A070826 -id:A070826 - cp's OEIS frontend

A136356 Increasing sequence obtained by union of two sequences A136353 and {b(n)}, where b(n) is the smallest composite number m such that m-1 is prime and the set of distinct prime factors of m consists of the first n primes.

4, 6, 9, 15, 30, 105, 420, 1155, 2310, 15015, 30030, 255255, 1021020, 4849845, 19399380, 111546435, 669278610, 9704539845, 38818159380, 100280245065, 601681470390, 14841476269620, 18551845337025, 152125131763605

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A070826 and A118750.

			a(4)=15 because k=2 and prime factors are 3 and 5; 15 is odd and n-2=13, prime.

Cf. A136349-A136355, A136357-A136358, A070826, A118750.

a[n_]:=(c=Product[Prime[k],{k,n}]; For[m=1,!(!PrimeQ[c*m]&&PrimeQ[c*m-1]&&Length[FactorInteger[c*m]]==n),m++ ]; c*m);
b[n_]:=(c=Product[Prime[k],{k,2,n+1}]; For[m=1,!(!PrimeQ[c(2m-1)]&&PrimeQ[c(2m-1)-2]&&Length[FactorInteger[c(2*m-1)]]==n),m++ ]; c(2m-1));
Take[Union[Table[a[k],{k,24}],Table[b[k],{k,24}]],24] (* Farideh Firoozbakht, Aug 13 2009 *)

Edited, corrected and extended by Farideh Firoozbakht, Aug 13 2009

A136357 Increasing sequence obtained by union of two sequences A136354 and {b(n)}, where b(n) is the smallest composite number m such that m+1 is prime and the set of distinct prime factors of m consists of the first n primes.

Original entry on oeis.org

4, 6, 9, 15, 30, 105, 210, 2310, 3465, 15015, 120120, 765765, 4084080, 33948915, 106696590, 334639305, 892371480, 3234846615, 71166625530, 100280245065, 200560490130, 3710369067405, 29682952539240, 1369126185872445

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A070826 and A118750.

			a(4)=15 because k=2 with prime factors 3 and 5 and 15 is followed by 17, prime;
a(5)=30 because k=3 with prime factors 2, 3, 5 and 30 is followed by 31, prime.

Cf. A136349-A136356, A136358, A070826, A118750.

a[n_]:=(c=Product[Prime[k],{k,n}]; For[m=1,!(!PrimeQ[c*m]&&PrimeQ[c*m+1]&& Length[FactorInteger[c*m]]==n),m++ ]; c*m);
b[n_]:=(c=Product[Prime[k],{k,2, n+1}]; For[m=1,!(!PrimeQ[c(2*m-1)]&&PrimeQ[c(2*m-1)+2]&&Length[FactorInteger [c(2*m-1)]]==n),m++ ]; c(2*m-1));
Take[Union[Table[a[k],{k,24}],Table[b[k],{k, 24}]],24] (* Farideh Firoozbakht, Aug 13 2009 *)

Edited, corrected and extended by Farideh Firoozbakht, Aug 13 2009

A164979 Slowest growing sequence of primes having the semiprime-pairwise property: for any i,j, a(i)+a(j) is semiprime.

Original entry on oeis.org

2, 7, 19, 67, 127, 6619, 126127, 345979, 476407, 1658119, 15182459419, 105169832587, 287583971287

Offset: 1

Zak Seidov, Sep 03 2009

By Dirichlet's theorem and Linnik's theorem, a(n) exists for all n. - Charles R Greathouse IV, Jun 03 2025

Subsequence of A045375.

Cf. A001358, A114845, A116656.

lista(pmax) = {my(v = [2], ans); print1(v[1], ", "); forprime(p=3, pmax, ans = 1; for(i=1, #v, if(bigomega(p + v[i]) != 2, ans = 0; break)); if(ans, print1(p, ", "); v=concat(v, p)));} \\ Amiram Eldar, Jun 27 2024

a(n) = A114845(n)/2.

a(n) << A070826(n)^5. - Charles R Greathouse IV, Jun 03 2025

a(12) from Amiram Eldar, Jun 27 2024

a(13) from Jinyuan Wang, May 29 2025

A196529 Half of greatest common divisor of products of first n prime numbers and first n composite numbers.

Original entry on oeis.org

1, 3, 3, 3, 15, 15, 105, 105, 105, 105, 105, 105, 1155, 1155, 1155, 15015, 15015, 15015, 15015, 15015, 15015, 255255, 255255, 255255, 4849845, 4849845, 4849845, 4849845, 4849845, 4849845, 111546435, 111546435, 111546435, 111546435, 111546435, 111546435

Offset: 1

Reinhard Zumkeller, Oct 03 2011

nonn

a(n) = gcd(A002110(n),A036691(n)) / 2.

			a(3) = gcd(2*3*5,4*6*8)/2 = gcd(30,192)/2 = 6/2 = 3;
a(4) = gcd(2*3*5*7,4*6*8*9)/2 = gcd(210,1728)/2 = 6/2 = 3;
a(5) = gcd(2*3*5*7*11,4*6*8*9*10)/2 = gcd(2310,17280)/2 = 30/2 = 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A196527, A070826.

nn=40;With[{prs=Prime[Range[nn]],comps=Take[Complement[Range[Prime[nn]], Prime[ Range[nn]]],nn]},Rest[Table[GCD[Times@@Take[prs,n], Times@@Take[ comps,n]]/2,{n,nn}]]] (* Harvey P. Dale, Oct 16 2011 *)

A258566 Triangle in which n-th row contains all possible products of n-1 of the first n primes in descending order.

Original entry on oeis.org

1, 3, 2, 15, 10, 6, 105, 70, 42, 30, 1155, 770, 462, 330, 210, 15015, 10010, 6006, 4290, 2730, 2310, 255255, 170170, 102102, 72930, 46410, 39270, 30030, 4849845, 3233230, 1939938, 1385670, 881790, 746130, 570570, 510510

Offset: 1

Philippe Deléham, Jun 03 2015

Triangle read by rows, truncated rows of the array in A185973.

Reversal of A077011.

			Triangle begins:
      1;
      3,     2;
     15,    10,    6;
    105,    70,   42,   30;
   1155,   770,  462,  330,  210;
  15015, 10010, 6006, 4290, 2730, 2310;
  ...

Row sums: A024451.
T(n,1) = A070826(n).
T(n,n) = A002110(n-1).
For 2 <= n <= 9, T(n,2) = A118752(n-2). [corrected by Peter Munn, Jan 13 2018]
T(n,k) = A121281(n,k), but the latter has an extra column (0).
Cf. A077011, A185973, A286947.

T:= n-> (m-> seq(m/ithprime(j), j=1..n))(mul(ithprime(i), i=1..n)):
seq(T(n), n=1..10);  # Alois P. Heinz, Jun 18 2015

T[1, 1] = 1; T[n_, n_] := T[n, n] = Prime[n-1]*T[n-1, n-1];
T[n_, k_] := T[n, k] = Prime[n]*T[n-1, k];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 26 2016 *)

T(1,1) = 1, T(n,k) = A000040(n)*T(n-1,k) for k < n, T(n,n) = A000040(n-1) * T(n-1,n-1).

A340467 a(n) is the n-th squarefree number having n prime factors.

Original entry on oeis.org

2, 10, 66, 462, 4290, 53130, 903210, 17687670, 406816410, 11125544430, 338431883790, 11833068917670, 457077357006270, 20384767656323070, 955041577211912190, 49230430891074322890, 2740956243836856315270, 168909608387276001835590, 11054926927790884163355330

Offset: 1

Alois P. Heinz, Jan 08 2021

nonn

a(n) is the n-th product of n distinct primes.
All terms are even.
This sequence differs from A073329 which has also nonsquarefree terms.

			a(1) = A000040(1) = 2.
a(2) = A006881(2) = 10.
a(3) = A007304(3) = 66.
a(4) = A046386(4) = 462.
a(5) = A046387(5) = 4290.
a(6) = A067885(6) = 53130.
a(7) = A123321(7) = 903210.
a(8) = A123322(8) = 17687670.
a(9) = A115343(9) = 406816410.
a(10) = A281222(10) = 11125544430.

Alois P. Heinz, Table of n, a(n) for n = 1..350

Main diagonal of A340316.
Cf. A001221, A001222, A005117, A070826, A072047, A073329, A101695, A340313.
Cf. A000040, A006881, A007304, A046386, A046387, A067885, A123321, A123322, A115343, A281222.

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A340467(n):
    if n == 1: return 2
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 31 2024

a(n) = A340316(n,n).
a(n) = A005117(m) <=> A072047(m) = n = A340313(m).
A001221(a(n)) = A001222(a(n)) = n.
a(n) < A070826(n+1), the least odd number with exactly n distinct prime divisors.

A346424 Number of partitions of the 2n-multiset {0,...,0,1,2,...,n}.

Original entry on oeis.org

1, 2, 11, 74, 592, 5317, 52902, 572402, 6670707, 83025806, 1096662664, 15292076689, 224145880470, 3440981816071, 55153081768896, 920494136057715, 15959177281931953, 286834809549486462, 5334308665713522860, 102476857445135062727, 2030589375575413246579

Offset: 0

Alois P. Heinz, Jul 16 2021

nonn

Also number of factorizations of 2^n * Product_{i=1..n} prime(i+1); a(2) = 11: 2*2*3*5, 3*4*5, 2*5*6, 6*10, 2*3*10, 5*12, 4*15, 2*2*15, 3*20, 2*30, 60.

			a(0) = 1: {}.
a(1) = 2: 01, 0|1.
a(2) = 11: 00|1|2, 001|2, 1|002, 0|0|1|2, 0|01|2, 0|1|02, 01|02, 00|12, 0|0|12, 0|012, 0012.

Alois P. Heinz, Table of n, a(n) for n = 0..509

Main diagonal of A346426.

Cf. A000040, A000041, A000079, A000110, A001055, A048993, A070826, A292508, A346519.

s:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=0,
      combinat[numbpart](n), add(b(n-j, i-1), j=0..n)))
    end:
a:= n-> add(S(n, j)*b(n, j), j=0..n):
seq(a(n), n=0..21);

s[n_] := s[n] = Expand[If[n == 0, 1,
     x*Sum[s[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0,
     PartitionsP[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
a[n_] := Sum[S[n, j]*b[n, j], {j, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Apr 06 2022, after Alois P. Heinz *)

a(n) = A001055(A000079(n)*A070826(n+1)).
a(n) = Sum_{j=0..n} A048993(n,j)*A292508(n,j+1).
a(n) = A346426(n,n).

A346519 Number of partitions of the 2n-multiset {0,...,0,1,2,...,n} into distinct multisets.

Original entry on oeis.org

1, 2, 9, 59, 442, 3799, 36332, 379831, 4288933, 51867573, 667168482, 9076862555, 130018298663, 1953284957029, 30675458303547, 502166867458649, 8547908294767932, 150965367603029126, 2760941474553823577, 52196915577464262360, 1018499212583077293854

Offset: 0

Alois P. Heinz, Jul 21 2021

nonn

Also number of factorizations of 2^n * Product_{i=1..n} prime(i+1) into distinct factors; a(2) = 9: 3*4*5, 2*5*6, 6*10, 2*3*10, 5*12, 4*15, 3*20, 2*30, 60.

			a(0) = 1: {}.
a(1) = 2: 01, 0|1.
a(2) = 9: 00|1|2, 001|2, 1|002, 0|01|2, 0|1|02, 01|02, 00|12, 0|012, 0012.

Alois P. Heinz, Table of n, a(n) for n = 0..510

Main diagonal of A346520.

Cf. A000009, A000040, A000079, A045778, A048993, A070826, A346424.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
s:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i=0, g(n), add(b(n-j, i-1), j=0..n)))
    end:
a:= n-> add(S(n, j)*b(n, j), j=0..n):
seq(a(n), n=0..20);

g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j]*Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
s[n_] := s[n] = Expand[If[n == 0, 1, x*Sum[s[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, g[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
a[n_] := Sum[S[n, j]*b[n, j], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 06 2022, after Alois P. Heinz *)

a(n) = A045778(A000079(n)*A070826(n+1)).
a(n) = Sum_{j=0..n} Stirling2(n,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000009(n-i).
a(n) = A346520(n,n).

A346521 Total number of partitions of all n-multisets {0,...,0,1,2,...,j} into distinct multisets for 0 <= j <= n.

Original entry on oeis.org

1, 2, 5, 15, 46, 161, 624, 2669, 12483, 63261, 344631, 2005058, 12390086, 80945545, 556896913, 4021109557, 30382294412, 239589006143, 1967343509525, 16786587081641, 148561276135546, 1361378815644787, 12897870827339021, 126158299918183469, 1272377007364596242

Offset: 0

Alois P. Heinz, Jul 21 2021

nonn

Also total number of factorizations of 2^(n-j) * Product_{i=1..j} prime(i+1) into distinct factors for 0 <= j <= n; a(2) = 5: 4, 2*3, 6, 3*5, 15; a(3) = 15: 2*4, 8, 3*4, 2*6, 12, 2*3*5, 5*6, 3*10, 2*15, 30, 3*5*7, 7*15, 5*21, 3*35, 105.

			a(2) = 5: 00, 01, 0|1, 12, 1|2.
a(3) = 15: 000, 0|00, 001, 00|1, 0|01, 012, 0|12, 02|1, 01|2, 0|1|2, 123, 1|23, 13|2, 12|3, 1|2|3.

Alois P. Heinz, Table of n, a(n) for n = 0..576

Antidiagonal sums of A346520.

Cf. A000009, A000040, A000079, A000110, A045778, A070826, A346428.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
s:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i=0, g(n), add(b(n-j, i-1), j=0..n)))
    end:
A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k):
a:= n-> add(A(n-j, j), j=0..n):
seq(a(n), n=0..24);

g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j]*Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
s[n_] := s[n] = Expand[If[n == 0, 1, x*Sum[s[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, g[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
A[n_, k_] := Sum[S[k, j]*b[n, j], {j, 0, k}];
a[n_] := Sum[A[n - j, j], {j, 0, n}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jul 31 2021, after Alois P. Heinz *)

a(n) = Sum_{j=0..n} A346520(n-j,j).

a(n) = Sum_{j=0..n} A045778(A000079(n-j)*A070826(j+1)).

A083339 a(n) is the number of distinct prime factors of n that occur in partitions into two primes when n is even and into three primes when n is odd.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2, 1, 0, 0, 0, 1, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 1, 2, 0, 2, 0, 0, 0, 1, 1, 0, 0, 2, 1, 2, 0, 0, 0, 2, 0, 2, 1, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3

Offset: 1

Reinhard Zumkeller, Apr 24 2003

nonn

Number of distinct prime factors of n that occur in prime-partitions confirming Goldbach's conjectures. (The original name of this sequence.)

Conjecture: Apart from k=2, A070826(k): 1, 3, 15, 105, 1155, 15015, 255255, gives the positions of records (each equal to k-1). This follows from the conjectured formula. - Antti Karttunen, Sep 14 2017

			For n = 14 = 2*7 = 3 + 11 = 7 + 7, only one factor of 14 occurs, thus a(14) = 1.
For n = 15 = 3*5 = 2 + 2 + 11 = 3 + 5 + 7 = 5 + 5+ 5, both factors of 15 occur, thus a(15) = 2.
For n = 105 = 3*5*7, with 35 different partitions into three primes, the partition 97 + 5 + 3 contains the prime factors 3 and 5, while the partition 79 + 19 + 7 contains 7, thus all three prime factors of 115 occur and a(115) = 3.
For n = 1155 = 3*5*7*11, among 891 different partitions into three primes, the following four partitions: 1129 + 23 + 3 = 1129 + 19 + 7 = 1109 + 41 + 5 = 1103 + 41 + 11 each have either 3, 5, 7 or 11 as one of their parts, thus a(1155) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..2048
Antti Karttunen, Scheme-program for computing A083338 and A083339
Eric Weisstein's World of Mathematics, Goldbach Conjecture.
Index entries for sequences related to Goldbach conjecture

Cf. A001221, A083338, A045917, A054860, A070826, A100484.

Table[Count[Union@ Flatten@ Select[IntegerPartitions[n, {2 + Boole[OddQ@ n]}], AllTrue[#, PrimeQ] &], p_ /; Divisible[n, p]], {n, 105}] (* Michael De Vlieger, Sep 16 2017 *)

If n is even, a(n) = A010051(n/2), if n is an odd prime, a(n) = 0, and for odd composites (conjecturally), a(n) = A001221(n). - Antti Karttunen, Sep 14 2017

Name edited and two further examples added by Antti Karttunen, Sep 14 2017

cp's OEIS Frontend

A136356 Increasing sequence obtained by union of two sequences A136353 and {b(n)}, where b(n) is the smallest composite number m such that m-1 is prime and the set of distinct prime factors of m consists of the first n primes.

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A136357 Increasing sequence obtained by union of two sequences A136354 and {b(n)}, where b(n) is the smallest composite number m such that m+1 is prime and the set of distinct prime factors of m consists of the first n primes.

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A164979 Slowest growing sequence of primes having the semiprime-pairwise property: for any i,j, a(i)+a(j) is semiprime.

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PARI

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A196529 Half of greatest common divisor of products of first n prime numbers and first n composite numbers.

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A258566 Triangle in which n-th row contains all possible products of n-1 of the first n primes in descending order.

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A340467 a(n) is the n-th squarefree number having n prime factors.

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A346424 Number of partitions of the 2n-multiset {0,...,0,1,2,...,n}.

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A346519 Number of partitions of the 2n-multiset {0,...,0,1,2,...,n} into distinct multisets.

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