A062457 -id:A062457 - cp's OEIS frontend

A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

2, 12, 18, 36, 120, 270, 360, 540, 600, 750, 1080, 1350, 1500, 1680, 1800, 2250, 2700, 3000, 4500, 5040, 5400, 5670, 6750, 8400, 9000, 11340, 11760, 13500, 15120, 22680, 25200, 26250, 27000, 28350, 35280, 36960, 39690, 42000, 45360, 52500, 56700, 58800, 72030

Offset: 1

Gus Wiseman, Mar 24 2025

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, sum A056239.

			The terms together with their prime indices begin:
      2: {1}
     12: {1,1,2}
     18: {1,2,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    270: {1,2,2,2,3}
    360: {1,1,1,2,2,3}
    540: {1,1,2,2,2,3}
    600: {1,1,1,2,3,3}
    750: {1,2,3,3,3}
   1080: {1,1,1,2,2,2,3}
   1350: {1,2,2,2,3,3}
   1500: {1,1,2,3,3,3}
   1680: {1,1,1,1,2,3,4}
   1800: {1,1,1,2,2,3,3}

Counting partitions by the LHS gives A008284, rank statistic A061395.
Without the RHS we have A055932, counted by A000009.
Counting partitions by the RHS gives A091602, rank statistic A051903.
Counting partitions by the middle statistic gives A116608/A365676, rank stat A001221.
Without the LHS we have A212166, counted by A239964.
Without the middle statistic we have A381542, counted by A240312.
Partitions of this type are counted by A382302.
A000040 lists the primes, differences A001223.
A001222 counts prime factors, distinct A001221.
A047993 counts balanced partitions, ranks A106529.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents partition conjugation in terms of Heinz numbers.
Cf. A000720, A048767, A062457, A066328, A130091, A317090, A363719, A363740, A380955, A381632.

Select[Range[2,1000],PrimePi[FactorInteger[#][[-1,1]]]==PrimeNu[#]==Max@@FactorInteger[#][[All,2]]&]

A061395(a(n)) = A001221(a(n)) = A051903(a(n)).

A069459 a(n) = prime(n)^n - 1.

Original entry on oeis.org

1, 8, 124, 2400, 161050, 4826808, 410338672, 16983563040, 1801152661462, 420707233300200, 25408476896404830, 6582952005840035280, 925103102315013629320, 73885357344138503765448, 12063348350820368238715342, 3876269050118516845397872320, 1271991467017507741703714391418

Offset: 1

Reinhard Zumkeller, Mar 24 2002

nonn

a(n) = A062457(n) - 1.

			a(16) = A062457(n) - 1 = A000040(16)^16 - 1 = 53^16-1 =
= 3876269050118516845397872320 =
= 2^6*3^3*5*13*17*281*232073*31129845205681.

G. C. Greubel, Table of n, a(n) for n = 1..300

Cf. A062006, A069460, A069461, A069462, A000040.

[NthPrime(n)^n - 1: n in [1..25]]; // G. C. Greubel, Apr 22 2018

Table[Prime[n]^n - 1, {n, 1, 25}] (* G. C. Greubel, Apr 22 2018 *)

for(n=1, 25, print1(prime(n)^n - 1, ", ")) \\ G. C. Greubel, Apr 22 2018

A242421 Fixed points of A153212: After a(1) = 1, numbers of the form p_i1^i1 * p_i2^(i2-i1) * p_i3^(i3-i2) * ... * p_ik^(ik-i_{k-1}), where p_i's are distinct primes present in the prime factorization of n, with i1 < i2 < i3 < ... < ik, and k = A001221(n) and ik = A061395(n).

Original entry on oeis.org

1, 2, 6, 9, 30, 45, 50, 125, 210, 294, 315, 350, 441, 686, 875, 2310, 2401, 3234, 3465, 3630, 3850, 4851, 5445, 6050, 7546, 7986, 9625, 11979, 15125, 26411, 29282, 30030, 35490, 42042, 45045, 47190, 49686, 50050, 53235, 59150, 63063, 65910, 70785, 74529, 78650, 98098, 98865, 103818, 109850, 115934, 125125, 147875, 155727, 161051, 171366, 196625, 257049, 274625, 343343, 380666, 405769, 510510

Offset: 1

Antti Karttunen, May 16 2014

nonn

This sequence is closed with respect to A122111, i.e., for any n, A122111(a(n)) is either the same as a(n) or some other term a(k) of the sequence.
These numbers encode partitions in whose Young diagrams all pairs of successive horizontal and vertical segments (those pairs sharing "a common convex corner") are of equal length. Cf. the example-illustration at A153212.
Note: The seventh primorial, 510510 (= A002110(7)) occurs here as a term a(62).

			2 = p_1^1 is present, as the first prime index delta and exponent are equal.
3 = p_2^1 is not present, as 1 <> 2.
6 = p_1^1 * p_2^(2-1) is present.
9 = p_2^2 is present, as 2 = 2.
30 = p_1^1 * p_2^(2-1) * p_3^(3-2) is present, as all primorials are.
50 = p_1^1 * p_3^(3-1) is present also.

Subsequences: A002110 (primorial numbers), A062457.

Cf. A242422, A088902, A241912, A209861, A209636, A129594.

A307539 Heinz numbers of square integer partitions, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)...prime(y_k).

Original entry on oeis.org

1, 2, 9, 125, 2401, 161051, 4826809, 410338673, 16983563041, 1801152661463, 420707233300201, 25408476896404831, 6582952005840035281, 925103102315013629321, 73885357344138503765449, 12063348350820368238715343, 3876269050118516845397872321

Offset: 0

Gus Wiseman, Apr 13 2019

nonn

			The square partition (4,4,4,4) has Heinz number prime(4)^4 = 7^4 = 2401.

Alois P. Heinz, Table of n, a(n) for n = 0..303

After a(0) = 1, same as A062457.
Cf. A002024, A047993, A056239, A096771, A106529, A112798, A115720, A174090, A257990, A263297.
Cf. A088218, A330394.

a:= n-> mul(ithprime(i), i=[n$n]):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 03 2020

Mathematica
```
Table[If[n==0,1,Prime[n]]^n,{n,0,10}]
```

a(n) = A330394(A088218(n)). - Alois P. Heinz, Mar 03 2020

A077254 a(n) = prime(n)^n mod n.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 3, 1, 8, 1, 9, 1, 2, 1, 8, 1, 8, 1, 10, 1, 13, 15, 14, 1, 7, 9, 1, 9, 22, 19, 3, 1, 26, 9, 4, 1, 9, 7, 5, 1, 15, 1, 19, 9, 17, 41, 23, 1, 31, 1, 11, 1, 29, 1, 23, 9, 8, 13, 41, 1, 39, 41, 55, 1, 53, 31, 63, 13, 8, 1, 69, 1, 2, 9, 49, 5, 16, 25, 6, 1, 80, 39, 16, 1, 29, 83

Offset: 1

Reinhard Zumkeller, Oct 31 2002

nonn

a(A077255(n)) = 1.

			a(13) = prime(13)^13 mod 13 = 41^13 mod 13 = 925103102315013629321 mod 13 = 2.

Zak Seidov, Table of n, a(n) for n = 1..10000

a(n) = A062457(n) mod n, A077256, A000040, A000027.

a:= n-> ithprime(n) &^ n mod n:
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 07 2012

Table[PowerMod[Prime[n], n, n], {n, 100}] (* Zak Seidov, Dec 07 2012 *)

A197987 a(n) = prime(n)^(n+1).

Original entry on oeis.org

4, 27, 625, 16807, 1771561, 62748517, 6975757441, 322687697779, 41426511213649, 12200509765705829, 787662783788549761, 243569224216081305397, 37929227194915558802161, 3177070365797955661914307, 566977372488557307219621121, 205442259656281392806087233013

Offset: 1

Bruno Berselli, Oct 20 2011

Subsequence of A000961, A120458.
First five elements are also consecutive members of A133018. - Omar E. Pol, Oct 20 2011
Third diagonal of A319075. - Omar E. Pol, Sep 13 2018

			The fourth prime number is 7, so a(4) = 7^(4+1) = 7^5 = 16807. - _Omar E. Pol_, Oct 20 2011

Bruno Berselli, Table of n, a(n) for n = 1..50

Cf. A000961, A062457, A093360, A120458, A133018, A319075.

Magma
```
[NthPrime(n)^(n+1): n in [1..16]];
```

Table[Prime[n]^(n+1),{n,20}] (* Harvey P. Dale, Dec 16 2012 *)

for(n=1, 16, print1(prime(n)^(n+1)", "));

a(n) = A000040(n)^(n+1). - Omar E. Pol, Oct 20 2011

A381632 Numbers such that (greatest prime exponent) = (sum of distinct prime indices).

Original entry on oeis.org

2, 9, 24, 54, 72, 80, 108, 125, 216, 224, 400, 704, 960, 1215, 1250, 1568, 1664, 2000, 2401, 2500, 2688, 2880, 4352, 4800, 5000, 5103, 6075, 7290, 7744, 8064, 8448, 8640, 8960, 9375, 9728, 10000, 10976, 14400, 14580, 18816, 19968, 21632, 23552, 24000, 24057

Offset: 1

Gus Wiseman, Mar 24 2025

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, sum A056239.

			The terms together with their prime indices begin:
      2: {1}
      9: {2,2}
     24: {1,1,1,2}
     54: {1,2,2,2}
     72: {1,1,1,2,2}
     80: {1,1,1,1,3}
    108: {1,1,2,2,2}
    125: {3,3,3}
    216: {1,1,1,2,2,2}
    224: {1,1,1,1,1,4}
    400: {1,1,1,1,3,3}
    704: {1,1,1,1,1,1,5}
    960: {1,1,1,1,1,1,2,3}

For (length) instead of (sum of distinct) we have A000961.
Including number of parts gives A062457 (degenerate).
Counting partitions by the LHS gives A091602, rank statistic A051903.
Counting partitions by the RHS gives A116861, rank statistic A066328.
Partitions of this type are counted by A381079.
A001222 counts prime factors, distinct A001221.
A047993 counts partitions with max part = length, ranks A106529.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, complement A351293.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A000720, A048767, A130091, A246655, A317090, A380955, A381437, A381439.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@Last/@FactorInteger[#]==Total[Union[prix[#]]]&]

A051903(a(n)) = A066328(a(n)).

A118672 Numbers divisible by prime(i)^i for some i.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 106, 108, 110, 112, 114, 116, 117, 118, 120

Offset: 1

Franklin T. Adams-Watters, May 19 2006

nonn

Any multiple of an element of this sequence is in the sequence. The primitive elements of this sequence are A062457.

The asymptotic density of this sequence is 1 - Product_{k>=1} (1 - 1/prime(k)^k) = 0.55929756713969708790... - Amiram Eldar, Apr 06 2021

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

Cf. A062457, A000040.

Complement of A325128.

N:= 1000: # to get all terms <= N
S:= {}:
for i from 1 do
  v:= ithprime(i)^i;
  if v > N then break fi;
  S:= S union {seq(j,j=v..N,v)};
od:
sort(convert(S,list)); # Robert Israel, Mar 27 2018

seq[max_] := Module[{s = {}, p = 2, i = 1, q = 2}, While[q < max, s = Join[s, Range[q, max, q]]; p = NextPrime[p]; i++; q = p^i]; Union[s]]; seq[120] (* Amiram Eldar, Apr 06 2021 *)

An incorrect g.f. was deleted by N. J. A. Sloane, Sep 13 2009

A262207 a(n) = prime(n)^n mod n^n.

Original entry on oeis.org

0, 1, 17, 97, 1676, 21241, 214259, 5020449, 34808102, 7233300201, 46070142226, 7806783217105, 165239209697109, 1608006723911113, 48560388990668468, 4867006141797699265, 530779430908845468654, 18442832496573633213385

Offset: 1

Altug Alkan, Sep 15 2015

Inspired by A002380, A067602, A138654.
a(3), a(4), a(7) and a(48) are prime numbers.
There are no further prime numbers up to a(1000). - Harvey P. Dale, Jun 15 2025

			For n = 1, a(n) = prime(1)^1 mod 1^1 = 2^1 mod 1 = 2 mod 1 = 0.
For n = 2, a(n) = prime(2)^2 mod 2^2 = 3^2 mod 4 = 9 mod 4 = 1.
For n = 3, a(n) = prime(3)^3 mod 3^3 = 5^3 mod 27 = 125 mod 27 = 17.

Harvey P. Dale, Table of n, a(n) for n = 1..386

Cf. A002380, A067602, A138654, A000312, A062457, A121623.

Table[Mod[Prime[n]^n, n^n], {n, 18}] (* Michael De Vlieger, Sep 15 2015 *)
Table[PowerMod[Prime[n],n,n^n],{n,20}] (* Harvey P. Dale, Jun 15 2025 *)

a(n) = (prime(n)^n) % (n^n);
vector(18, n, a(n))

a(n) = A062457(n) mod A000312(n). - Michel Marcus, Sep 15 2015

A291140 Sum of the n-th powers of the first n primes.

Original entry on oeis.org

2, 13, 160, 3123, 181258, 6732437, 493478344, 24995572327, 2255433009730, 470444892889497, 38714638073629150, 7749166585021832891, 1203906832960860262108, 121893712541593098356317, 17161342484454585041813494

Offset: 1

Vojtech Strnad, Aug 18 2017

nonn

			a(3) = 2^3 + 3^3 + 5^3 = 160.

Vojtech Strnad, Table of n, a(n) for n = 1..200

Cf. A031971, A062457, A087480.

f:= n -> add(ithprime(i)^n,i=1..n):
map(f, [$1..20]); # Robert Israel, Aug 20 2017

Table[Total[Prime[Range@ n]^n], {n, 15}] (* Michael De Vlieger, Aug 19 2017 *)

a(n) = sum(i=1, n, prime(i)^n) \\ Felix Fröhlich, Aug 18 2017

a(n) = prime(1)^n + prime(2)^n + ... + prime(n)^n.

cp's OEIS Frontend

A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

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Mathematica

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A069459 a(n) = prime(n)^n - 1.

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Magma

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PARI

A242421 Fixed points of A153212: After a(1) = 1, numbers of the form p_i1^i1 * p_i2^(i2-i1) * p_i3^(i3-i2) * ... * p_ik^(ik-i_{k-1}), where p_i's are distinct primes present in the prime factorization of n, with i1 < i2 < i3 < ... < ik, and k = A001221(n) and ik = A061395(n).

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A307539 Heinz numbers of square integer partitions, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

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Maple

Mathematica

Formula

A077254 a(n) = prime(n)^n mod n.

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Maple

Mathematica

A197987 a(n) = prime(n)^(n+1).

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Programs

Magma

Mathematica

PARI

Formula

A381632 Numbers such that (greatest prime exponent) = (sum of distinct prime indices).

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A118672 Numbers divisible by prime(i)^i for some i.

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A307539 Heinz numbers of square integer partitions, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)...prime(y_k).