A355524 -id:A355524 - cp's OEIS frontend

A355536 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime, row n is empty.

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

Offset: 2

Gus Wiseman, Jul 12 2022

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 version where zero is prepended to the prime indices is A287352.
One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.

			Triangle begins (showing n, prime indices, differences*):
   2:    (1)       .
   3:    (2)       .
   4:   (1,1)      0
   5:    (3)       .
   6:   (1,2)      1
   7:    (4)       .
   8:  (1,1,1)    0 0
   9:   (2,2)      0
  10:   (1,3)      2
  11:    (5)       .
  12:  (1,1,2)    0 1
  13:    (6)       .
  14:   (1,4)      3
  15:   (2,3)      1
  16: (1,1,1,1)  0 0 0

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

Row-lengths are A001222 minus one.
The prime indices are A112798, sum A056239.
Row-sums are A243055.
Constant rows have indices A325328.
The Heinz numbers of the rows plus one are A325352.
Strict rows have indices A325368.
Row minima are A355524.
Row maxima are A286470, also A355526.
An adjusted version is A358169, reverse A355534.
Cf. A066312, A124010, A129654, A243056, A287352, A325394, A355523, A355528, A355531.

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

A355534 Irregular triangle read by rows where row n lists the augmented differences of the reversed prime indices of n.

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, Jul 12 2022

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 augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.

			Triangle begins:
   2: 1
   3: 2
   4: 1 1
   5: 3
   6: 2 1
   7: 4
   8: 1 1 1
   9: 1 2
  10: 3 1
  11: 5
  12: 2 1 1
  13: 6
  14: 4 1
  15: 2 2
  16: 1 1 1 1
For example, the reversed prime indices of 825 are (5,3,3,2), which have augmented differences (3,1,2,2).

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

Crossrefs found in the link are not repeated here.
Row-lengths are A001222.
Row-sums are A252464
Other similar triangles are A287352, A091602.
Constant rows have indices A307824.
The Heinz numbers of the rows are A325351.
Strict rows have indices A325366.
Row minima are A355531, non-augmented A355524, also A355525.
Row maxima are A355535, non-augmented A286470, also A355526.
The non-augmented version is A355536, also A355533.
A112798 lists prime indices, sum A056239.
Cf. A066312, A124010, A129654, A243055, A243056, A325352, A325394, A355523, A355528, A355531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aug[y_]:=Table[If[i

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}]

A355533 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime(k), then row n is just (k).

Original entry on oeis.org

1, 2, 0, 3, 1, 4, 0, 0, 0, 2, 5, 0, 1, 6, 3, 1, 0, 0, 0, 7, 1, 0, 8, 0, 2, 2, 4, 9, 0, 0, 1, 0, 5, 0, 0, 0, 3, 10, 1, 1, 11, 0, 0, 0, 0, 3, 6, 1, 0, 1, 0, 12, 7, 4, 0, 0, 2, 13, 1, 2, 14, 0, 4, 0, 1, 8, 15, 0, 0, 0, 1, 0, 2, 0

Offset: 2

Gus Wiseman, Jul 12 2022

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 version where zero is prepended to the prime indices before taking differences is A287352.
One could argue that row n = 1 is empty, but adding it changes only the offset, with no effect on the data.

			Triangle begins (showing n, prime indices, differences*):
   2:    (1)       1
   3:    (2)       2
   4:   (1,1)      0
   5:    (3)       3
   6:   (1,2)      1
   7:    (4)       4
   8:  (1,1,1)    0 0
   9:   (2,2)      0
  10:   (1,3)      2
  11:    (5)       5
  12:  (1,1,2)    0 1
  13:    (6)       6
  14:   (1,4)      3
  15:   (2,3)      1
  16: (1,1,1,1)  0 0 0
For example, the prime indices of 24 are (1,1,1,2), with differences (0,0,1).

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

Crossrefs found in the link are not repeated here.
Row sums are A243056.
The version for prime indices prepended by 0 is A287352.
Constant rows have indices A325328.
Strict rows have indices A325368.
Number of distinct terms in each row are 1 if prime, otherwise A355523.
Row minima are A355525, augmented A355531.
Row maxima are A355526, augmented A355535.
The augmented version is A355534, Heinz number A325351.
The version with prime-indexed rows empty is A355536, Heinz number A325352.
A112798 lists prime indices, sum A056239.
Cf. A001222, A066312, A124010, A286469, A286470, A325160, A325390, A355524.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],{PrimePi[n]},Differences[primeMS[n]]],{n,2,30}]

Row lengths are 1 or A001222(n) - 1 depending on whether n is prime.

A355531 Minimal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 14 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 augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

			The reversed prime indices of 825 are (5,3,3,2), with augmented differences (3,1,2,2), so a(825) = 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 A008578.
Positions of 1's are 2 followed by A013929.
The non-augmented maximal version is A286470, also A355526.
The non-augmented version is A355524, also A355525.
Row minima of A355534, which has Heinz number A325351.
The maximal version is A355535.
A001222 counts prime indices.
A112798 lists prime indices, sum A056239.
Cf. A124010, A129654, A243055, A243056, A307824, A325366, A325394, A355528, A355533, A355536.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aug[y_]:=Table[If[i

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}]

A355527 Squarefree numbers having at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent prime indices of n is 1.

Original entry on oeis.org

6, 15, 30, 35, 42, 66, 70, 77, 78, 102, 105, 114, 138, 143, 154, 165, 174, 186, 195, 210, 221, 222, 231, 246, 255, 258, 282, 285, 286, 318, 323, 330, 345, 354, 366, 385, 390, 402, 426, 429, 435, 437, 438, 442, 455, 462, 465, 474, 498, 510, 534, 546, 555, 570

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:
    6: {1,2}
   15: {2,3}
   30: {1,2,3}
   35: {3,4}
   42: {1,2,4}
   66: {1,2,5}
   70: {1,3,4}
   77: {4,5}
   78: {1,2,6}
  102: {1,2,7}
  105: {2,3,4}
  114: {1,2,8}
  138: {1,2,9}
  143: {5,6}
  154: {1,4,5}
  165: {2,3,5}
  174: {1,2,10}
  186: {1,2,11}
  195: {2,3,6}
  210: {1,2,3,4}

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 minimal difference <= 1 we have A055932.
For maximal instead of minimal difference = 1 we have A066312.
For minimal difference > 1 we have A325160.
If zero is considered a prime index we get A355530.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A287352, A355533, A355534, A355536 list the differences of prime indices.
A355524 or A355525 give minimal difference between prime indices.
Cf. A000005, A000040, A056239, A120944, A130091, A238353, A238354, A286470, A325161, A352822, A355526, A355531.

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

Intersection of A005117 (squarefree) and A104210 (has consecutive primes).

A355532 Maximal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 14 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 augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

			The reversed prime indices of 825 are (5,3,3,2), with augmented differences (3,1,2,2), so a(825) = 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.
Prepending 1 to the positions of 1's gives A000079.
Positions of first appearances are A008578.
Positions of 2's are A065119.
The non-augmented version is A286470, also A355526.
The non-augmented minimal version is A355524, also A355525.
The minimal version is A355531.
Row maxima of A355534, which has Heinz number A325351.
A001222 counts prime indices, distinct A001221.
A112798 lists prime indices, sum A056239.
Cf. A066312, A124010, A129654, A243055, A243056, A307824, A325366, A325394, A355533, A355536.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aug[y_]:=Table[If[i

A355523 Number of distinct differences between adjacent prime indices of n.

Original entry on oeis.org

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

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.

			For example, the prime indices of 22770 are {1,2,2,3,5,9}, with differences (1,0,1,2,4), so a(22770) = 4.

Crossrefs found in the link are not repeated here.
Counting m such that A056239(m) = n and a(m) = k gives A279945.
With multiplicity we have A252736(n) = A001222(n) - 1.
The maximal difference is A286470, minimal A355524.
A008578 gives the positions of 0's.
A287352 lists differences between 0-prepended prime indices.
A355534 lists augmented differences between prime indices.
A355536 lists differences between prime indices.
Cf. A066312, A238353, A238710, A286469, A320348, A325388, A325406, A351294, A352827, A355525-A355533.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Differences[primeMS[n]]]],{n,1000}]

A355523(n) = if(1==n, 0, my(pis = apply(primepi,factor(n)[,1]), difs = vector(#pis-1, i, pis[i+1]-pis[i])); (#Set(difs)+!issquarefree(n))); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

cp's OEIS Frontend

A355536 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime, row n is empty.

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A355534 Irregular triangle read by rows where row n lists the augmented differences of the reversed prime indices of n.

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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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A355533 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime(k), then row n is just (k).

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A355531 Minimal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

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

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A355527 Squarefree numbers having at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent prime indices of n is 1.

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