A256617 -id:A256617 - cp's OEIS frontend

A083553 Product of prime(n+1)-1 and prime(n)-1.

2, 8, 24, 60, 120, 192, 288, 396, 616, 840, 1080, 1440, 1680, 1932, 2392, 3016, 3480, 3960, 4620, 5040, 5616, 6396, 7216, 8448, 9600, 10200, 10812, 11448, 12096, 14112, 16380, 17680, 18768, 20424, 22200, 23400, 25272, 26892, 28552, 30616, 32040

Offset: 1

Labos Elemer, May 22 2003

nonn

The conductor of x*prime(n) + y*prime(n+1); that is, for all k >= a(n), there exist nonnegative integers x and y such that k = x*prime(n) + y*prime(n+1). - T. D. Noe, Sep 22 2004

			n=25: a(25) = (97-1)*(101-1) = 9600.

David Bressoud and Stan Wagon, A Course in Computational Number Theory, Key College Pub., 2000, p. 46.

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

Cf. A000040, A006093, A058263, A083538-A083555, A099407 (terms halved), A172042 [= A000010(a(n))], A256617.
One more than A037165.
Column 3 of A379010.

f[x_] := Prime[x]-1; Table[f[w+1]*f[w], {w, 1, 128}]

A083553(n) = ((prime(1+n)-1)*(prime(n)-1)); \\ Antti Karttunen, Dec 14 2024

a(n) = A006093(n+1)*A006093(n) = (prime(n+1)-1)*(prime(n)-1).
a(n) = A037165(n) + 1.
a(n) = 2*A099407(n). - Antti Karttunen, Dec 14 2024

A307517 Numbers with at least two not necessarily distinct prime factors less than the largest prime factor.

Original entry on oeis.org

12, 20, 24, 28, 30, 36, 40, 42, 44, 45, 48, 52, 56, 60, 63, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 130, 132, 135, 136, 138, 140, 144, 148, 150, 152, 153, 154, 156, 160, 164, 165, 168

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with at least two not necessarily distinct parts less than the largest part. The enumeration of these partitions by sum is given by A000094.

			The sequence of terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   30: {1,2,3}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   70: {1,3,4}
   72: {1,1,1,2,2}
   76: {1,1,8}

Cf. A000094, A000245, A000961, A001222, A052126, A056239, A061395, A064989, A105441, A112798, A243055, A256617, A307516, A325196.

q:= n-> (l-> add(l[i][2], i=1..nops(l)-1)>1)(sort(ifactors(n)[2])):
select(q, [$1..200])[];  # Alois P. Heinz, Apr 12 2019

Select[Range[100],PrimeOmega[#/Power@@FactorInteger[#][[-1]]]>1&]

A325198 Positive numbers whose maximum prime index minus minimum prime index is 2.

Original entry on oeis.org

10, 20, 21, 30, 40, 50, 55, 60, 63, 80, 90, 91, 100, 105, 120, 147, 150, 160, 180, 187, 189, 200, 240, 247, 250, 270, 275, 300, 315, 320, 360, 385, 391, 400, 441, 450, 480, 500, 525, 540, 551, 567, 600, 605, 637, 640, 713, 720, 735, 750, 800, 810, 900, 945

Offset: 1

Gus Wiseman, Apr 11 2019

nonn

Also Heinz numbers of integer partitions whose maximum minus minimum part is 2 (counted by A008805). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   10: {1,3}
   20: {1,1,3}
   21: {2,4}
   30: {1,2,3}
   40: {1,1,1,3}
   50: {1,3,3}
   55: {3,5}
   60: {1,1,2,3}
   63: {2,2,4}
   80: {1,1,1,1,3}
   90: {1,2,2,3}
   91: {4,6}
  100: {1,1,3,3}
  105: {2,3,4}
  120: {1,1,1,2,3}
  147: {2,4,4}
  150: {1,2,3,3}
  160: {1,1,1,1,1,3}
  180: {1,1,2,2,3}
  187: {5,7}

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

Positions of 2's in A243055.
A061395(a(n)) - A055396((n)) = 2.
Cf. A000961, A008805, A046660, A056239, A093641, A112798, A118914, A174090, A195086, A256617, A325180, A325197.

N:= 1000: # for terms <= N
q:= 2: r:= 3:
Res:= NULL:
do
  p:= q; q:= r; r:= nextprime(r);
  if p*r > N then break fi;
  for i from 1 do
    pi:= p^i;
    if pi*r > N then break fi;
    for j from 0 do
      piqj:= pi*q^j;
      if piqj*r > N then break fi;
      Res:= Res, seq(piqj*r^k,k=1 .. floor(log[r](N/piqj)))
    od
  od
od:
sort([Res]); # Robert Israel, Apr 12 2019

Select[Range[100],PrimePi[FactorInteger[#][[-1,1]]]-PrimePi[FactorInteger[#][[1,1]]]==2&]

A325226 Number of prime factors of n that are less than the largest, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 0, 1, 1, 1, 2, 0, 1, 1, 3, 0, 2, 0, 2, 2, 1, 0, 4, 0, 1, 1, 2, 0, 1, 1, 3, 1, 1, 0, 3, 0, 1, 2, 0, 1, 2, 0, 2, 1, 2, 0, 3, 0, 1, 1, 2, 1, 2, 0, 4, 0, 1, 0, 3, 1, 1, 1, 3, 0, 3, 1, 2, 1, 1, 1, 5, 0, 1, 2, 2, 0, 2, 0, 3, 2

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

			The prime factors of 300 are {2,2,3,5,5} of which {2,2,3} are less than the largest, so a(300) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Positions of 0's are A000961. Positions of 1's are A325230. Positions of terms > 1 are A307517.

Cf. A000094, A000245, A001222, A052126, A053585, A061395, A062977, A064989, A071178, A105441, A108951, A256617.

Table[PrimeOmega[n/Power@@FactorInteger[n][[-1]]],{n,100}]

A071178(n) = if(1==n, 0, factor(n)[omega(n), 2]);
A325226(n) = (bigomega(n) - A071178(n)); \\ Antti Karttunen, Nov 17 2019

a(n) = A001222(n/A053585(n)).

a(n) = A001222(n) - A071178(n) = A062977(A108951(n)). - Antti Karttunen, Nov 17 2019

Data section extended up to term a(105) by Antti Karttunen, Nov 17 2019

A378884 Numbers that are not powers of primes and whose two smallest prime divisors are consecutive primes.

Original entry on oeis.org

6, 12, 15, 18, 24, 30, 35, 36, 42, 45, 48, 54, 60, 66, 72, 75, 77, 78, 84, 90, 96, 102, 105, 108, 114, 120, 126, 132, 135, 138, 143, 144, 150, 156, 162, 165, 168, 174, 175, 180, 186, 192, 195, 198, 204, 210, 216, 221, 222, 225, 228, 234, 240, 245, 246, 252, 255, 258

Offset: 1

Amiram Eldar, Dec 09 2024

Subsequence of A104210 and first differs from at an n = 15: A104210(15) = 70 = 2 * 5 * 7 is not a term of this sequence.
All the positive multiples of 6 (A008588 \ {0}) are terms.
Numbers k such that nextprime(lpf(k)) = A151800(A020639(k)) | k.
The asymptotic density of this sequence is Sum_{k>=1} (Product_{j=1..k-1} (1-1/prime(j)))/(prime(k)*prime(k+1)) = 0.2178590011934... .

			12 = 2^2 * 3 is a term since 2 and 3 are consecutive primes.
70 = 2 * 5 * 7 is not a term since 2 and 5 are not consecutive primes.
165 = 3 * 5 * 11 is a term since 3 and 5 are consecutive primes.

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

Subsequence of A024619, A104210 and A378885.
Subsequences: A006094, A256617.
Cf. A008588, A020639, A151800.

q[k_] := Module[{p = FactorInteger[k][[;; , 1]]}, Length[p] > 1 && p[[2]] == NextPrime[p[[1]]]]; Select[Range[300], q]

is(k) = if(k == 1, 0, my(p = factor(k)[,1]); #p > 1 && p[2] == nextprime(p[1]+1));

A307516 Numbers whose maximum prime index and minimum prime index differ by more than 1.

Original entry on oeis.org

10, 14, 20, 21, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114, 115, 116, 117, 118

Offset: 1

Gus Wiseman, Apr 12 2019

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. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose maximum and minimum parts differ by more than 1. The enumeration of these partitions by sum is given by A000094.

Differs from A069900 first at n = 43.

			The sequence of terms together with their prime indices begins:
   10: {1,3}
   14: {1,4}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   38: {1,8}
   39: {2,6}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   46: {1,9}
   50: {1,3,3}
   51: {2,7}
   52: {1,1,6}
   55: {3,5}

Positions of numbers > 1 in A243055. Complement of A000961 and A256617.

Cf. A000094, A000245, A001222, A052126, A056239, A061395, A064989, A069900, A105441, A112798, A307517, A325196.

with(numtheory):
q:= n-> (l-> pi(l[-1])-pi(l[1])>1)(sort([factorset(n)[]])):
select(q, [$2..200])[];  # Alois P. Heinz, Apr 12 2019

Select[Range[100],PrimePi[FactorInteger[#][[-1,1]]]-PrimePi[FactorInteger[#][[1,1]]]>1&]

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A083553 Product of prime(n+1)-1 and prime(n)-1.

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A307517 Numbers with at least two not necessarily distinct prime factors less than the largest prime factor.

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A325198 Positive numbers whose maximum prime index minus minimum prime index is 2.

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A325226 Number of prime factors of n that are less than the largest, counted with multiplicity.

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A378884 Numbers that are not powers of primes and whose two smallest prime divisors are consecutive primes.

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A307516 Numbers whose maximum prime index and minimum prime index differ by more than 1.

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