A025584 -id:A025584 - cp's OEIS frontend

A378620 Lesser prime index of twin primes with nonsquarefree mean.

2, 5, 7, 17, 20, 28, 35, 41, 43, 45, 49, 52, 57, 64, 69, 81, 83, 98, 109, 120, 140, 144, 152, 171, 173, 176, 178, 182, 190, 206, 215, 225, 230, 236, 253, 256, 262, 277, 286, 294, 296, 302, 307, 315, 318, 323, 336, 346, 373, 377, 390, 395, 405, 428, 430, 444

Offset: 1

Gus Wiseman, Dec 10 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

This is a subset of A029707 (twin prime indices). The other twin primes are A068361, so A029707 is the disjoint union of A068361 and A378620.

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

The lesser of twin primes is A001359, index A029707 (complement A049579).
The greater of twin primes is A006512, index A107770 (complement appears to be A168543).
A subset of A029707 (twin prime lesser indices).
Prime indices of the primes listed by A061368.
Indices of twin primes with squarefree mean are A068361.
A000040 lists the primes, differences A001223, (run-lengths A333254, A373821).
A005117 lists the squarefree numbers, differences A076259.
A006562 finds balanced primes.
A013929 lists the nonsquarefree numbers, differences A078147.
A014574 is the intersection of A006093 and A008864.
A038664 finds the first position of a prime gap of 2n.
A046933 counts composite numbers between primes.
A120327 gives the least nonsquarefree number >= n.
Cf. A007674, A072284, A068360.
Cf. A057627, A061398, A061399, A067535, A070321, A071403, A112925.
Cf. A000720, A025584, A067774, A122535, A155752, A179067.

Select[Range[100],Prime[#]+2==Prime[#+1]&&!SquareFreeQ[Prime[#]+1]&]
PrimePi/@Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&&!SquareFreeQ[Mean[#]]&][[;;,1]] (* Harvey P. Dale, Jul 13 2025 *)

prime(a(n)) = A061368(n).

A062303 Number of ways writing the n-th prime as a sum of two nonprimes.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 5, 6, 7, 8, 9, 9, 11, 13, 14, 15, 16, 17, 18, 19, 21, 24, 25, 26, 26, 27, 27, 33, 34, 36, 37, 40, 41, 42, 44, 45, 47, 49, 50, 53, 54, 54, 55, 59, 64, 65, 66, 66, 68, 69, 72, 74, 76, 78, 79, 80, 81, 82, 85, 91, 92, 93, 93, 99, 101, 105, 106, 106, 108

Offset: 1

Labos Elemer, Jul 05 2001

nonn

			n=10,p(10)=29 has 14 partitions of form a+b=29; 1+28=4+25=8+21=9+20=14+15 are the 5 relevant partitions, so a(10)=5.

Cf. A061358, A062602, A062610, A000040, A014092, A025584.

Table[c = 0; Do[If[i + j == Prime[n] && ! PrimeQ[i] && ! PrimeQ[j], c = c + 1], {i, Prime[n] - 1}, {j, i}]; c, {n, 72}] (* Jayanta Basu, Apr 22 2013 *)
cnpQ[{a_,b_}]:=(!PrimeQ[a]&&CompositeQ[b])||(!PrimeQ[b]&&CompositeQ[a]); Join[{1},Table[Length[Select[IntegerPartitions[Prime[n],{2}],cnpQ]],{n,2,80}]] (* Harvey P. Dale, Sep 30 2018 *)

A062610(A000040(n)) = number of [nonprime+composite] partitions of p(n).

Offset and name corrected by Sean A. Irvine, Mar 25 2023

A083264 Numbers k such that the difference d of the largest and smallest prime factors of k is a composite divisor of k.

Original entry on oeis.org

198, 396, 510, 594, 792, 966, 990, 1020, 1188, 1386, 1530, 1566, 1584, 1782, 1932, 1980, 2040, 2178, 2376, 2550, 2590, 2772, 2898, 2970, 3060, 3132, 3168, 3198, 3564, 3570, 3864, 3960, 4080, 4158, 4230, 4356, 4590, 4698, 4752, 4830, 4950, 5100, 5180

Offset: 1

Labos Elemer, May 12 2003

From David A. Corneth, Jul 14 2018: (Start)
No term k is a perfect power (or 1). If k is a perfect power then it's divisible by 0, a contradiction. Hence a term k has at least two prime factors.
All terms are even. Suppose a term k is odd. Then the smallest prime factor is > 2. Since k has at least two prime factors which are odd, the difference between the largest and smallest prime factor is even hence k is even. A contradiction, hence all terms are even.
All terms are of the form 2 * (p - 2) * p * m where p - 2 is composite, p is prime and m has all, if any, of its prime factors between 2 and p (inclusive). (End)

			198 = 2*3*3*11 = 2*9*11 is in the sequence where d = 11 - 2 = 9 is composite.

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

Cf. A025584, A033845, A071141, A006530, A020639, A083263.

ffi[x_] := Flatten[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; lf[x_] := Length[FactorInteger[x]]; ma[x_] := Max[ba[x]]; mi[x_] := Min[ba[x]] Do[s=ma[ba[n]]-mi[ba[n]]; If[Mod[n, s]==0&&Greater[s, 2]&&!PrimeQ[s], Print[n]], {n, 1, 20000}]
dllpfQ[n_]:=Module[{c=Transpose[FactorInteger[n]][[1]],d},d=Last[c]-First[ c];If[d==0,d=1];Divisible[n,d]&&d>2&&CompositeQ[d]]; Select[ Range[ 6000],dllpfQ] (* Harvey P. Dale, Sep 26 2014 *)

isok(n) = if (n>1, my(f=factor(n)[,1], d = vecmax(f) - vecmin(f)); (d > 1) && !isprime(d) && !(n % d)); \\ Michel Marcus, Jul 09 2018

Solutions to x mod d = 0 where d = A006530(x) - A020639(x) is composite.

Name, Formula, and Example simplified by Jon E. Schoenfield, Jul 14 2018

A201828 The smallest A(m) such that the interval (A(m)n, A(m+1)n) contains exactly one element of A, where A is the sequence of primes p for which p-2 is not prime.

Original entry on oeis.org

37, 37, 2, 2, 2, 2, 907, 2, 2833, 907, 2, 8269, 2749, 2953, 5413, 7699, 2137, 27103, 28513, 74377, 45673, 56629, 79147, 33529, 15259, 96847, 101599, 57649, 44983, 300973, 706309, 715357, 351847, 38557, 308809, 720607, 901447, 2229889, 867253, 2370937, 1276867

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Jan 09 2013

nonn

This sequence is the "A-analog" of A195871.
This is a possible model sequence to understand the role of twin primes in sequences like A195871.  In particular, if after a large number N_tw, there are no twin primes, what primes will take their place in A195871?  Our observations and expectations are expressed in the following conjecture.
Conjecture: For n>=13, every a(n) is the lesser of a pair of cousin primes p and p+4, cf. A023200.  Note that it is only conjectured that there are infinitely many pairs of cousin primes.
The limit of a(n) as n goes to infinity is infinity.

			Let n=2. We have the following intervals of the form (2*p,2*q), where p,q are consecutive primes in A025584:(4,6),(6,22),(22,34),(34,46),(46,58),(58,74),(74,82),..., containing 0,2,2,2,2,3,1,... primes from A025584. The interval (74,82) is the first to contain exactly one prime from A025584, so a(2)=74/2=37.

Cf. A025584, A195871, A023200.

myPrime=Select[#,!PrimeQ[#-2]&]&[Prime[Range[500000]]];  binarySearch[lst_,find_]:=Module[{lo=1,up=Length[lst],v},(While[lo<=up,v=Floor[(lo+up)/2];If[lst[[v]]-find==0,Return[v]];If[lst[[v]]0&]]]+offset-1]];   z=1;(*example for "contains exactly ONE myPrime in the interval"*)Table[myPrime[[NestWhile[#1+1&,1,!((nextMyPrime[n myPrime[[#1]],z]n myPrime[[#1+1]]))&]]],{n,2,30}]

npr(n) = {local(p); p=n+1; while(!isprime(p) || isprime(p-2), p=p+1); p}
cnt(a,b) = {local(r); r=0; for(p=a, b, if(isprime(p) && !isprime(p-2), r=r+1)); r}
a201828(n) = {local(a,b); a=2; b=3; while(cnt(a*n, b*n) != 1, a=b; b=npr(b)); a} \\ Michael B. Porter, Feb 18 2013

cp's OEIS Frontend

A378620 Lesser prime index of twin primes with nonsquarefree mean.

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A062303 Number of ways writing the n-th prime as a sum of two nonprimes.

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A083264 Numbers k such that the difference d of the largest and smallest prime factors of k is a composite divisor of k.

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A201828 The smallest A(m) such that the interval (A(m)*n, A(m+1)*n) contains exactly one element of A, where A is the sequence of primes p for which p-2 is not prime.

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A201828 The smallest A(m) such that the interval (A(m)n, A(m+1)n) contains exactly one element of A, where A is the sequence of primes p for which p-2 is not prime.