A303945 -id:A303945 - cp's OEIS frontend

A304465 If n is prime, set a(n) = 1. Otherwise, start with the multiset of prime factors of n, and given a multiset take the multiset of its multiplicities. Repeating this until a multiset of size 1 is reached, set a(n) to the unique element of this multiset.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 3, 1, 5, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 6, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 4, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3

Offset: 1

Gus Wiseman, May 13 2018

nonn

a(1) = 0 by convention.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Nov 08 2018

			Starting with the multiset of prime factors of 2520 we have {2,2,2,3,3,5,7} -> {1,1,2,3} -> {1,1,2} -> {1,2} -> {1,1} -> {2}, so a(2520) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001222, A001597, A005117, A007916, A055932, A056239, A112798, A181819, A182850, A182857, A275870, A296150, A303945, A304464.

Table[Switch[n,1,0,?PrimeQ,1,,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Length[#]>1&]//First],{n,100}]

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A304465(n) = if(1==n,0,my(t=isprimepower(n)); if(t,t, t=omega(n); if(bigomega(n)==t),t,A304465(A181819(n)))); \\ Antti Karttunen, Nov 08 2018

a(p^n) = n where p is any prime number.
a(product of n distinct primes) = n.
a(1) = 0; and for n > 1, if n = prime^k, a(n) = k, otherwise, if n is squarefree [i.e., A001221(n) = A001222(n)], a(n) = A001221(n), otherwise a(n) = a(A181819(n)). - Antti Karttunen, Nov 08 2018

More terms from Antti Karttunen, Nov 08 2018

A304464 Start with the normalized multiset of prime factors of n > 1. Given a multiset, take the multiset of its multiplicities. Repeat this until a multiset of size 1 is obtained. a(n) is the unique element of this multiset.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 13 2018

nonn

a(1) = 0 by convention.

			Starting with the normalized multiset of prime factors of 360, we obtain {1,1,1,2,2,3} -> {1,2,3} -> {1,1,1} -> {3}, so a(360) = 3.

Cf. A000005, A001222, A001597, A005117, A007916, A055932, A056239, A112798, A181819, A182850, A182857, A275870, A296150, A303945, A304465.

Table[If[n===1,0,NestWhile[Sort[Length/@Split[#]]&,If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]],Length[#]>1&]//First],{n,100}]

a(prime(n)) = n.
a(p^n) = n where p is any prime number and n > 1.
a(product of n > 1 distinct primes) = n.

A304455 Number of steps in the reduction to a multiset of size 1 of the multiset of prime factors of n, obtained by repeatedly taking the multiset of multiplicities.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 12 2018

nonn

			The a(2520) = 5 steps are {2,2,2,3,3,5,7} -> {1,1,2,3} -> {1,1,2} -> {1,2} -> {1,1} -> {2}.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A000961, A001222, A001597, A005117, A007916, A055932, A056239, A112798, A130091, A181819, A182850, A182853, A182857, A303945, A304464, A304465, A320118.

Table[Length[Select[FixedPointList[Sort[Length/@Split[#]]&,If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[p,{k}]]]]],Length[#]>1&]],{n,100}]

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A304455(n) = if(n<=2,0, n=A181819(n); if(2==n,0,1+A304455(n))); \\ Antti Karttunen, Dec 06 2018

For n > 2, a(n) = A182850(n) - 1.
a(prime(n)) = 0.
a(A246547(n)) = 1.

More terms from Antti Karttunen, Dec 06 2018

A303974 Regular triangle where T(n,k) is the number of aperiodic multisets of size k that fit within some normal multiset of size n.

Original entry on oeis.org

1, 2, 1, 3, 3, 3, 4, 6, 10, 6, 5, 10, 22, 23, 15, 6, 15, 40, 57, 62, 27, 7, 21, 65, 115, 165, 129, 63, 8, 28, 98, 205, 356, 385, 318, 120, 9, 36, 140, 336, 676, 914, 1005, 676, 252, 10, 45, 192, 518, 1176, 1885, 2524, 2334, 1524, 495, 11, 55, 255, 762, 1918, 3528, 5495, 6319, 5607, 3261, 1023

Offset: 1

Gus Wiseman, May 03 2018

A multiset is normal if it spans an initial interval of positive integers. It is aperiodic if its multiplicities are relatively prime.

			Triangle begins:
1
2    1
3    3    3
4    6   10    6
5   10   22   23   15
6   15   40   57   62   27
7   21   65  115  165  129   63
8   28   98  205  356  385  318  120
9   36  140  336  676  914 1005  676  252
The a(4,3) = 10 multisets: (112), (113), (122), (123), (124), (133), (134), (223), (233), (234).
The a(5,4) = 23 multisets:
(1112), (1222),
(1113), (1123), (1223), (1233), (1333), (2223), (2333),
(1124), (1134), (1224), (1234), (1244), (1334), (1344), (2234), (2334), (2344),
(1235), (1245), (1345), (2345).

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Row sums are A303976.

Cf. A000740, A000837, A001597, A007716, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length/@GatherBy[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&],Length],{n,10}]

T(n,k)={sumdiv(k, d, moebius(k/d)*sum(i=1, d, binomial(d-1, i-1)*binomial(n-k+i, i)))} \\ Andrew Howroyd, Sep 18 2018

T(n,k) = Sum_{d|k} mu(k/d) * Sum_{i=1..d} binomial(d-1, i-1)*binomial(n-k+i, i). - Andrew Howroyd, Sep 18 2018

Terms a(56) and beyond from Andrew Howroyd, Sep 18 2018

A303976 Number of different aperiodic multisets that fit within some normal multiset of size n.

Original entry on oeis.org

1, 3, 9, 26, 75, 207, 565, 1518, 4044, 10703, 28234, 74277, 195103, 511902, 1342147, 3517239, 9214412, 24134528, 63204417, 165505811, 433361425, 1134664831, 2970787794, 7777975396, 20363634815, 53313819160, 139579420528, 365427311171, 956707667616, 2504704955181

Offset: 1

Gus Wiseman, May 03 2018

nonn

A multiset is normal if it spans an initial interval of positive integers. It is aperiodic if its multiplicities are relatively prime.

			The a(4) = 26 aperiodic multisets:
(1), (2), (3), (4),
(12), (13), (14), (23), (24), (34),
(112), (113), (122), (123), (124), (133), (134), (223), (233), (234),
(1112), (1123), (1222), (1223), (1233), (1234).

Andrew Howroyd, Table of n, a(n) for n = 1..500

Row sums of A303974.

Cf. A000740, A000837, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&]],{n,10}]

seq(n)={Vec(sum(d=1, n, moebius(d)*x^d/(1 - x - x^d*(2-x)) + O(x*x^n))/(1-x))} \\ Andrew Howroyd, Feb 04 2021

a(n) = Sum_{k=1..n} Sum_{d|k} mu(k/d) * Sum_{i=1..d} binomial(d-1, i-1)*binomial(n-k+i, i). - Andrew Howroyd, Sep 18 2018

G.f.: Sum_{d>=1} mu(d)*x^d/((1 - x - x^d*(2-x))*(1-x)). - Andrew Howroyd, Feb 04 2021

Terms a(13) and beyond from Andrew Howroyd, Sep 18 2018

A304450 Numbers that are not perfect powers and whose prime factors span an initial interval of prime numbers.

Original entry on oeis.org

2, 6, 12, 18, 24, 30, 48, 54, 60, 72, 90, 96, 108, 120, 150, 162, 180, 192, 210, 240, 270, 288, 300, 360, 384, 420, 432, 450, 480, 486, 540, 600, 630, 648, 720, 750, 768, 810, 840, 864, 960, 972, 1050, 1080, 1152, 1200, 1260, 1350, 1440, 1458, 1470, 1500, 1536

Offset: 1

Gus Wiseman, May 12 2018

nonn

The multiset of prime indices of a(n) is the a(n)-th row of A112798. This multiset is normal, meaning it spans an initial interval of positive integers, and aperiodic, meaning its multiplicities are relatively prime.

			Sequence of all normal aperiodic multisets begins
2:   {1}
6:   {1,2}
12:  {1,1,2}
18:  {1,2,2}
24:  {1,1,1,2}
30:  {1,2,3}
48:  {1,1,1,1,2}
54:  {1,2,2,2}
60:  {1,1,2,3}
72:  {1,1,1,2,2}
90:  {1,2,2,3}
96:  {1,1,1,1,1,2}
108: {1,1,2,2,2}
120: {1,1,1,2,3}
150: {1,2,3,3}
162: {1,2,2,2,2}
180: {1,1,2,2,3}
192: {1,1,1,1,1,1,2}
210: {1,2,3,4}
240: {1,1,1,1,2,3}
270: {1,2,2,2,3}
288: {1,1,1,1,1,2,2}
300: {1,1,2,3,3}
360: {1,1,1,2,2,3}
384: {1,1,1,1,1,1,1,2}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000005, A000740, A000837, A000961, A001597, A001694, A005117, A007916, A052409, A052410, A055932, A056239, A112798, A303431, A303546, A303945.

Select[Range[1000],FactorInteger[#][[-1,1]]==Prime[Length[FactorInteger[#]]]&&GCD@@FactorInteger[#][[All,2]]===1&]

ok(n)={my(f=factor(n)[,1]); #f && !ispower(n) && #f==primepi(f[#f])} \\ Andrew Howroyd, Aug 26 2018

Intersection of A007916 and A055932.

A304623 Regular triangle where T(n,k) is the number of aperiodic multisets with maximum k that fit within some normal multiset of weight n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 6, 11, 8, 1, 10, 21, 27, 16, 1, 12, 38, 61, 63, 32, 1, 18, 57, 120, 162, 143, 64, 1, 22, 87, 205, 347, 409, 319, 128, 1, 28, 122, 333, 651, 950, 1000, 703, 256, 1, 32, 164, 506, 1132, 1926, 2504, 2391, 1535, 512, 1, 42, 217, 734, 1840

Offset: 1

Gus Wiseman, May 15 2018

A multiset is normal if it spans an initial interval of positive integers, and is aperiodic if its multiplicities are relatively prime.

			Triangle begins:
1
1    2
1    4    4
1    6   11    8
1   10   21   27   16
1   12   38   61   63   32
1   18   57  120  162  143   64
1   22   87  205  347  409  319  128
The a(4,3) = 11 multisets are (3), (13), (23), (113), (123), (133), (223), (233), (1123), (1223), (1233).

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A303976.

Cf. A000740, A000837, A001597, A007716, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945, A303974.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length/@GatherBy[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&],Max],{n,10}]

T(n,k) = sum(j=1, n, sumdiv(j, d, sum(i=max(1, j+k-n), d, moebius(j/d)*binomial(k-1, i-1)*binomial(d-1, i-1)))) \\ Andrew Howroyd, Jan 20 2023

T(n,k) = Sum_{j=1..n} Sum_{d|j} Sum_{i=max(1, j+k-n)..d} mu(j/d)*binomial(k-1, i-1)*binomial(d-1, i-1). - Andrew Howroyd, Jan 20 2023

cp's OEIS Frontend

A304465 If n is prime, set a(n) = 1. Otherwise, start with the multiset of prime factors of n, and given a multiset take the multiset of its multiplicities. Repeating this until a multiset of size 1 is reached, set a(n) to the unique element of this multiset.

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A304464 Start with the normalized multiset of prime factors of n > 1. Given a multiset, take the multiset of its multiplicities. Repeat this until a multiset of size 1 is obtained. a(n) is the unique element of this multiset.

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A304455 Number of steps in the reduction to a multiset of size 1 of the multiset of prime factors of n, obtained by repeatedly taking the multiset of multiplicities.

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A303974 Regular triangle where T(n,k) is the number of aperiodic multisets of size k that fit within some normal multiset of size n.

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A303976 Number of different aperiodic multisets that fit within some normal multiset of size n.

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A304450 Numbers that are not perfect powers and whose prime factors span an initial interval of prime numbers.

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A304623 Regular triangle where T(n,k) is the number of aperiodic multisets with maximum k that fit within some normal multiset of weight n.

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