A215217 -id:A215217 - cp's OEIS frontend

A308339 Sphenic number indices of A215217.

21, 30, 49, 51, 75, 82, 96, 106, 120, 130, 133, 136, 141, 148, 152, 157, 161, 173, 177, 180, 186, 187, 189, 202, 207, 209, 213, 217, 221, 226, 236, 240, 242, 244, 248, 261, 264, 277, 285, 286, 294, 305, 306, 311, 317, 320, 322, 327, 333, 349, 355, 364, 368

Offset: 1

Peter Dolland, May 20 2019

nonn

A215217 is a subsequence of A007304 (by definition). The index sequence is more compact.

			For n = 3: a(3) = 49 and A215217(3) = 429 = A007304(49).

Peter Dolland, Table of n, a(n) for n = 1..200

Cf. A007304, A215217.

twinLowX [] = []
twinLowX [_] = []
twinLowX (n : (m : ns))
    | m == n + 1 = 1 : (map succ (twinLowX (m : ns)))
    | otherwise = (map succ (twinLowX (m : ns)))
a308339 n = (twinLowX a007304_list) !! (n - 1)
-- Peter Dolland, May 31 2019

A215217(n) = A007304(a(n)).

A066509 a(n) is the first of a triple of consecutive integers, each of which is both the product of three distinct primes and also the product of three primes counted with multiplicity.

Original entry on oeis.org

1309, 1885, 2013, 2665, 3729, 5133, 6061, 6213, 6305, 6477, 6853, 6985, 7257, 7953, 8393, 8533, 8785, 9213, 9453, 9821, 9877, 10281, 10945, 11605, 12453, 12565, 12801, 12857, 12993, 13053, 14133, 14313, 14329, 14465, 14817, 15085, 15265, 15805, 16113, 16133

Offset: 1

Jason Earls, Jan 04 2002

nonn

A subsequence of A052214 and thus of A005238. - M. F. Hasler, Jan 05 2013
Also, the start of pairs of adjacent sphenic twins, i.e., a(n) = A215217(k) such that A215217(k+1) = A215217(k)+1. Therefore these triples might be called "sphenic triples". They form a subsequence of A242606. - M. F. Hasler, May 18 2014
Minimal difference is 4 which occurs at indices n = {316, 547, 566, 604, 666, 695, 821, 874, 979, ...}. - Zak Seidov, Jul 04 2020

			a(5) = 3729 because it along with 3730 and 3731 are all the product of three distinct primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 1309.

Subsequence of A052214 and hence of A005238.

Subsequence of A215217, A007675, A242606 and A168626.

f[n_]:=Last/@FactorInteger[n]=={1,1,1};lst={};Do[If[f[n]&&f[n+1]&&f[n+2],AppendTo[lst,n]],{n,9!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 04 2010 *)
SequencePosition[Table[If[PrimeNu[n]==PrimeOmega[n]==3,1,0],{n,17000}],{1,1,1}][[;;,1]] (* Harvey P. Dale, Feb 28 2025 *)

Trip(n) = { local(f); f=factor(n); if (matsize(f)[1] != 3, return(0)); for(i=1, 3, if (f[i, 2] != 1, return(0))); return(1); } { n=0; for (m=1, 10^10, if (!Trip(m) || !Trip(m+1) || !Trip(m+2), next); write("b066509.txt", n++, " ", m); if (n==1000, return) ) } \\ Harry J. Smith, Feb 19 2010

A066509(n,show_all=0,a=2*3*5,s=[1,1,1]~)={until( !n-- || !a++, until(, factor(a+2)[,2]!=s && (a+=3) && next; factor(a+1)[,2]!=s && (a+=2) && next; factor(a)[,2]==s && break; factor(a+3)[,2]==s && a++ && break; a+=4);show_all && print1(a",")); a} \\ M. F. Hasler, Jan 05 2013

is3dp(n)=my(f=factor(n));matsize(f)==[3,2]&&vecmax(f[,2])==1
list(lim)=my(v=List(),t);forprime(p=17,lim\15, forprime(q=5,min(p-1,lim\3), forprime(r=3,min(q-1,lim\(p*q)), t=p*q*r; if(t%4==1 && is3dp(t+1) && is3dp(t+2), listput(v,t))))); Set(v) \\ Charles R Greathouse IV, Jan 05 2013; updated Jan 22 2025

list(lim)=my(v=List(),ct); forfactored(n=1309,lim\1+2, if(n[2][,2]==[1,1,1]~, if(ct++==3, listput(v,n[1]-2)), ct=0)); Vec(v) \\ Charles R Greathouse IV, Aug 30 2022

a(n) == 1 (mod 4). - Zak Seidov, Mar 31 2020

Definition clarified by Harvey P. Dale, Feb 28 2025

A318896 Numbers k such that k and k+1 are the product of exactly four distinct primes.

Original entry on oeis.org

7314, 8294, 8645, 11570, 13629, 13845, 15105, 15554, 16554, 17390, 17654, 18290, 19005, 20405, 20769, 21489, 22010, 22154, 23001, 23114, 23529, 24530, 24765, 24870, 24969, 25346, 26690, 26894, 26961, 27434, 27965, 28105, 29145, 29210, 29414, 29469, 29666, 30414

Offset: 1

Seiichi Manyama, Sep 05 2018

nonn

This sequence is different from A140078. For example, A140078(4) = 9009 = 3^2 * 7 * 11 * 13 is not a term.

			n | a(n)                    | a(n)+1
--+-------------------------+-------------------------
1 | 7314 = 2 *  3 * 23 * 53 | 7315 = 5 * 7 * 11 *  19
2 | 8294 = 2 * 11 * 13 * 29 | 8295 = 3 * 5 *  7 *  79
3 | 8645 = 5 *  7 * 13 * 19 | 8646 = 2 * 3 * 11 * 131

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Subsequence of A140078.

Cf. A046386, A052215, A215217, A263990, A318964.

is(n) = omega(n)==4 && omega(n+1)==4 && bigomega(n)==4 && bigomega(n+1)==4 \\ Felix Fröhlich, Sep 05 2018

is(n) = factor(n)[, 2]~ == [1, 1, 1, 1] && factor(n+1)[, 2]~ == [1, 1, 1, 1] \\ David A. Corneth, Sep 06 2018

A086263 Smaller of two consecutive squarefree numbers having equal numbers of prime factors.

Original entry on oeis.org

2, 14, 21, 33, 34, 38, 57, 85, 86, 93, 94, 118, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 217, 218, 230, 253, 285, 298, 301, 302, 326, 334, 381, 393, 394, 429, 434, 445, 446, 453, 481, 501, 514, 526, 537, 542, 553, 565, 609, 622, 633, 634

Offset: 1

Reinhard Zumkeller, Jul 14 2003

nonn

a(k) is a term of A075039 iff a(k)+1 = a(k+1).

If a prime divides a(n) then it does not divide a(n) + 1. If a prime divides a(n) + 1, then it does not divide a(n). The sets of prime divisors of a(n) and a(n) + 1 are disjoint. - Torlach Rush, Jan 13 2018

			230 = 2*5*23 and 230+1 = 3*7*11, therefore 230 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A005117, A263990 (2 prime factors), A215217 (3 prime factors), A318896 (4 prime factors), A318964 (5 prime factors), A001221, A001222, A075039.

Select[Range[2, 634], SquareFreeQ[#] && SquareFreeQ[# + 1] && Length[FactorInteger[#]] == Length[FactorInteger[# + 1]] &] (* T. D. Noe, Jun 26 2013 *)
#[[1,1]]&/@Select[Partition[Table[{n,If[SquareFreeQ[n],1,0], PrimeOmega[ n]},{n,700}],2,1],#[[1,2]]==#[[2,2]]==1&&#[[1,3]]==#[[2,3]]&] (* Harvey P. Dale, Dec 13 2014 *)

for(n=1,10^3, if ( issquarefree(n) && issquarefree(n+1) && (omega(n)==omega(n+1)) , print1(n,", "))); \\ Joerg Arndt, Jun 26 2013

A001221(a(n)) = A001222(a(n)) = A001221(a(n)+1) = A001222(a(n)+1).

A318964 Numbers k such that both k and k+1 are the product of exactly five distinct primes.

Original entry on oeis.org

378014, 421134, 483405, 486590, 486794, 489345, 507129, 545258, 549185, 558789, 558830, 634809, 637329, 663585, 667029, 690234, 720290, 776985, 782690, 823745, 824109, 853005, 853034, 855645, 873885, 883245, 892905, 935714, 945230, 968253, 987734, 999005, 1005081, 1013726

Offset: 1

Seiichi Manyama, Sep 06 2018

nonn

			n | a(n)                           | a(n)+1
--+--------------------------------+--------------------------------
1 | 378014 = 2 * 7 * 13 * 31 *  67 | 378015 = 3 *  5 * 11 * 29 * 79
2 | 421134 = 2 * 3 *  7 * 37 * 271 | 421135 = 5 * 11 * 13 * 19 * 31
3 | 483405 = 3 * 5 * 13 * 37 *  67 | 483406 = 2 *  7 * 11 * 43 * 73

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Subsequence of A140079.

Cf. A046387, A052215, A215217, A263990, A318896.

is(n) = omega(n)==5 && omega(n+1)==5 && bigomega(n)==5 && bigomega(n+1)==5 \\ Felix Fröhlich, Sep 06 2018

A215015 Number of sphenic twins up to 10^n.

Original entry on oeis.org

0, 0, 11, 337, 4206, 43330, 417479, 3917508, 36358375, 336046778, 3105465308, 28739218426

Offset: 1

Martin Renner, Aug 06 2012

The sphenic twin pairs {230, 231}, {285, 286}, ... are counted once at a time.

			a(3) = 11 since there are 11 sphenic twins below 10^3 whose smaller members are 230, 285, 429, 434, 609, 645, 741, 805, 902, 969, 986.

Lucas A. Brown, Python program.

Cf. A007304, A215217, A066509, A215218, A215152.

sphQ[n_]:= FactorInteger[n][[;;,2]] == {1, 1, 1}; c = 0; p = 10; q1 = 0; seq = {}; Do[q2 = sphQ[k]; If[q1 && q2, c++]; If[k == p, AppendTo[seq, c]; p*=10]; q1 = q2, {k, 2, 10^5}]; seq (* Amiram Eldar, Dec 26 2019 *)

a(8)-a(10) from Amiram Eldar, Dec 26 2019

a(11)-a(12) from Lucas A. Brown, Feb 12 2024

A215152 Number of sphenic triples up to 10^n.

Original entry on oeis.org

0, 0, 0, 21, 445, 5457, 55576, 527138, 4824694, 43484124, 389855718

Offset: 1

Martin Renner, Aug 06 2012

The sphenic triples {1309, 1310, 1311}, {1885, 1886, 1887}, ... are counted once at a time.

Cf. A007304, A215217, A066509, A215218, A215015.

issemiprime(n) = factor(n)[,2]~ == [1,1];
issphenic(n) = factor(n)[,2]~ == [1,1,1];
list(nmax) = {my(c = 0, pow = 10, lim = 10^nmax/2+1); forstep(k = 1, lim, 2, if(issemiprime(k) && issphenic(2*k-1) && issphenic(2*k+1), c++); if(2*k-1 > pow, print1(c, ", "); pow *= 10));} \\ Amiram Eldar, Jan 15 2025

a(8)-a(10) from Bert Dobbelaere, Jul 15 2023

a(11) from Amiram Eldar, Jan 15 2025

A240716 Both 1 + 6 n and 6 + 35 n are prime.

Original entry on oeis.org

1, 5, 7, 13, 17, 23, 25, 35, 37, 55, 61, 91, 95, 101, 121, 131, 137, 143, 161, 175, 187, 221, 233, 245, 257, 271, 311, 335, 391, 395, 397, 443, 445, 451, 461, 475, 511, 527, 545, 557, 577, 583, 641, 653, 683, 685, 703, 737, 761, 773, 787, 797, 805

Offset: 1

Robert Israel, Apr 10 2014

nonn

The consecutive integers 35 + 210 a(n) and 36 + 210 a(n) are both products of three primes (distinct if n > 1).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A215217.

[n: n in [0..1000] | IsPrime(6*n+1) and IsPrime(35*n+6)]; // Vincenzo Librandi, Jul 01 2014

A240716 := select(t -> andmap(isprime,[1+6*t,6+35*t]),[$1..N]); # Robert Israel, Apr 10 2014

Select[Range[1000], PrimeQ[6 # + 1] && PrimeQ[35 # + 6] &] (* Vincenzo Librandi, Jul 01 2014 *)

A242068 First of two consecutive sphenic numbers with no semiprime between them.

Original entry on oeis.org

102, 170, 230, 238, 255, 282, 285, 366, 399, 429, 430, 434, 438, 598, 602, 606, 609, 615, 638, 642, 645, 651, 663, 741, 759, 805, 822, 826, 854, 902, 935, 969, 986, 1001, 1022, 1030, 1065, 1085, 1086, 1102, 1105, 1130, 1178, 1182, 1221, 1245, 1265, 1295, 1309, 1310, 1334, 1358, 1374, 1406, 1419, 1426, 1434

Offset: 1

Robert Israel, Aug 13 2014

nonn

Sphenic numbers are products of three distinct primes. Semiprimes are products of two primes, not necessarily distinct.

Contains A215217.

			102=2*3*17 and 105=3*5*7 are sphenic numbers, i.e., products of three distinct primes, and neither 103 (a prime) nor 104=2^3*13 is a semiprime, so 102 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..8683

Cf. A001358, A007304, A215217.

N:= 10000: # to get all terms where the next sphenic number <= N
Sphenics:= select(t -> (map(s->s[2],ifactors(t)[2])=[1,1,1]), {$1..N}):
Primes:= select(isprime,{2,seq(2*i+1,i=1..floor(N/2))}):
Semiprimes:= {seq(seq(p*q,q=select(`<=`,Primes,N/p)),p=Primes)}:
map(proc(i) if nops(Semiprimes intersect {$Sphenics[i]..Sphenics[i+1]}) = 0 then Sphenics[i] else NULL fi end proc, [$1..nops(Sphenics)-1]);

sw = Switch[FactorInteger[#][[All, 2]], {1, 1}, {#, 2}, {1, 1, 1}, {#, 3}, _, Nothing]& /@ Range[10^4];
sp = SequencePosition[sw, {{, 3}, {, 3}}][[All, 1]];
sw[[sp]][[All, 1]] (* Jean-François Alcover, Sep 26 2020 *)

cp's OEIS Frontend

A308339 Sphenic number indices of A215217.

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A066509 a(n) is the first of a triple of consecutive integers, each of which is both the product of three distinct primes and also the product of three primes counted with multiplicity.

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A318896 Numbers k such that k and k+1 are the product of exactly four distinct primes.

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A086263 Smaller of two consecutive squarefree numbers having equal numbers of prime factors.

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A318964 Numbers k such that both k and k+1 are the product of exactly five distinct primes.

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A215015 Number of sphenic twins up to 10^n.

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A215152 Number of sphenic triples up to 10^n.

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