A062457 -id:A062457 - cp's OEIS frontend

A319074 a(n) is the sum of the first n nonnegative powers of the n-th prime.

1, 4, 31, 400, 16105, 402234, 25646167, 943531280, 81870575521, 15025258332150, 846949229880161, 182859777940000980, 23127577557875340733, 1759175174860440565844, 262246703278703657363377, 74543635579202247026882160, 21930887362370823132822661921, 2279217547342466764922495586798

Offset: 1

Omar E. Pol, Sep 11 2018

nonn

			For n = 4 the 4th prime is 7 and the sum of the first four nonnegative powers of 7 is 7^0 + 7^1 + 7^2 + 7^3 = 1 + 7 + 49 + 343 = 400, so a(4) = 400.

Main diagonal of A319076.
Cf. A000040, A000203, A006093, A062457, A069459, A093360, A319075.
Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Cf. A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.

a(n) = sum(k=0, n-1, prime(n)^k); \\ Michel Marcus, Sep 13 2018

a(n) = Sum_{k=0..n-1} A000040(n)^k.
a(n) = Sum_{k=0..n-1} A319075(k,n).
a(n) = (A000040(n)^n - 1)/(A000040(n) - 1).
a(n) = (A062457(n) - 1)/A006093(n).
a(n) = A069459(n)/A006093(n).
a(n) = A000203(A000040(n)^(n-1)).
a(n) = A000203(A093360(n)).

A325129 Heinz numbers of integer partitions into nonsquares (A087153).

Original entry on oeis.org

1, 3, 5, 9, 11, 13, 15, 17, 19, 25, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 51, 55, 57, 59, 61, 65, 67, 71, 73, 75, 79, 81, 83, 85, 87, 89, 93, 95, 99, 101, 103, 107, 109, 111, 113, 117, 121, 123, 125, 127, 129, 131, 135, 137, 139, 141, 143, 145, 149, 153, 155

Offset: 1

Gus Wiseman, Apr 01 2019

nonn

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:
   1: {}
   3: {2}
   5: {3}
   9: {2,2}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  19: {8}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}
  45: {2,2,3}

Cf. A001156, A011757, A033461, A056239, A062457, A066328, A087153, A112798.

Cf. A324571, A324587, A324588, A325128, A325130.

Select[Range[100],!MemberQ[If[#==1,{},FactorInteger[#]],{p_,_}/;IntegerQ[Sqrt[PrimePi[p]]]]&]

A325132 Number of integer partitions of n where the multiplicity of each part k is at least prime(k).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 6, 6, 7, 7, 10, 8, 11, 12, 12, 14, 17, 16, 20, 22, 24, 26, 31, 31, 37, 39, 43, 46, 54, 53, 63, 65, 73, 75, 87, 87, 100, 102, 115, 117, 133, 134, 151, 155, 172, 176, 197, 202, 223, 231, 254, 262, 290, 298, 327, 341, 370

Offset: 0

Gus Wiseman, Apr 01 2019

nonn

The Heinz numbers of these partitions are given by A054744.

			The first few terms count the following integer partitions:
   0: ()
   2: (11)
   3: (111)
   4: (1111)
   5: (11111)
   6: (222)
   6: (111111)
   7: (1111111)
   8: (2222)
   8: (22211)
   8: (11111111)
   9: (222111)
   9: (111111111)
  10: (22222)
  10: (222211)
  10: (2221111)
  10: (1111111111)
  11: (2222111)
  11: (22211111)
  11: (11111111111)
  12: (222222)
  12: (2222211)
  12: (22221111)
  12: (222111111)
  12: (111111111111)

Cf. A052335, A054744, A062457, A087153, A117144, A324525, A324572.

Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>=Prime[i],{i,Union[#]}]&]],{n,0,30}]

G.f.: Product_{k>=1} (1 + x^(prime(k)*k) / (1 - x^k)). - Ilya Gutkovskiy, Nov 28 2020

A201614 Decimal expansion of Sum_{n = 1 .. infinity } (-1)^(n+1)/ prime(n)^n.

Original entry on oeis.org

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

Offset: 0

Michel Lagneau, Dec 03 2011

			0.39647840017672880132037721... = 1/2^1 - 1/3^2 + 1/5^3 - 1/7^4 + ...

Cf. A062457, A093358.

with(numtheory): Digits:=105:s:=sum( evalf(((-1)^(n+1))/ ithprime(n)^n),n=1..200):print(s):

digits = 105; NSum[(-1)^(n+1)/Prime[n//Round]^n, {n, 1, Infinity},  Method -> "AlternatingSigns", WorkingPrecision -> digits, NSumTerms -> digits] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 24 2014 *)

A271182 a(n) = prime(n)^(2*n) - prime(n)^(n-1).

Original entry on oeis.org

3, 78, 15600, 5764458, 25937409960, 23298084751188, 168377826535263360, 288441413566727295942, 3244150909895169974315088, 176994576151109738690640664532, 645590698195138072217104753157760, 43335257111193343900187118461545288548

Offset: 1

Wesley Ivan Hurt, Apr 07 2016

Cf. A000010, A000040, A000203, A062354, A062457.

[NthPrime(n)^(2*n)-NthPrime(n)^(n-1) : n in [1..12]];

A271182:=n->ithprime(n)^(2*n)-ithprime(n)^(n-1): seq(A271182(n), n=1..15);

Table[Prime[n]^(2*n) - Prime[n]^(n - 1), {n, 12}]

a(n) = prime(n)^(2*n) - prime(n)^(n-1); \\ Altug Alkan, Apr 07 2016

a(n) = sigma(prime(n)^n) * phi(prime(n)^n) = A062354(A062457(n)).

A271440 a(n) = sigma(prime(n)^n) - phi(prime(n)^n).

Original entry on oeis.org

2, 7, 56, 743, 30746, 773527, 49783736, 1837403019, 160181560802, 29532404308019, 1666577516860962, 360777399719461393, 45691067858241526814, 3477439299142731351087, 518913689466371066697746, 147680787468230866751370317, 43490064769447225534580532962

Offset: 1

Wesley Ivan Hurt, Apr 07 2016

Harvey P. Dale, Table of n, a(n) for n = 1..303

Cf. A000010 (phi), A000040 (primes), A000203 (sigma), A051612, A062457.

with(numtheory): A271440:=n->sigma(ithprime(n)^n)-phi(ithprime(n)^n): seq(A271440(n), n=1..30);

Table[DivisorSigma[1, Prime[n]^n] - EulerPhi[Prime[n]^n], {n, 20}]
DivisorSigma[1,#]-EulerPhi[#]&/@Table[Prime[n]^n,{n,20}] (* Harvey P. Dale, Feb 07 2025 *)

a(n) = sigma(prime(n)^n) - eulerphi(prime(n)^n); \\ Altug Alkan, Apr 08 2016

a(n) = (2*prime(n)^n-prime(n)^(n-1)-1) / (prime(n)-1).
a(n) = (prime(n)^(n+1)-prime(n)^(n-1)*(prime(n)-1)^2-1) / (prime(n)-1).
a(n) = A051612(A062457(n)) = A000203(A062457(n)) - A000010(A062457(n)).

A304917 a(n) = prime(n)^n - primorial(n - 1).

Original entry on oeis.org

1, 7, 119, 2371, 160841, 4824499, 410308643, 16983052531, 1801142961773, 420707010207331, 25408470426711601, 6582951805279545151, 925103094894275494511, 73885357039888240238239, 12063348337737606907045313, 3876269049503627062809380911

Offset: 1

David James Sycamore, May 20 2018

nonn

			a(1) = prime(1)^1 - primorial(0) = 2^1 - 1 = 1.

Cf. A002110, A062457, A305076 (n such that a(n) is prime).

N:=15:
for X from 1 to N do
Z:=mul(ithprime(i),i=1..(X-1));
Y:=(ithprime(X)^X-Z);
print(Y);
end do:
# Second Maple program
seq(ithprime(k)^k-mul(ithprime(i),i=1..k-1),k=1..15);  # Muniru A Asiru, Jul 08 2018

Fold[Append[#1, {#1 - #2, #2} & @@ {Prime[#2]^#2, Prime[#2 - 1] #1[[-1, -1]]}] &, {{1, 1}}, Range[2, 16]][[All, 1]] (* Michael De Vlieger, Jul 19 2018 *)

a(n) = prime(n)^n - factorback(primes(n - 1)) \\ David A. Corneth, May 21 2018

a(n) = A062457(n) - A002110(n-1).

A305076 Numbers k such that prime(k)^k - primorial(k - 1) is prime.

Original entry on oeis.org

2, 4, 5, 8, 9, 15, 29, 213, 666, 1360, 3932, 7916

Offset: 1

David James Sycamore, May 24 2018

Numbers k such that A304917(k) is prime.

a(12) > 4000 if it exists.

			n = 1 gives 2 - 1 = 1. n=2 gives 3^2 - 2 = 7, so 2 is the first term.

Cf. A062457, A002110, A304917.

N:=2000:
  for X from 1 to N do
Z:=mul(ithprime(i),i=1..(X-1));
Y:=(ithprime(X)^X - Z);
if isprime(Y) then print(X);
end if
end do:

Select[Range@ 700, PrimeQ[Prime[#]^# - Product[Prime@ i, {i, # - 1}]] &] (* Michael De Vlieger, Jul 19 2018 *)

isok(k) = isprime(prime(k)^k - prod(j=1, k-1, prime(j))); \\ Michel Marcus, Jun 09 2018

a(12) from Michael S. Branicky, Jun 11 2024

A130606 a(n) = prime(n+1)^n - prime(n)^n where prime(n) is the n-th prime number.

Original entry on oeis.org

1, 16, 218, 12240, 210242, 19310760, 483533066, 61327422240, 12705993314406, 398921053680600, 152509144883055582, 15980538294526150800, 793161021967277155922, 182781628843528905568920, 61073803538208251485772814

Offset: 1

Cino Hilliard, Jun 17 2007

nonn

			For n=2, prime(2+1)^2 - prime(2)^2 = 5^2 - 3^2 = 4^2, the second entry.

a := proc (n) options operator, arrow; ithprime(n+1)^n-ithprime(n)^n end proc: seq(a(n), n = 1 .. 15); # Emeric Deutsch, Jul 09 2007

n[x_]:=Module[{pn=Prime[x]},(NextPrime[pn])^x-pn^x]; n/@Range[20]  (* Harvey P. Dale, Apr 11 2011 *)

g1(n) = for(x=1,n,y=prime(x+1)^x-prime(x)^x;print1(y","))

a(n) = A093360(n+1) - A062457(n). - R. J. Mathar, Nov 25 2008

More terms from Emeric Deutsch, Jul 09 2007

A130607 a(n) = prime(n+1)^n + prime(n)^n.

Original entry on oeis.org

5, 34, 468, 17042, 532344, 28964378, 1304210412, 95294548322, 16308298637332, 1240335520281002, 203326098675865244, 29146442306206221362, 2643367226597304414564, 330552343531805913099818, 85200500239848987963203500, 25435446457194919247155743362, 3513844792272393084250431362040

Offset: 1

Cino Hilliard, Jun 17 2007

nonn

Prime(n) is the n-th prime number.

			For n=2, prime(2+1)^2 - prime(2)^2 = 5^2 + 3^2 = 34, the second term.

Cf. A062457, A093360.

Table[Prime[n+1]^n + Prime[n]^n, {n, 1, 20}] (* Amiram Eldar, Jun 30 2024 *)

g2(n) = for(x=1,n,y=prime(x+1)^x+prime(x)^x;print1(y","))

a(n) = A062457(n) + A093360(n+1). - Amiram Eldar, Jun 30 2024

More terms from Amiram Eldar, Jun 30 2024

cp's OEIS Frontend

A319074 a(n) is the sum of the first n nonnegative powers of the n-th prime.

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A325129 Heinz numbers of integer partitions into nonsquares (A087153).

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A325132 Number of integer partitions of n where the multiplicity of each part k is at least prime(k).

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A201614 Decimal expansion of Sum_{n = 1 .. infinity } (-1)^(n+1)/ prime(n)^n.

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Maple

Mathematica

A271182 a(n) = prime(n)^(2*n) - prime(n)^(n-1).

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Magma

Maple

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A271440 a(n) = sigma(prime(n)^n) - phi(prime(n)^n).

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A304917 a(n) = prime(n)^n - primorial(n - 1).

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Maple

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A305076 Numbers k such that prime(k)^k - primorial(k - 1) is prime.

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Maple