Carlos Eduardo Olivieri - cp's OEIS frontend

A273323 a(n) = greatest number k <= n^2 having exactly n divisors, or 0 if no such k exists.

1, 3, 9, 15, 16, 32, 0, 56, 36, 80, 0, 140, 0, 192, 144, 216, 0, 300, 0, 336, 0, 0, 0, 540, 0, 0, 0, 0, 0, 720, 0, 840, 0, 0, 0, 1260, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Carlos Eduardo Olivieri, May 20 2016

The sequence is zero for n>36, since A005179(n) > n^2 for all n > 36.

			a(4) = 15, because 15 is the greatest number <= 4^2 with exactly 4 divisors.

Cf. A000005, A005179, A035033.

Seq := {}; For[n = 1, n < 50, n++, AppendTo[Seq, a = Max[Select[Range[n^2], DivisorSigma[0, #] == n &]]; If[a == -Infinity, 0, a]]]; Seq

A264786 Let { d_1, d_2, ..., d_k } be the divisors of n. Then a(n) = d_k^1 + d_(k-1)^2 + ... + d_1^k.

Original entry on oeis.org

1, 3, 4, 9, 6, 24, 8, 33, 19, 44, 12, 226, 14, 72, 68, 161, 18, 429, 20, 534, 98, 152, 24, 3858, 51, 204, 136, 856, 30, 6534, 32, 1089, 182, 332, 210, 22965, 38, 408, 236, 12886, 42, 14262, 44, 2148, 1868, 584, 48, 128338, 99, 2333, 368, 3214, 54, 21810, 302

Offset: 1

Carlos Eduardo Olivieri, Nov 29 2015

			For n = 4: a(4) = 4^1 + 2^2 + 1^3 = 9.
For n = 5: a(5) = 5^1 + 1^2 = 6.
For n = 6: a(6) = 6^1 + 3^2 + 2^3 + 1^4 = 24.

Carlos Eduardo Olivieri, A polynomial from divisors of n

Cf. A027750, A180851.

a[n_] := Sum[Sort[Divisors[n], #1 > #2 &][[i]]^i, {i, DivisorSigma[0, n]}]; Table[a[n], {n, 60}]

a(n) = my(d = divisors(n)); sum(k=1, #d, d[k]^(#d-k+1)); \\ Michel Marcus, Jan 01 2016

A261256 Let S_k denote the sequence of numbers j such that A001222(j) - A001221(j) = k. Then a(n) is the n-th term of S_n.

Original entry on oeis.org

4, 24, 72, 160, 432, 896, 2592, 5632, 12800, 26624, 61440, 124416, 278528, 622592, 1376256, 2949120, 5971968, 12058624, 25690112, 60817408, 130023424, 262144000, 528482304, 1107296256, 2264924160, 4586471424, 9395240960, 19864223744, 40265318400, 83751862272

Offset: 1

Carlos Eduardo Olivieri, Aug 12 2015

nonn

S_0 would correspond to the squarefree numbers (A005117), that is, numbers j such that A001222(j) = A001221(j). Note that S_0 is excluded from the scheme. - Michel Marcus, Sep 21 2015

			For n = 1, S_1 = {4, 9, 12, 18, 20, 25, ...}, so a(1) = S_1(1) = 4.
For n = 2, S_2 = {8, 24, 27, 36, 40, 54, ...}, so a(2) = S_2(2) = 24.
For n = 3, S_3 = {16, 48, 72, 80, 81, 108, ...}, so a(3) = S_3(3) = 72.
For n = 4, S_4 = {32, 96, 144, 160, 216, 224, ...}, so a(4) = S_4(4) = 160.
For n = 5, S_5 = {64, 192, 288, 320, 432, 448, ...}, so a(5) = S_5(5) = 432.

Charlie Neder, Table of n, a(n) for n = 1..500

Cf. A001221, A001222.
Cf. A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093.
Cf. A046660, A257851, A264959.

a261256 n = a257851 n (n - 1)  -- Reinhard Zumkeller, Nov 29 2015

OutSeq = {}; For[i = 1, i <= 16, i++, l = Select[Range[10^2*2^i], PrimeOmega[#] - PrimeNu[#] == i &]; AppendTo[OutSeq, l[[i]]]]; OutSeq

a(n) = {ik = 1; nbk = 0; while (nbk != n, ik++; if (bigomega(ik) == omega(ik) + n, nbk++);); ik;} \\ Michel Marcus, Oct 06 2015

a(n+1) > 2*a(n).
a(n) >= 2^prime(n) for n < 5.
a(n) = A257851(n,n-1). - Reinhard Zumkeller, Nov 29 2015
a(n) = b(n)*2^(n+1), where b(n) consists of the values of k/2^excess(k) over odd k, sorted in ascending order. In particular, a(n) <= prime(n)*2^(n+1), with equality only when n = 2. - Charlie Neder, Jan 31 2019

a(17)-a(21) from Jon E. Schoenfield, Sep 12 2015

More terms from Charlie Neder, Jan 31 2019

A260985 Numbers k such that A001222(k) - A001221(k) is an odd prime.

Original entry on oeis.org

16, 48, 64, 72, 80, 81, 108, 112, 162, 176, 192, 200, 208, 240, 256, 272, 288, 304, 320, 336, 360, 368, 392, 405, 432, 448, 464, 496, 500, 504, 528, 540, 560, 567, 592, 600, 624, 625, 648, 656, 675, 688, 704, 729, 752, 756, 768, 792, 800, 810, 816, 832, 848

Offset: 1

Carlos Eduardo Olivieri, Aug 06 2015

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0626525..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

			16 is in the sequence because A001222(16) - A001221(16) = 3.
80 is in the sequence because A001222(80) - A001221(80) = 3.
192 is in the sequence because A001222(192) - A001221(192) = 5.

Cf. A001221, A001222, A046660, A059956, A112526.
Subsequence of A013929.
Subsequences: A195087, A195089, A195091.

Select[Range[10^3], !PrimeQ[#] && PrimeQ[p = PrimeOmega[#] - PrimeNu[#]] && OddQ[p] &]

isok(n) = (d=bigomega(n)-omega(n)) && (d != 2) && isprime(d); \\ Michel Marcus, Aug 07 2015

from sympy import isprime, primefactors
def omega(n): return 0 if n==1 else len(primefactors(n))
def bigomega(n): return 0 if n==1 else bigomega(n//min(primefactors(n))) + 1
def ok(n):
    d = bigomega(n) - omega(n)
    return d%2 and isprime(d)
print([n for n in range(1, 1001) if ok(n)]) # Indranil Ghosh, Apr 25 2017

A259172 Numbers in A259145 that are neither prime nor semiprime.

Original entry on oeis.org

561, 595, 1105, 1235, 1245, 1495, 1547, 1885, 2405, 2555, 2717, 2849, 3115, 3495, 3655, 3657, 3689, 3815, 4521, 4795, 4945, 5035, 5385, 5395, 5453, 5457, 5709, 5865, 6083, 6141, 6251, 6285, 6365, 6391, 6501, 6695, 6755, 6969, 7021, 7887, 8113, 8255, 8355

Offset: 1

Carlos Eduardo Olivieri, Jun 19 2015

Regarding the distribution: Let K be the union of primes and semiprimes in A259145. Let S be the set of other terms. The growth rate of the cardinality of S with respect to the cardinality of K is significantly slower. For instance, if we take the first 50000 terms of A259145, about 32.5 percent are contained in S. If we take the first 350000 terms, about 38.2 percent are contained in S.
a(n) that are in A002997 (Carmichael numbers) for a(n) <= 10^6 are 561, 1105, 8911, 10585, 29341, 825265.
a(n) that are in A051015 (Zeisel numbers) for a(n) <= 3*10^6 are 1885, 353977, 2953711.

Carlos Eduardo Olivieri, Table of n, a(n) for n = 1..8906
Eric Naslund, Integers with a predetermined prime factorization, arXiv:1203.2363 [math.NT], 2012.

Cf. A001221, A001358, A002997, A007053, A051015, A125527, A259145.

Subsequence of A000469, A033942, A050384 (conjuctered).

Select[Range[25000], PrimeQ[#^2 - EulerPhi[#]] && PrimeNu[#] > 2 &]

A001221(a(n)) > 2.

A000005(a(n)) = 2^k, k >= 3.

A259145 Numbers k such that k^2 - phi(k) is prime, where phi() is A000010.

Original entry on oeis.org

2, 3, 7, 13, 33, 35, 65, 67, 77, 79, 91, 133, 139, 151, 163, 193, 221, 247, 249, 287, 299, 321, 337, 341, 349, 377, 379, 437, 457, 481, 533, 541, 551, 561, 581, 591, 595, 611, 613, 643, 721, 727, 763, 769, 779, 789, 803, 817, 843, 851, 869, 917, 919, 991

Offset: 1

Carlos Eduardo Olivieri, Jun 19 2015

Conjecture: a(n) is a cyclic number (see A003277) for all n.

A065508 is the subsequence of prime terms. - Michel Marcus, Jun 19 2015

			a(1) = 2, since phi(2) = 1, thus 2^2 - 1 = 3 (prime).
a(3) = 7, since phi(7) = 6, thus 7^2 - 6 = 43 (prime).
a(5) = 33, since phi(33) = 20, thus 33^2 - 20 = 1069 (prime).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A003277, A065508, A258435.

[n: n in [1..1000] | IsPrime(n^2 - EulerPhi(n))]; // Vincenzo Librandi, Jun 21 2015

Select[Range[2000], PrimeQ[#^2 - EulerPhi[#]] &]

main(size)={ v=vector(size); i=0; m=1; while(iAnders Hellström, Jul 08 2015 */

A258436 Primes p of form x^2 - phi(x) such that (p-1)/tau(p-1) is also prime.

Original entry on oeis.org

157, 1069, 61837, 190573, 840109, 1950349, 2485453, 20616397, 38844349, 57648589, 133091053, 144685357, 188582029, 222029869, 276773389, 346282477, 399067213, 472656589, 827175949, 929558797, 1137622957, 1352220109, 1369037389

Offset: 1

Carlos Eduardo Olivieri, May 30 2015

Intersection of A252021 and A258435.

Cf. A252021, A258435.

lst = Table[n^2 - EulerPhi[n], {n, 100000}]; Select[lst, PrimeQ[#] && PrimeQ[ ( # - 1)/DivisorSigma[0, # - 1] ] &]

lista(nn) = {for (n=1, nn, if (isprime(p=n^2-eulerphi(n)) && (pp=p-1) && (type(r=pp/numdiv(pp))=="t_INT") && isprime(r), print1(p, ", ")););} \\ Michel Marcus, Jul 08 2015

A258435 Primes of form x^2 - phi(x) in increasing order.

Original entry on oeis.org

3, 7, 43, 157, 1069, 1201, 4177, 4423, 5869, 6163, 8209, 17581, 19183, 22651, 26407, 37057, 48649, 60793, 61837, 82129, 89137, 102829, 113233, 115981, 121453, 141793, 143263, 190573, 208393, 230929, 283609, 292141, 303097, 314401, 337069

Offset: 1

Carlos Eduardo Olivieri, May 30 2015

			a(1) = 3, because  2^2 - 1 = 3, and 1^2 - 1 = 0 is not a prime.
a(2) = 7, since 3^2 = 9, phi(3) = 2, so 9-2 = 7 (prime).
a(3) = 43, since 7^2 = 49, phi(7) = 6, so 49-6 = 43 (prime).
a(6) = 1201, since 35^2 = 1225, phi(35) = 24, so 1225-24 = 1201 (prime).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subset of A258434.
For phi see A000010.
A074268 is a subsequence. - Michel Marcus, Jun 19 2015
Cf. A259145.

[a: n in [1..1000] | IsPrime(a) where a is n^2-EulerPhi(n) ]; // Vincenzo Librandi, Jun 03 2015

lst = Table[n^2 - EulerPhi[n], {n, 1000}]; Select[lst, PrimeQ]
Select[Table[n^2 - EulerPhi[n], {n, 1000}], PrimeQ] (* Vincenzo Librandi, Jun 03 2015 *)

lista(nn) = {for (n=1, nn, if (isprime(p=n^2 -eulerphi(n)), print1(p, ", ")););} \\ Michel Marcus, Jul 08 2015

More terms from Vincenzo Librandi, Jun 03 2015

Edited by Wolfdieter Lang, Jun 16 2015

A258400 Perfect powers m^k such that m, k and m+k are primes.

Original entry on oeis.org

8, 9, 25, 32, 121, 289, 841, 1681, 2048, 3481, 5041, 10201, 11449, 18769, 22201, 32041, 36481, 38809, 51529, 57121, 72361, 78961, 96721, 120409, 131072, 175561, 185761, 212521, 271441, 323761, 358801, 380689, 410881, 434281, 654481, 674041, 683929, 734449

Offset: 1

Carlos Eduardo Olivieri, May 28 2015

nonn

Necessarily either m or k = 2, thus if a(n) is even, it is a power of 2 with odd prime exponent, otherwise (if a(n) is odd), it is a square of odd prime.
For each term m^k, there will be another k^m.
a(3), a(5), a(11) are of the form n! + 1.
Let F(m,k) = m*k, such that m^k = a(n), so A108605 is a subsequence of F. For example a(1) = 2^3 and F(2,3) = A108605(1).

			a(1) = 8, because 8 = 2^3 and 2+3 = 5.
a(4) = 32, because 32 = 2^5 and 2+5 = 7.
a(5) = 121, because 121 = 11^2 and 11+2 = 13.
a(25) = 131072, because 131072 = 2^17 and 2+17 = 19.

Subsequence of A001597, A000961.

Cf. A000079, A001248, A108605, A109611.

SmallestDivisor[n_] := If[n == 1, 1, Divisors[n][[2]]]; perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; ppl = Select[Range[200000], perfectPowerQ]; base[n_] := ppl[[n]]^(1/exp[n]); exp[n_] := SmallestDivisor[GCD @@ FactorInteger[ppl[[n]]][[All, 2]] ]; pp2l = Table[ {base[n], exp[n]}, {n, Length[ppl]}]; p[n_] := pp2l[[n]][[1]]; q[n_] := pp2l[[n]][[2]]; lt = Select[Range[Length[pp2l]], PrimeQ[p[#]] && PrimeQ[q[#]] && PrimeQ[p[#] + q[#]] &]; ppl[[lt]]
Select[Range[10^6], Length[f = FactorInteger@ #] == 1 && PrimeQ@ f[[1, 2]] && PrimeQ@ Total@ f[[1]] &] (* Giovanni Resta, Jun 23 2015 *)

a(28)-a(38) from Giovanni Resta, Jun 23 2015

A258468 a(n) = lcm(n, n - tau(n)).

Original entry on oeis.org

0, 0, 3, 4, 15, 6, 35, 8, 18, 30, 99, 12, 143, 70, 165, 176, 255, 36, 323, 140, 357, 198, 483, 48, 550, 286, 621, 308, 783, 330, 899, 416, 957, 510, 1085, 108, 1295, 646, 1365, 160, 1599, 714, 1763, 836, 585, 966, 2115, 912, 2254, 1100, 2397, 1196

Offset: 1

Carlos Eduardo Olivieri, May 31 2015

For tau see A000005.

			a(5) = 15, since tau(5) = 2, lcm(5, 3) = 15.
a(7) = 35, since tau(7) = 2, lcm(7, 5) = 35.
a(10) = 30, since tau(10) = 4, lcm (10, 6) = 30.

Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Cf. A000005, A009230, A005563.

Table[LCM[n, n - DivisorSigma[0, n]], {n, 200}]

vector(100, n, lcm(n, n-numdiv(n))) \\ Michel Marcus, May 31 2015

a(n) = lcm(n, n - tau(n)).

a(n) = n * (n - 2) = A005563(n-2) if n is prime.

Edited by Wolfdieter Lang, Jun 16 2015

cp's OEIS Frontend

User: Carlos Eduardo Olivieri

A273323 a(n) = greatest number k <= n^2 having exactly n divisors, or 0 if no such k exists.

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A264786 Let { d_1, d_2, ..., d_k } be the divisors of n. Then a(n) = d_k^1 + d_(k-1)^2 + ... + d_1^k.

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A261256 Let S_k denote the sequence of numbers j such that A001222(j) - A001221(j) = k. Then a(n) is the n-th term of S_n.

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A260985 Numbers k such that A001222(k) - A001221(k) is an odd prime.

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A259172 Numbers in A259145 that are neither prime nor semiprime.

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A259145 Numbers k such that k^2 - phi(k) is prime, where phi() is A000010.

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A258436 Primes p of form x^2 - phi(x) such that (p-1)/tau(p-1) is also prime.

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A258435 Primes of form x^2 - phi(x) in increasing order.

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