A355527 -id:A355527 - cp's OEIS frontend

A104210 Positive integers divisible by at least 2 consecutive primes.

6, 12, 15, 18, 24, 30, 35, 36, 42, 45, 48, 54, 60, 66, 70, 72, 75, 77, 78, 84, 90, 96, 102, 105, 108, 114, 120, 126, 132, 135, 138, 140, 143, 144, 150, 154, 156, 162, 165, 168, 174, 175, 180, 186, 192, 195, 198, 204, 210, 216, 221, 222, 225, 228, 231, 234, 240

Offset: 1

Leroy Quet, Mar 13 2005

nonn

If a perfect square is in this sequence, then so is its square root (e.g., 144 and 12). - Alonso del Arte, May 07 2012

The numbers of terms not exceeding 10^k, for k=1,2,..., are 1, 22, 242, 2456, 24632, 246414, 2464272, 24643281, 246433426, ... Apparently, the asymptotic density of this sequence is 0.24643... - Amiram Eldar, Apr 10 2021

			35 is divisible by both 5 and 7, and 5 and 7 are consecutive primes.
77 is divisible by both 7 and 11, and 7 and 11 are consecutive primes.
110 is not in the sequence because, although it is divisible by 2, 5 and 11, it is not divisible by 3 or 7.

Cf. A003961, A296210 (characteristic function), A319630 (complement), A379230 [= A252748(a(n))].
Positions of terms larger than 1 in A300820 and in A322361.
Subsequences: A006094, A349169 (conjectured, after its initial 1), A349176, A355527 (squarefree terms), A372566, A378884, A379232.

N:= 1000: # for terms <= N
R:= {}:
p:= 2:
do
  q:= p; p:= nextprime(p);
  if p*q > N then break fi;
  R:= R union {seq(i,i=p*q..N,p*q)}
od:
sort(convert(R,list)); # Robert Israel, Apr 13 2020

fQ[n_] := Block[{lst = PrimePi /@ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]}, Count[ Drop[lst, 1] - Drop[lst, -1], 1] > 0]; Select[ Range[244], fQ[ # ] &] (* Robert G. Wilson v, Mar 16 2005 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
is_A104210(n) = (gcd(n,A003961(n))>1); \\ Antti Karttunen, Dec 24 2024

{k such that gcd(k, A003961(k)) > 1}. - Antti Karttunen, Dec 24 2024

More terms from Robert G. Wilson v, Mar 16 2005

A355524 Minimal difference between adjacent prime indices of n > 1, or 0 if n is prime.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 3, 6, 1, 0, 0, 7, 4, 0, 0, 1, 0, 0, 0, 8, 0, 0, 0, 0, 5, 0, 0, 0, 2, 0, 6, 9, 0, 0, 0, 10, 0, 0, 3, 1, 0, 0, 7, 1, 0, 0, 0, 11, 0, 0, 1, 1, 0, 0, 0, 12, 0, 0, 4, 13, 8

Offset: 2

Gus Wiseman, Jul 10 2022

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 prime indices of 9842 are {1,4,8,12}, with differences (3,4,4), so a(9842) = 3.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are A077017 w/o the first term.
Positions of terms > 0 are A120944.
Positions of zeros are A130091.
Triangle A238353 counts m such that A056239(m) = n and a(m) = k.
For maximal difference we have A286470 or A355526.
Positions of terms > 1 are A325161.
If singletons (k) have minimal difference k we get A355525.
Positions of 1's are A355527.
Prepending 0 to the prime indices gives A355528.
A115720 and A115994 count partitions by their Durfee square.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A000005, A066312, A238354, A238710, A286469, A351294, A352822, A355530, A355531, A355532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],0,Min@@Differences[primeMS[n]]],{n,2,100}]

A355526 Maximal difference between adjacent prime indices of n, or k if n is the k-th prime.

Original entry on oeis.org

1, 2, 0, 3, 1, 4, 0, 0, 2, 5, 1, 6, 3, 1, 0, 7, 1, 8, 2, 2, 4, 9, 1, 0, 5, 0, 3, 10, 1, 11, 0, 3, 6, 1, 1, 12, 7, 4, 2, 13, 2, 14, 4, 1, 8, 15, 1, 0, 2, 5, 5, 16, 1, 2, 3, 6, 9, 17, 1, 18, 10, 2, 0, 3, 3, 19, 6, 7, 2, 20, 1, 21, 11, 1, 7, 1, 4, 22, 2, 0, 12

Offset: 2

Gus Wiseman, Jul 10 2022

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 prime indices of 9842 are {1,4,8,12}, with differences (3,4,4), so a(9842) = 4.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are 4 followed by A000040.
Positions of 0's are A025475, minimal version A013929.
Positions of 1's are 2 followed by A066312, minimal version A355527.
Triangle A238710 counts m such that A056239(m) = n and a(m) = k.
Prepending 0 to the prime indices gives A286469, minimal version A355528.
See also A286470, minimal version A355524.
The minimal version is A355525, triangle A238709.
The augmented version is A355532.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A000005, A047966, A091602, A238353, A279945, A325160, A325161, A352822, A351294, A355530.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],PrimePi[n],Max@@Differences[primeMS[n]]],{n,2,100}]

A355525 Minimal difference between adjacent prime indices of n, or k if n is the k-th prime.

Original entry on oeis.org

1, 2, 0, 3, 1, 4, 0, 0, 2, 5, 0, 6, 3, 1, 0, 7, 0, 8, 0, 2, 4, 9, 0, 0, 5, 0, 0, 10, 1, 11, 0, 3, 6, 1, 0, 12, 7, 4, 0, 13, 1, 14, 0, 0, 8, 15, 0, 0, 0, 5, 0, 16, 0, 2, 0, 6, 9, 17, 0, 18, 10, 0, 0, 3, 1, 19, 0, 7, 1, 20, 0, 21, 11, 0, 0, 1, 1, 22, 0, 0, 12

Offset: 2

Gus Wiseman, Jul 10 2022

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 prime indices of 9842 are {1,4,8,12}, with differences (3,4,4), so a(9842) = 3.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are 4 followed by A000040.
Positions of 0's are A013929, see also A130091.
Triangle A238709 counts m such that A056239(m) = n and a(m) = k.
For maximal instead of minimal difference we have A286470.
Positions of terms > 1 are A325160, also A325161.
See also A355524, A355528.
Positions of 1's are A355527.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A238352 counts partitions by fixed points, rank statistic A352822.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A066312, A120944, A238353, A279945, A286469, A325351, A325352, A351294, A355526, A355530, A355531, A355532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],PrimePi[n],Min@@Differences[primeMS[n]]],{n,2,100}]

A355528 Minimal difference between adjacent 0-prepended prime indices of n > 1.

Original entry on oeis.org

1, 2, 0, 3, 1, 4, 0, 0, 1, 5, 0, 6, 1, 1, 0, 7, 0, 8, 0, 2, 1, 9, 0, 0, 1, 0, 0, 10, 1, 11, 0, 2, 1, 1, 0, 12, 1, 2, 0, 13, 1, 14, 0, 0, 1, 15, 0, 0, 0, 2, 0, 16, 0, 2, 0, 2, 1, 17, 0, 18, 1, 0, 0, 3, 1, 19, 0, 2, 1, 20, 0, 21, 1, 0, 0, 1, 1, 22, 0, 0, 1, 23

Offset: 2

Gus Wiseman, Jul 10 2022

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 0-prepended prime indices of 9842 are {0,1,4,8,12}, with differences (1,3,4,4), so a(9842) = 1.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are 4 followed by A000040.
Positions of positive terms are A005117, complement A013929.
A similar statistic is counted by A238353.
The maximal version is A286469, without prepending A355526.
Without prepending we have A355524 or A355525.
Positions of ones are A355530.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A112798 lists prime indices, with sum A056239.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A064428, A066312, A091602, A120944, A238354, A286470, A325161, A352822, A355527, A355531, A355532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Min@@Differences[Prepend[primeMS[n],0]],{n,2,100}]

A355530 Squarefree numbers that are either even or have at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent 0-prepended prime indices of n is 1.

Original entry on oeis.org

2, 6, 10, 14, 15, 22, 26, 30, 34, 35, 38, 42, 46, 58, 62, 66, 70, 74, 77, 78, 82, 86, 94, 102, 105, 106, 110, 114, 118, 122, 130, 134, 138, 142, 143, 146, 154, 158, 165, 166, 170, 174, 178, 182, 186, 190, 194, 195, 202, 206, 210, 214, 218, 221, 222, 226, 230

Offset: 1

Gus Wiseman, Jul 10 2022

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.
A number is squarefree if it is not divisible by any perfect square > 1.
A number has consecutive prime factors if it is divisible by both prime(k) and prime(k+1) for some k.

			The terms together with their prime indices begin:
   2: {1}
   6: {1,2}
  10: {1,3}
  14: {1,4}
  15: {2,3}
  22: {1,5}
  26: {1,6}
  30: {1,2,3}
  34: {1,7}
  35: {3,4}
  38: {1,8}
  42: {1,2,4}
  46: {1,9}
  58: {1,10}
  62: {1,11}
  66: {1,2,5}
  70: {1,3,4}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
All terms are in A005117, complement A013929.
For maximal instead of minimal difference we have A055932 or A066312.
Not prepending zero gives A355527.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A056239 adds up prime indices.
A238352 counts partitions by fixed points, rank statistic A352822.
A279945 counts partitions by number of distinct differences.
A287352, A355533, A355534, A355536 list the differences of prime indices.
A355524 gives minimal difference if singletons go to 0, to index A355525.
Cf. A000005, A000040, A120944, A238354, A286469, A286470, A325160, A325161, A355526, A355531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Min@@Differences[Prepend[primeMS[#],0]]==1&]

Equals A005117 /\ (A005843 \/ A104210).

cp's OEIS Frontend

A104210 Positive integers divisible by at least 2 consecutive primes.

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A355524 Minimal difference between adjacent prime indices of n > 1, or 0 if n is prime.

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A355526 Maximal difference between adjacent prime indices of n, or k if n is the k-th prime.

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A355525 Minimal difference between adjacent prime indices of n, or k if n is the k-th prime.

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A355528 Minimal difference between adjacent 0-prepended prime indices of n > 1.

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A355530 Squarefree numbers that are either even or have at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent 0-prepended prime indices of n is 1.

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