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.

Showing 1-7 of 7 results.

A349259 Numbers where A349258 reaches a record value.

Original entry on oeis.org

1, 2, 6, 8, 24, 120, 128, 384, 1920, 3456, 17280, 32768, 98304, 491520, 884736, 4423680, 30965760, 71663616, 358318080, 2147483648, 6442450944, 32212254720, 57982058496, 289910292480, 2029372047360, 4696546738176, 23482733690880, 164379135836160, 587068342272000
Offset: 1

Views

Author

Amiram Eldar, Nov 12 2021

Keywords

Comments

The corresponding record values are 0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, ... (see the link for more values).

Examples

			The first 6 terms of A349258 are 0, 1, 1, 1, 1 and 2, The record values, 0, 1 and 2, occur at 1, 2 and 6, the first 3 terms of this sequence.

Links

Crossrefs

Cf. A349258.
Subsequence of A025487.

Programs

  • Mathematica
    f[p_,e_] := 2^DigitCount[e, 2, 1] - 1; c[1] = 0; c[n_] := Plus @@ f @@@ FactorInteger[n]; cm = -1; s = {}; Do[c1 = c[n]; If[c1 > cm, cm = c1; AppendTo[s, n]], {n, 1, 10^5}]; s

A349260 a(n) is the least number k such that A349258(k) = n.

Original entry on oeis.org

1, 2, 6, 8, 24, 120, 216, 128, 384, 1920, 3456, 17280, 120960, 432000, 279936, 32768, 98304, 491520, 884736, 4423680, 30965760, 110592000, 71663616, 358318080, 2508226560, 8957952000, 62705664000, 689762304000, 3072577536000, 5598720000000, 470184984576, 2147483648
Offset: 0

Views

Author

Amiram Eldar, Nov 12 2021

Keywords

Examples

			a(2) = 6 since A349258(6) = 2 and A349258(k) != 2 for all k < 6.

Links

Crossrefs

Cf. A349258, A349259.

Programs

  • Mathematica
    f[p_, e_] := 2^DigitCount[e, 2, 1] - 1; d[1] = 0; d[n_] := Plus @@ f @@@ FactorInteger[n]; seq[len_, nmax_] := Module[{s = Table[0, {len}], k = 0, n = 1, i}, While[k < len && n < nmax, i = d[n] + 1; If[i <= len && s[[i]] == 0, k++; s[[i]] = n]; n++]; TakeWhile[s, # > 0 &]]; seq[15, 10^6]

A349261 a(n) is the least number k such that A349258(k) = A349258(k+1) = n.

Original entry on oeis.org

2, 14, 125, 135, 2079, 21735, 2730375, 916352, 5955200, 4122495, 444741759, 7391633535, 98228219264
Offset: 1

Views

Author

Amiram Eldar, Nov 12 2021

Keywords

Examples

			2 is a term since A349258(2) = A349258(3) = 1.
14 is a term since A349258(14) = A349258(15) = 2.

Crossrefs

Cf. A349258.
Similar sequences: A075036, A093548, A115186, A343818.

Programs

  • Mathematica
    f[p_, e_] := 2^DigitCount[e, 2, 1] - 1; c[1] = 0; c[n_] := Plus @@ f @@@ FactorInteger[n]; seq[len_, nmax_] := Module[{s = Table[0, {len}], k = 0, n = 1, i}, While[n < nmax && k < len, i = c[n]; If[c[n + 1] == i && i <= len && s[[i]] == 0, k++; s[[i]] = n]; n++]; TakeWhile[s, # > 0 &]]; seq[8, 3*10^6]

A349262 a(n) is the start of the least run of exactly n consecutive numbers with the same value of A349258.

Original entry on oeis.org

1, 14, 20, 2, 91, 6850, 2302, 141, 56014, 184171, 2800171, 27805034, 35297611, 8313366182, 1791416073, 3618621410
Offset: 1

Views

Author

Amiram Eldar, Nov 12 2021

Keywords

Comments

a(17) > 10^11, if it exists.

Examples

			a(2) = 14 since A349258(14) = A349258(15) = 2, but A349258(13) != 2 and A349258(16) != 2.

Crossrefs

Cf. A349258.
Similar sequences: A006558, A045983, A048932, A067813, A077657, A318166.

Programs

  • Mathematica
    f[p_, e_] := 2^DigitCount[e, 2, 1] - 1; d[1] = 0; d[n_] := Plus @@ f @@@ FactorInteger[n]; seq[len_, nmax_] := Module[{s = Table[0, {len}], dprev = 0, n = 2, c = 1, k = 1}, s[[1]] = 1; While[k < len && n < nmax, d1 = d[n]; If[d1 == dprev, c++, If[c > 0 && c <= len && s[[c]] == 0, k++; s[[c]] = n - c]; c = 1]; n++; dprev = d1]; TakeWhile[s, # > 0 &]]; seq[8, 10^4]

A349281 a(n) is the number of prime powers (not including 1) that are (1+e)-divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 3, 4, 2, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 3, 2, 3, 1, 4, 3, 2, 1, 4, 2, 2, 2, 3, 1, 4, 2, 3, 2, 2, 2, 3, 1, 3, 3, 4, 1, 3, 1, 3, 3
Offset: 1

Views

Author

Amiram Eldar, Nov 13 2021

Keywords

Comments

(1+e)-divisors are defined in A049599.
First differs from A106490 at n = 64.
The total number of prime powers (not including 1) that divide n is A001222(n).
If p|n and p^e is the highest power of p that divides n, then the powers of p that are (1+e)-divisors of n are of the form p^d where d|e.

Examples

			8 has 3 (1+e)-divisors, 1, 2 and 8. Two of these divisors, 2 and 8 = 2^3 are prime powers. Therefore, a(8) = 2.

Links

Crossrefs

Cf. A000005, A000961, A001222, A046099, A049599, A077761, A106490, A246655, A349258.

Programs

  • Mathematica
    f[p_,e_] := DivisorSigma[0, e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a,100]
  • PARI
    A349281(n) = vecsum(apply(e->numdiv(e),factor(n)[,2])); \\ Antti Karttunen, Nov 13 2021

Formula

Additive with a(p^e) = A000005(e).
a(n) <= A001222(n), with equality if and only if n is cubefree (A046099).
a(n) <= A049599(n)-1, with equality if and only if n is a prime power (including 1, A000961).
Sum_{k=1..n} a(n) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.51780076119050171903..., where f(x) = -x + (1-x) * Sum_{k>=1} x^k/(1-x^k). - Amiram Eldar, Sep 29 2023

A349326 a(n) is the number of prime powers (not including 1) that are bi-unitary divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 1, 3, 1, 5, 2, 2, 2, 2, 1, 2, 2, 4, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 3, 1, 2, 2, 5, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 2, 3, 1, 4, 3, 2, 1, 3, 2, 2, 2, 4, 1, 3, 2, 2, 2, 2, 2, 6, 1, 2, 2, 2, 1, 3, 1, 4, 3
Offset: 1

Views

Author

Amiram Eldar, Nov 15 2021

Keywords

Comments

The total number of prime powers (not including 1) that divide n is A001222(n).
The least number k such that a(k) = m is A122756(m).

Examples

			12 has 4 bi-unitary divisors, 1, 3, 4 and 12. Two of these divisors, 3 and 4 = 2^2 are prime powers. Therefore a(12) = 2.

Links

Crossrefs

Cf. A000961, A001222, A122756, A162641, A222266, A246655, A286324, A268335, A350390.
Similar sequences: A001222, A349258, A349281.
Cf. A083342, A179119.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], e, e - 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecsum(apply(x -> if(x%2, x, x-1), factor(n)[, 2])); \\ Amiram Eldar, Sep 29 2023

Formula

Additive with a(p^e) = e if e is odd, and e-1 if e is even.
a(n) <= A001222(n), with equality if and only if n is an exponentially odd number (A268335).
a(n) <= A286324(n) - 1, with equality if and only if n is a prime power (including 1, A000961).
a(n) = A001222(n) - A162641(n). - Amiram Eldar, May 18 2023
From Amiram Eldar, Sep 29 2023: (Start)
a(n) = A001222(A350390(n)) (the number of prime factors of the largest exponentially odd number dividing n, counted with multiplicity).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B_2 - C), where B_2 = A083342 and C = A179119. (End)

A372503 The number of prime powers that are noninfinitary divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0
Offset: 1

Views

Author

Amiram Eldar, May 04 2024

Keywords

Comments

First differs from A318499 at n = 32.

Links

Crossrefs

Cf. A000120, A001222, A036537, A048967, A162643, A318499, A349258.

Programs

  • Mathematica
    f[p_, e_] :=  e + 1 - 2^DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecsum(apply(x -> x + 1 - 1 << hammingweight(x), factor(n)[, 2]));

Formula

Additive with a(p^e) = e - 2^A000120(e) + 1 = A048967(e).
a(n) = A001222(n) - A349258(n).
a(n) = 0 if and only if n is in A036537.
a(n) > 0 if and only if n is in A162643.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.4917971717413486467..., where f(x) = 1/(1-x) - (1-x) * Product_{k>=0} (1 + 2*x^(2^k)).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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