A024619 -id:A024619 - cp's OEIS frontend

A376309 Run-lengths of the sequence of first differences of prime-powers.

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

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			The sequence of prime-powers (A246655) is:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...
The sequence of first differences (A057820) of prime-powers is:
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, ...
with runs:
  (1,1,1),(2),(1,1),(2,2),(3),(1),(2),(4),(2,2,2,2),(1),(5),(4),(2),(4), ...
with lengths A376309 (this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..20000

For runs of prime-powers increasing by one we have A174965.
For primes instead of prime-powers we have A333254.
For squarefree numbers instead of prime-powers we have A376306.
For compression instead of run-lengths we have A376308.
For run-sums instead of run-lengths we have A376310.
For positions of first appearances we have A376341, sorted A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A024619 and A361102 list non-prime-powers, first differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A025475, A037201, A053289, A110969, A120430, A251092, A373822, A375706.

Length/@Split[Differences[Select[Range[100],PrimePowerQ]]]

up_to = 20000;
A376309list(up_to) = { my(v=vector(up_to), ppp=2, pd=1, d, rl=0, k=2, i=0); while(i<#v, k++; if(isprimepower(k), d = k-ppp; ppp = k; if(d == pd, rl++, i++; v[i] = rl; rl = 1; pd = d))); (v); };
v376309 = A376309list(up_to);
A376309(n) = v376309[n]; \\ Antti Karttunen, Jan 18 2025

More terms from Antti Karttunen, Jan 18 2025

A376310 Run-sums of the sequence of first differences of prime-powers.

Original entry on oeis.org

3, 2, 2, 4, 3, 1, 2, 4, 8, 1, 5, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 12, 4, 2, 4, 6, 2, 10, 2, 4, 2, 24, 4, 2, 4, 6, 4, 8, 5, 1, 12, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 14, 12, 4, 2, 4, 6, 2, 18, 4, 6, 8, 4, 8, 10, 2

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			The sequence of prime-powers (A246655) is:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...
The sequence of first differences (A057820) of prime-powers is:
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, ...
with runs:
  (1,1,1),(2),(1,1),(2,2),(3),(1),(2),(4),(2,2,2,2),(1),(5),(4),(2),(4), ...
with sums A376310 (this sequence).

For primes instead of prime-powers we have A373822, halved A373823.
For squarefree numbers instead of prime-powers we have A376307.
For compression instead of run-sums we have A376308.
For run-lengths instead of run-sums we have A376309.
For positions of first appearances we have A376341, sorted A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A024619 and A361102 list non-prime-powers, first differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A007916, A025475, A037201, A053289, A054265, A110969, A120430, A174965, A251092, A375706.

Total/@Split[Differences[Select[Range[100],PrimePowerQ]]]

A066031 Composite numbers n the sum of whose prime factors divides n, but which are not themselves powers of primes.

Original entry on oeis.org

30, 60, 70, 84, 90, 105, 120, 140, 150, 168, 180, 231, 234, 240, 252, 260, 270, 280, 286, 300, 315, 336, 350, 360, 450, 456, 468, 480, 490, 504, 520, 525, 528, 532, 540, 560, 572, 588, 600, 627, 646, 672, 693, 700, 702, 720, 735, 750, 756, 805, 810, 897

Offset: 1

Joseph L. Pe, Dec 12 2001

nonn

Primes and powers of primes have been excluded from the sequence because they trivially satisfy the condition "the sum of the prime factors of n divides n". Call a term of the sequence "primitive" if it is not a multiple of some previous term; for example, 70 is primitive while 60 is not. Are there infinitely many primitive terms? See A064623.

Intersection of A089352 and A024619. - Michel Marcus, Feb 03 2016

			The sum of the prime factors of 70 is 2 + 5 + 7 = 14, which divides 70.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jean-Marie de Koninck, Florian Luca, Integers divisible by sums of powers of their prime factors, Journal of Number Theory, Volume 128, Issue 3, March 2008, Pages 557-563.
J. Pe, The Prime-perfect Numbers

Cf. A024619, A089352.

Select[ Range[2, 900], IntegerQ[ # / Apply[ Plus, First[ Transpose[ FactorInteger[ # ]]]]] && Mod[ #, # - EulerPhi[ # ]] != 0 & ]

isok(n) = if (omega(n)<2, return(0)); my(f = factor(n)) ; (n % vecsum(f[,1])) == 0; \\ Michel Marcus, Feb 03 2016

More terms from Robert G. Wilson v, Dec 12 2001

A071140 Numbers n such that sum of distinct primes dividing n is divisible by largest prime dividing n; n is neither a prime, nor a true power of prime.

Original entry on oeis.org

30, 60, 70, 90, 120, 140, 150, 180, 240, 270, 280, 286, 300, 350, 360, 450, 480, 490, 540, 560, 572, 600, 646, 700, 720, 750, 810, 900, 960, 980, 1080, 1120, 1144, 1200, 1292, 1350, 1400, 1440, 1500, 1620, 1750, 1798, 1800, 1920, 1960, 2160, 2240, 2250

Offset: 1

Labos Elemer, May 13 2002

nonn

a(n) are the numbers such that the difference between the largest and the smallest prime divisor equals the sum of the other distinct prime divisors. - Michel Lagneau, Nov 13 2011
The statement above is only true for 966 of the first 1000 terms. The first counterexample is a(140) = 15015. - Donovan Johnson, Apr 10 2013
Lagneau's definition can be simplified to the largest prime divisor equals the sum of the other distinct prime divisors. - Christian N. K. Anderson, Apr 15 2013

			n = 70 = 2*5*7 has a form of 2pq, where p and q are twin primes; n = 3135 = 3*5*11*19, sum = 3+5+11+19 = 38 = 2*19, divisible by 19.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

Subsequence of A024619.

a071140 n = a071140_list !! (n-1)
a071140_list = filter (\x -> a008472 x `mod` a006530 x == 0) a024619_list
-- Reinhard Zumkeller, Apr 18 2013

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Greater[s, 1], Print[{n, ba[n]}]], {n, 2, 1000000}]
(* Second program: *)
Select[Range@ 2250, And[Length@ # > 1, Divisible[Total@ #, Last@ #]] &[FactorInteger[#][[All, 1]] ] &] (* Michael De Vlieger, Jul 18 2017 *)

A008472(n)/A006530(n) is an integer and n has at least 3 distinct prime factors.

A008472(a(n)) mod A006530(a(n)) = 0 and A010055(a(n)) = 0. - Reinhard Zumkeller, Apr 18 2013

A363729 Numbers that are not a power of a prime but whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

90, 270, 525, 550, 756, 810, 1666, 1911, 1950, 2268, 2430, 2625, 2695, 2700, 2750, 5566, 6762, 6804, 6897, 7128, 7290, 8100, 8500, 9310, 9750, 10285, 10478, 11011, 11550, 11662, 12250, 12375, 12495, 13125, 13377, 13750, 14014, 14703, 18865, 19435, 20412, 21384

Offset: 1

Gus Wiseman, Jun 24 2023

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 mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The prime indices of 6897 are {2,5,5,8}, with mean 5, median 5, and modes {5}, so 6897 is in the sequence.
The terms together with their prime indices begin:
     90: {1,2,2,3}
    270: {1,2,2,2,3}
    525: {2,3,3,4}
    550: {1,3,3,5}
    756: {1,1,2,2,2,4}
    810: {1,2,2,2,2,3}
   1666: {1,4,4,7}
   1911: {2,4,4,6}
   1950: {1,2,3,3,6}
   2268: {1,1,2,2,2,2,4}
   2430: {1,2,2,2,2,2,3}

For just primes instead of prime powers we have A363722.
Including prime-powers gives A363727, counted by A363719.
These partitions are counted by A363728.
For unequal instead of equal we have A363730, counted by A363720.
A000961 lists the prime powers, complement A024619.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with multiple modes, counted by A362610.
A360005 gives twice the median of prime indices.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Just two statistics:
- (mean) = (median): A359889, counted by A240219.
- (mean) != (median): A359890, counted by A359894.
- (mean) = (mode): counted by A363723, see A363724, A363731.
- (median) = (mode): counted by A363740.
Cf. A215366, A327473, A327476, A359893, A359908, A360009, A360248, A360550, A363721, A363725, A363741.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Select[Range[1000],!PrimePowerQ[#]&&{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]

A376341 Position of first appearance of n in A057820, the sequence of first differences of prime-powers, or 0 if n does not appear.

Original entry on oeis.org

1, 5, 10, 13, 19, 25, 199, 35, 118, 48, 28195587, 61, 3745011205066703, 80, 6635, 312, 1079, 207, 3249254387600868788, 179, 43580, 216, 21151968922, 615, 762951923, 403, 1962, 466, 12371, 245, 1480223716, 783, 494890212533313, 1110, 2064590, 1235, 375744164943287809536

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

For odd n either a(n) or a(n)+1 is in A024622 (unless a(n) = 0), corresponding to cases where the smaller or the larger term in the pair of consecutive prime powers, respectively, is a power of 2. - Pontus von Brömssen, Sep 27 2024

			a(4) = 13, because the first occurrence of 4 in A057820 is at index 13. The corresponding first pair of consecutive prime powers with difference 4 is (19, 23), and a(4) = A025528(23) = 13.
a(61) = A024622(96), because the first pair of consecutive prime powers with difference 61 is (2^96, 2^96+61), and A025528(2^96+61) = A024622(96).

Pontus von Brömssen, Table of n, a(n) for n = 1..60

For compression instead of first appearances we have A376308.
For run-lengths instead of first appearances we have A376309.
For run-sums instead of first appearances we have A376310.
For squarefree numbers instead of prime-powers we have A376311.
The sorted version is A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A024619 and A361102 list non-prime-powers, first differences A375708.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A116861 counts partitions by compressed sum, by compressed length A116608.
Cf. A001597, A006549, A007916, A024622, A025475, A025528, A037201, A038664, A053289, A110969, A120430, A174965, A373400, A375706.

mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Differences[Select[Range[100],#==1||PrimePowerQ[#]&]];
Table[Position[q,k][[1,1]],{k,mnrm[q]}]

A057820(a(n)) = n whenever a(n) > 0. - Pontus von Brömssen, Sep 24 2024

Definition modified by Pontus von Brömssen, Sep 26 2024

More terms from Pontus von Brömssen, Sep 27 2024

A378370 Distance between n and the least prime power >= n, allowing 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 27 2024

nonn

Prime powers allowing 1 are listed by A000961.

Sequences obtained by adding n to each term are placed in parentheses below.
For prime instead of prime power we have A007920 (A007918), strict A013632.
For perfect power we have A074984 (A377468), opposite A069584 (A081676).
For squarefree we have A081221 (A067535).
The restriction to the prime numbers is A377281 (A345531).
The strict version is A377282 = a(n) + 1.
For non prime power instead of prime power we have A378371 (A378372).
The opposite version is A378457, strict A276781.
A000015 gives the least prime power >= n, opposite A031218.
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A151800 gives the least prime > n.
Prime-powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A001597, A053707, A065514, A065890, A343249, A376596, A376597, A377051, A377054, A377289, A377781.

Table[NestWhile[#+1&,n,#>1&&!PrimePowerQ[#]&]-n,{n,100}]

a(n) = A000015(n) - n.

a(n) = A377282(n - 1) - 1 for n > 1.

A378767 Numbers k that are not prime powers which are divisible by a cube greater than 1.

Original entry on oeis.org

24, 40, 48, 54, 56, 72, 80, 88, 96, 104, 108, 112, 120, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 248, 250, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 344, 351, 352, 360, 368, 375, 376, 378, 384

Offset: 1

Michael De Vlieger, Dec 06 2024

Products m = j*k such that omega(k) = omega(m) > omega(j), where rad(j) | k but j does not divide k, with rad = A007947 and omega = A001221.
Proper subset of A126706.
This sequence is distinct from A362148, since this sequence also contains 216, 432, etc.

			Prime decomposition of select a(n) = m, showing m = j*k:
a(1) = 24 = 2^3 * 3 = 4 * 6.
a(2) = 40 = 2^3 * 5 = 4 * 10.
a(3) = 48 = 2^4 * 3 = 8 * 6.
a(4) = 54 = 2 * 3^3 = 9 * 6.
a(5) = 56 = 2^3 * 7 = 4 * 14.
a(6) = 72 = 2^3 * 3^2 = 4 * 18.
a(9) = 96 = 2^5 * 3 = 8 * 12 = 16 * 6.
a(130) = 864 = 2^5 * 3^2 = 8 * 108 = 9 * 96 = 16 * 54, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A024619, A046099, A126706, A362148, A372695.

Select[Select[Range[2^10], AnyTrue[FactorInteger[#][[All, -1]], # > 2 &] &], Not@*PrimePowerQ]

{a(n)} = { k : omega(k) > 1, there exists p | k such that p^3 | k }.
Intersection of A046099 and A024619.
Union of A362148 and A372695.

A222084 Number of the least divisors of n whose LCM is equal to n.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava, Feb 07 2013

nonn

If we write n as the product of its prime factors, n = p1^a1*p2^a2*p3^a3*...*pr^ar, then tau#(n) gives the number of divisors from 1 to max(p1^a1, p2^a2, p3^a3, ..., pr^ar).
In general tau#(n) <= tau(n).
Also, tau#(n) = tau(n) is A000961, tau#(n) < tau(n) is A024619.
For any prime number p tau(p) = tau#(p) = 2.
tau#(n) = 3 only for semiprimes (A001358).

			For n=40, the divisors are (1, 2, 4, 5, 8, 10, 20, 40), so tau(40)=8.
lcm(1, 2, 4, 5, 8) = 40, but lcm(1, 2, 4, 5) = 20 < 40, so tau#(40)=5.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A000005, A000961, A001358, A003418, A005179, A024619, A034444, A077610, A222085.

with(numtheory);
A222084:=proc(q)
local a,b,c,j,n; print(1);
for n from 2 to q do a:=ifactors(n)[2]; b:=nops(a); c:=0;
  for j from 1 to b do if a[j][1]^a[j][2]>c then c:=a[j][1]^a[j][2]; fi; od;
  a:=op(sort([op(divisors(n))])); b:=nops(divisors(n));
  for j from 1 to b do if a[j]=c then break; fi; od; print(j); od; end:
A222084(100000);

Table[Count[ Divisors[n] , q_Integer /; q <= Max[Power @@@ FactorInteger[n]]], {n, 87}] (* Wouter Meeussen, Feb 09 2013 *)

a(n) = {my(d = divisors(n), k = 1); while (lcm(vector(k, j, d[j])) != n, k++); k;} \\ Michel Marcus, Mar 13 2018

A346608 Indices k such that A047994(k) != A344005(k).

Original entry on oeis.org

12, 15, 20, 21, 24, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48, 51, 52, 55, 56, 57, 60, 63, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 84, 85, 87, 88, 90, 91, 92, 93, 95, 96, 99, 100, 102, 104, 105, 108, 110, 111, 112, 114, 115, 116, 117, 119, 120, 123, 124, 126, 129, 130, 132, 133, 135, 136, 138, 140

Offset: 1

N. J. A. Sloane, Aug 09 2021

nonn

Conjectures: (i) For all k in this sequence, A047994(k) >= A344005(k).
(ii) Equals composite numbers with {18, 2*p (p prime), p^i (p prime, i >= 2)} deleted.
The second conjecture asserts that this is equal to A265128 with {0, 1, 18} deleted.
I believe I have a proof of both conjectures, although I have not yet written out all the details.
Numbers k that are in A265128, but do not appear here are: 1, 18, 50, 54, 98, 162, 242, 250, 338, 486, 578, 686, ... probably given by {1} UNION A354929. Hence conjecture: the sequence consists of numbers that are neither a power of prime, or 2 * power of prime. - Antti Karttunen, Jun 14 2022
Is this the set of all k such that Phi_k(-1) = Phi_k(0) = Phi_k(1) where Phi_k denotes the k-th cyclotomic polynomial? - Jeppe Stig Nielsen, Jun 26 2023

N. J. A. Sloane, Table of n, a(n) for n = 1..8020

Cf. A047994, A265128, A344005, A345992, A354928 (complement).
Positions of nonzeros in A346607. Positions of zeros in A354924.
Setwise difference A265128 \ ({0,1} U A138929). (conjectured).
Intersection of A024619 and A230078 (conjectured).

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005
A354924(n) = (A047994(n)==A344005(n));
isA346608(n) = !A354924(n); \\ Antti Karttunen, Jun 13 2022

isA346608conjectured(n) = ((n>1) && !isprimepower(n) && ((n%2) || !isprimepower(n/2)));

cp's OEIS Frontend

A376309 Run-lengths of the sequence of first differences of prime-powers.

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A376310 Run-sums of the sequence of first differences of prime-powers.

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A066031 Composite numbers n the sum of whose prime factors divides n, but which are not themselves powers of primes.

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A071140 Numbers n such that sum of distinct primes dividing n is divisible by largest prime dividing n; n is neither a prime, nor a true power of prime.

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A363729 Numbers that are not a power of a prime but whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

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A376341 Position of first appearance of n in A057820, the sequence of first differences of prime-powers, or 0 if n does not appear.

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A378370 Distance between n and the least prime power >= n, allowing 1.

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A378767 Numbers k that are not prime powers which are divisible by a cube greater than 1.

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