points

Cf. A006339, A046112, A062876, A102781, A111333.

from sympy import prime
def A062875(n): return 5**(prime(n)-1>>1) # Chai Wah Wu, Aug 02 2024

a(n) = A046112(A111333(n)) = A006339(A102781(n)).

a(n) = 5^(A111333(n)-1) = 5^A102781(n).

Edited and extended by Ray Chandler, Jan 05 2012

A062876 Numbers of lattice points corresponding to incrementally largest circle radii in A062875.

Original entry on oeis.org

4, 12, 20, 28, 44, 52, 68, 76, 92, 116, 124, 148, 164, 172, 188, 212, 236, 244, 268, 284, 292, 316, 332, 356, 388, 404, 412, 428, 436, 452, 508, 524, 548, 556, 596, 604, 628, 652, 668, 692, 716, 724, 764, 772, 788, 796, 844, 892, 908, 916, 932, 956, 964

Offset: 1

Eric W. Weisstein

For n = 1 and n >= 3, a(n) is the smallest nonsquarefree number divisible by prime(n). - David James Sycamore, Jun 15 2024

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Circle Lattice Points

Cf. A046112, A062875.
Cf. A017113, A111333.
Cf. A000040, A103929

[4] cat [4*NthPrime(n): n in [2..60]]; // Vincenzo Librandi, May 08 2015

Join[{4}, Table[4 Prime[n], {n, 2, 50}]] (* Vincenzo Librandi, May 08 2015 *)

a(n)=if(n>1,4*prime(n),4) \\ Charles R Greathouse IV, May 08 2015

from sympy import prime
def A062876(n): return prime(n)<<2 if n>1 else 4 # Chai Wah Wu, Aug 02 2024

a(n) = A017113(A111333(n)-1) = 8*A111333(n) - 4.

For n >= 2 a(n) = 4*A000040(n) (a term in A013929). - David James Sycamore, Jun 15 2024

Edited and extended by Ray Chandler, Jan 05 2012

A341351 a(n) = A048673(A181815(n)).

Original entry on oeis.org

1, 2, 5, 3, 14, 8, 41, 23, 4, 122, 13, 68, 11, 365, 38, 203, 32, 1094, 113, 18, 608, 6, 63, 95, 3281, 338, 53, 1823, 17, 188, 284, 9842, 1013, 158, 5468, 50, 563, 25, 851, 29525, 88, 3038, 28, 313, 473, 16403, 149, 1688, 74, 2552, 88574, 263, 9113, 7, 83, 938, 1418, 49208, 446, 5063, 221, 7655, 265721, 788, 27338, 20

Offset: 1

Antti Karttunen and Michael De Vlieger, Feb 10 2021

nonn

Maxima are in A007051 and appear at n in A025488, which are the indices of 2^k in A025487. 2^k is idempotent via A181815 but transformed by A003961 to 3^n, which are rendered by A048673 to (3^n + 1)/2.

Local minima are in A111333 and appear at n in A098719, which are the indices of P(k) = A002110(k) in A025487. P(k) is transformed by A181815 to p_k = A000040(k), which become p_(k+1) under A003961. Therefore these become (p_(k+1)+1)/2 via A048673.

Michael De Vlieger, Annotated logarithmic plot of a(n) for 1 <= n <= A098719(8) accentuating records in red and local minima in blue.
Michael De Vlieger, log-log plot of a(n) for 1 <= n <= A098719(16).
Michael De Vlieger, Notes on this sequence.
Index entries for sequences that are permutations of the natural numbers

Cf. A002110, A025487, A025488, A048673, A098719, A108951, A111333, A181812, A181815.
Cf. A341352 (inverse).
Cf. A007051 (record values).

a025487[n_] := {{1}}~Join~Block[{lim = Product[Prime@ i, {i, n}], ww = NestList[Append[#, 1] &, {1}, n - 1]}, Map[Block[{w = #, k = 1}, Sort@ Prepend[If[Length@ # == 0, #, #[[1]]], Product[Prime@ i, {i, Length@ w}]] &@ Reap[Do[If[# < lim, Sow[#]; k = 1, If[k >= Length@ w, Break[], k++]] &@ Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, #]] &@ Set[w, If[k == 1, MapAt[# + 1 &, w, k], PadLeft[#, Length@ w, First@ #] &@ Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1]]], {i, Infinity}]][[-1]] ] &, ww]]; Map[(1 + If[# == 1, 1, Apply[Times, NextPrime[#1]^#2 & @@@ FactorInteger[#]]])/2 &@ Apply[Times, Prime@ Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &@ If[# == 1, {0}, Function[f, ReplacePart[ConstantArray[0, PrimePi@ f[[-1, 1]] ], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #]] &, Union@ Flatten@ a025487@ 5] (* Michael De Vlieger, Feb 11 2021 *)

PARI
```
A341351(n) = A048673(A181815(n));
```

a(n) = A048673(A181815(n)).

For all n >= 1, A181812(a(n)) = A025487(n).

A359164 Difference between Kimberling's paraphrases and its Möbius transform.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 4, 4, 1, 1, 5, 1, 3, 5, 6, 1, 2, 3, 7, 5, 4, 1, 8, 1, 1, 7, 9, 6, 5, 1, 10, 8, 3, 1, 11, 1, 6, 11, 12, 1, 2, 4, 13, 10, 7, 1, 14, 8, 4, 11, 15, 1, 8, 1, 16, 14, 1, 9, 17, 1, 9, 13, 18, 1, 5, 1, 19, 18, 10, 9, 20, 1, 3, 14, 21, 1, 11, 11, 22, 16, 6, 1, 23, 10, 12, 17, 24, 12

Offset: 1

Antti Karttunen, Dec 22 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65539

Cf. A003602, A008683, A111333, A349136.

Cf. also A349135, A359165.

A003602(n) = (1+(n>>valuation(n,2)))/2;
A349136(n) = sumdiv(n,d,moebius(d)*A003602(n/d));
A359164(n) = (A003602(n) - A349136(n));

a(n) = A003602(n) - A349136(n).
a(n) = Sum_{d|n, dA349136(d).
a(n) = -Sum_{d|n, dA008683(n/d)*A003602(d).
a((2^e)*prime(n)) = A111333(n) for all n >= 1 and e >= 1.

cp's OEIS Frontend

A062875 Records in A046112 (or A006339).

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A062876 Numbers of lattice points corresponding to incrementally largest circle radii in A062875.

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Magma

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A341351 a(n) = A048673(A181815(n)).

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Mathematica

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A359164 Difference between Kimberling's paraphrases and its Möbius transform.

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