A103667 -id:A103667 - cp's OEIS frontend

A103664 Primes p such that the number of divisors of p-1 is less than the number of divisors of p+1.

2, 3, 5, 11, 17, 23, 29, 47, 53, 59, 71, 79, 83, 89, 107, 131, 139, 149, 167, 173, 179, 191, 197, 223, 227, 233, 239, 251, 263, 269, 293, 311, 317, 347, 359, 367, 383, 389, 419, 431, 439, 443, 449, 461, 467, 479, 499, 503, 509, 557, 563, 569, 587, 593, 599, 607

Offset: 1

Hugo Pfoertner, Feb 19 2005

nonn

Mathematica coding by Wouter Meeussen and Robert G. Wilson v.

			a(1)=2 because d(1)=1 < d(3)=2; a(2)=3 because d(2)=2 < d(4)=3.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A008328 number of divisors of p-1, A008329 number of divisors of p+1, A103665, A103666, A103667.

with(numtheory): p:=proc(n) if isprime(n) and tau(n-1)Emeric Deutsch, Feb 22 2005

Select[Prime[Range[1, 140]], Length[Divisors[ # - 1]] < Length[Divisors[ # + 1]] &]
Select[Prime[Range[200]],DivisorSigma[0,#-1]Harvey P. Dale, May 31 2019 *)

A103665 Primes p such that the number of divisors of p-1 is greater than the number of divisors of p+1.

Original entry on oeis.org

13, 31, 37, 43, 61, 67, 73, 97, 101, 109, 113, 127, 151, 157, 163, 181, 193, 211, 229, 241, 257, 271, 277, 281, 283, 313, 331, 337, 353, 373, 379, 397, 401, 409, 421, 433, 457, 463, 487, 521, 523, 541, 547, 571, 577, 601, 613, 617, 631, 641, 661, 673, 677

Offset: 1

Hugo Pfoertner, Feb 19 2005

nonn

Mathematica coding by Wouter Meeussen and Robert G. Wilson v.

			a(1)=13 because d(12)=6 > d(14)=4.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A008328 number of divisors of p-1, A008329 number of divisors of p+1, A103664, A103666, A103667.

Select[Prime[Range[2, 140]], Length[Divisors[ # - 1]] > Length[Divisors[ # + 1]] &]
Select[Prime[Range[200]],DivisorSigma[0,#-1]>DivisorSigma[0,#+1]&] (* Harvey P. Dale, Aug 21 2022 *)

forprime (k=2,700,if(numdiv(k-1)>numdiv(k+1),print1(k,", ")))
\\ Hugo Pfoertner, Nov 30 2017

A103666 Primes p such that the largest prime divisor of p-1 is less than the largest prime divisor of p+1.

Original entry on oeis.org

5, 13, 17, 19, 37, 41, 43, 61, 67, 73, 97, 101, 109, 113, 137, 151, 157, 163, 181, 193, 197, 211, 229, 241, 251, 257, 271, 277, 281, 283, 313, 331, 337, 353, 379, 397, 401, 409, 421, 433, 443, 457, 463, 487, 491, 521, 523, 541, 547, 577, 601, 613, 617, 631

Offset: 1

Hugo Pfoertner, Feb 19 2005

nonn

			a(1)=5 because the largest prime divisor of 4 is less than the largest prime divisor of 6.

Karl Hovekamp, Table of n, a(n) for n=1..39168

Cf. A023503 greatest prime divisor of n-th prime - 1, A023509 greatest prime divisor of n-th prime + 1, A103667.

Select[Prime[Range[2,200]],Max[Transpose[FactorInteger[#-1]][[1]]]< Max[Transpose[FactorInteger[#+1]][[1]]]&]  (* Harvey P. Dale, Apr 26 2011 *)

A174869 a(n) is 0 if n is a power of 2, otherwise the smallest k > 0 such that A006530(n+k) < A006530(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 7, 2, 1, 4, 1, 1, 1, 0, 1, 14, 1, 4, 3, 2, 1, 8, 2, 1, 5, 2, 1, 2, 1, 0, 2, 1, 1, 28, 1, 1, 1, 8, 1, 3, 1, 1, 3, 2, 1, 16, 1, 4, 1, 2, 1, 10, 1, 4, 3, 2, 1, 4, 1, 1, 1, 0, 1, 4, 1, 2, 1, 2, 1, 56, 1, 1, 6, 1, 3, 2, 1, 1, 47, 2, 1, 6, 3, 1, 1, 2, 1, 6, 5, 3, 2, 1, 1, 32, 1, 2, 1, 8, 1, 2, 1

Offset: 1

Vladimir Shevelev, Mar 31 2010

nonn

a(n)=1 if the index n is an odd prime.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A006530, A103667, A023503, A023509, A174857, A020639, A002618, A036689.

A006530 := proc(n) option remember; if n = 1 then 1; else max(op(numtheory[factorset](n)) ) ; end if; end proc:
A174869 := proc(n) if n <= 2 then 0; else gpf := A006530(n) ; if gpf = 2 then 0; else for k from 1 do if A006530(n+k) < gpf then return k; end if; end do: end if; end if; end proc:
seq(A174869(n),n=1..120) ; # R. J. Mathar, Aug 10 2010

Block[{s = Array[FactorInteger[#][[-1, 1]] &, 120]}, Array[If[IntegerQ@ Log2[#], 0, Block[{k = 1, n = s[[#]]}, While[n <= s[[# + k]], k++; If[# + k > Length[s], AppendTo[s, FactorInteger[# + k][[-1, 1]] ]] ]; k]] &, 102, 2]] (* Michael De Vlieger, Apr 06 2021 *)

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A174869(n) = if(!bitand(n,n-1), 0, my(gpf=A006530(n)); for(k=1,oo,if(A006530(n+k)Antti Karttunen, Apr 06 2021

More terms from R. J. Mathar, Aug 10 2010

A180641 Primes P such that P > (largest prime factor of (P-1)) * (largest prime factor of (P+1)).

Original entry on oeis.org

7, 17, 19, 31, 41, 53, 71, 79, 89, 97, 101, 109, 127, 151, 163, 181, 191, 197, 199, 239, 241, 251, 257, 271, 307, 337, 349, 379, 401, 419, 431, 433, 449, 461, 463, 487, 491, 499, 521, 571, 577, 593, 599, 601, 631, 641, 647, 659, 683, 701, 727, 751, 769, 809

Offset: 1

Karl Hovekamp, Sep 14 2010

nonn

			Example: For n = 3, a(3)=19.
The prime P = 19
P-1 = 18 (largest prime factor of 18 is 3)
P+1 = 20 (largest prime factor of 20 is 5)
19 > 3*5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A180640. See also A103666, A103667.

lpfQ[n_]:=Module[{a=FactorInteger[n-1][[-1,1]],b=FactorInteger[n+1][[-1,1]]},n>a*b]; Select[Prime[Range[200]],lpfQ] (* Harvey P. Dale, Aug 16 2013 *)

lpf(n) = {f = factor(n); return (f[#f~, 1]);}
lista(nn) = {forprime(p=3, nn,if ((p > lpf(p-1)*lpf(p+1)), print1(p, ", ");););} \\ Michel Marcus, Jul 25 2013

A180640 Primes P such that P < (largest prime factor of (P-1)) * (largest prime factor of (P+1)).

Original entry on oeis.org

2, 3, 5, 11, 13, 23, 29, 37, 43, 47, 59, 61, 67, 73, 83, 103, 107, 113, 131, 137, 139, 149, 157, 167, 173, 179, 193, 211, 223, 227, 229, 233, 263, 269, 277, 281, 283, 293, 311, 313, 317, 331, 347, 353, 359, 367, 373, 383, 389, 397, 409, 421, 439, 443, 457, 467

Offset: 1

Karl Hovekamp, Sep 14 2010

nonn

			For n = 3, a(3)=11.
The prime P = 11
P-1 = 10 (largest prime factor of 10 is 5)
P+1 = 12 (largest prime factor of 12 is 3)
11 < 5*3.

Cf. A180641. See also A103666, A103667.

Select[Prime[Range[100]],#<(FactorInteger[#-1][[-1,1]] FactorInteger[#+1][[-1,1]])&]  (* Harvey P. Dale, Feb 22 2011 *)

isok(p) = (p==2) || (isprime(p) && (p < vecmax(factor(p-1)[, 1]) * vecmax(factor(p+1)[, 1]))); \\ Michel Marcus, Oct 29 2022

Initial term, i.e., 2, added by Harvey P. Dale, Feb 22 2011

A174870 Odd indices m for which A174869(m) is <>1.

Original entry on oeis.org

1, 9, 21, 25, 27, 33, 45, 57, 75, 77, 81, 85, 91, 93, 105, 115, 117, 121, 125, 133, 135, 141, 145, 147, 165, 169, 171, 175, 177, 187, 189, 201, 205, 213, 217, 221, 225, 231, 235, 243, 245, 247, 253, 261, 273, 275, 289, 297, 301, 315, 325, 333, 343, 345, 355, 357, 361, 363

Offset: 1

Vladimir Shevelev, Mar 31 2010

nonn

All but the first term in the sequence are composite numbers.

			The 9 refers to A174869(9) = 7. The 21 refers to A174869(21) = 3.

Cf. A174869, A006530, A103667, A023503, A023509, A174857, A020639 ,A002618, A036689.

More terms from R. J. Mathar, Aug 10 2010

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A103664 Primes p such that the number of divisors of p-1 is less than the number of divisors of p+1.

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A103665 Primes p such that the number of divisors of p-1 is greater than the number of divisors of p+1.

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A103666 Primes p such that the largest prime divisor of p-1 is less than the largest prime divisor of p+1.

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A174869 a(n) is 0 if n is a power of 2, otherwise the smallest k > 0 such that A006530(n+k) < A006530(n).

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A180641 Primes P such that P > (largest prime factor of (P-1)) * (largest prime factor of (P+1)).

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A180640 Primes P such that P < (largest prime factor of (P-1)) * (largest prime factor of (P+1)).

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A174870 Odd indices m for which A174869(m) is <>1.

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