A014284 -id:A014284 - cp's OEIS frontend

A158975 a(n) = sum of numbers k <= n such that all proper divisors of k are divisors of n.

1, 3, 6, 10, 11, 21, 18, 30, 27, 32, 29, 68, 42, 60, 66, 70, 59, 96, 78, 120, 108, 104, 101, 180, 126, 131, 137, 155, 130, 229, 161, 221, 203, 199, 221, 281, 198, 240, 246, 321, 239, 335, 282, 360, 403, 332, 329, 488, 378, 418, 389, 419, 382, 500, 462, 557, 448

Offset: 1

Jaroslav Krizek, Apr 01 2009

nonn

For primes p, a(p) = A158662(p) = A014284(A036234(p)).

			For n = 8 we have the following proper divisors of k <= n: {1}, {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1}, {1, 2, 4}. Only k = 6 has a proper divisor that is not a divisor of 8, viz. 3. Hence a(8) = 1 + 2 + 3 + 4 + 5 + 7 + 8 = 30.

Cf. A000040, A158662, A014284, A036234, 158973.

[ &+[ k: k in [1..n] | forall(t){ d: d in Divisors(k) | d eq k or d in Divisors(n) } ]: n in [1..57] ];

Edited and extended by Klaus Brockhaus, Apr 06 2009

A225854 Frequency of prime numbers between consecutive partial sums of primes.

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 3, 5, 4, 6, 6, 8, 6, 9, 6, 9, 10, 10, 8, 12, 12, 11, 12, 12, 15, 14, 14, 14, 14, 17, 17, 16, 17, 19, 19, 22, 16, 24, 21, 20, 20, 20, 28, 22, 26, 21, 24, 28, 23, 31, 23, 30, 28, 28, 32, 28, 31, 30, 27, 36, 29, 32, 31, 39, 33, 38, 36, 36, 37

Offset: 1

Victor Phan, May 17 2013

nonn

Gives the numbers of primes between adjacent numbers in the sequence A014284, that is, primes in the half-open interval [A014284(k), A014284(k+1)).

The plot of this sequence follows a linear curve.

			List the numbers with an increment of 1 beginning at n=1, and stop when the number of numbers reaches a prime, in this case the list would be {1,2} since its size is 2. Find the number of primes in that interval and add it to the sequence. In this case, there is 1 prime in the list. Continue counting from the last number in the previous list and apply the same rules, the next list will be {3,4,5} of size 3 and contains 2 prime numbers. The list after that will be {6,7,8,9,10} of size 5 and contains 1 prime number.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A014284.

numberOfLines = 100; (*How many elements desired in the sequence*) a = {0}; distribution = {}; last = 0; For[j = 1, j <= numberOfLines, j++, frequency = 0; b = {}; For[i = 1, i <= Prime[j], i++, b = Append[b, last + i]; If[PrimeQ[b[[i]]], frequency += 1];];last += Prime[j]; distribution = Append[distribution, frequency];]; Print["Distribution = ", distribution]; ListPlot[distribution]; (*original program*)
seq[n_] := Block[{a=0, b=2, p=2, v}, Table[v = PrimePi@b-PrimePi@a; p = NextPrime@p; a = b; b += p; v, {n}]]; seq[100] (* faster version, Giovanni Resta, May 18 2013 *)

A264858 Integers k such that A007504(k) + 1 is a square.

Original entry on oeis.org

0, 17, 539, 652, 6420, 350857847

Offset: 1

Altug Alkan, Nov 26 2015

Integers k such that the sum of the first k primes + 1 is a square.
Integers k such that A014284(k+1) is a square.
In A110996, it is commented that a(6) > 250000, if it exists.
a(6) > 50000000, if it exists. - Jon E. Schoenfield, Nov 29 2015

			a(2) = 17 because A007504(17) + 1 = 440 + 1 = 441 is a square.

Cf. A007504, A014284, A033997, A089228, A110996.

t = Accumulate[Prime@ Range@ 100000]; Prepend[Select[Range@ 7000, IntegerQ@ Sqrt[t[[#]] + 1] &], 0] (* Michael De Vlieger, Nov 29 2015, after Zak Seidov at A007504 *)
Join[{0},Position[Accumulate[Prime[Range[6500]]]+1,?(IntegerQ[Sqrt[ #]]&)]] // Flatten (* _Harvey P. Dale, Feb 12 2018 *)

lista(nn) = { print1(0, ", "); s = 1; for(k=1, nn, s += prime(k); if(issquare(s), print1(k, ", ")););}

a(6) from Jinyuan Wang, Aug 09 2023

A358223 Inverse Möbius transform of A181819, prime shadow function.

Original entry on oeis.org

1, 3, 3, 6, 3, 9, 3, 11, 6, 9, 3, 18, 3, 9, 9, 18, 3, 18, 3, 18, 9, 9, 3, 33, 6, 9, 11, 18, 3, 27, 3, 29, 9, 9, 9, 36, 3, 9, 9, 33, 3, 27, 3, 18, 18, 9, 3, 54, 6, 18, 9, 18, 3, 33, 9, 33, 9, 9, 3, 54, 3, 9, 18, 42, 9, 27, 3, 18, 9, 27, 3, 66, 3, 9, 18, 18, 9, 27, 3, 54, 18, 9, 3, 54, 9, 9, 9, 33, 3, 54

Offset: 1

Antti Karttunen, Nov 30 2022

Multiplicative and dependent only on the prime signature (A046523) because also A181819 is.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n.

Cf. A014284, A046523, A181819.

f[n_] := f[n] = Times @@ Prime@ FactorInteger[n][[All, -1]]; Array[DivisorSum[#, f] - 1 &, 90] (* Michael De Vlieger, Nov 30 2022 *)

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A358223(n) = sumdiv(n,d,A181819(d));

a(n) = Sum_{d|n} A181819(d).

Multiplicative with a(p^e) = 1 + Sum_{k=1..e} prime(k) = A014284(e+1). - Amiram Eldar, Oct 23 2023

A364798 Triangular numbers that for some k >= 1 are also the sum of the first k noncomposite numbers (1 together with the primes).

Original entry on oeis.org

1, 3, 6, 78, 448516225, 448254714630193471

Offset: 1

Ilya Gutkovskiy, Aug 08 2023

a(n) = A000217(i) = A014284(j) for appropriate i, j.

			78 is a term because 78 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 1 + 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19.

Intersection of A000217 and A014284.

Cf. A008578, A066527, A110996, A364799.

Select[Accumulate[Join[{1}, Prime[Range[10000]]]], IntegerQ[Sqrt[8 # + 1]] &]

a(6) from Jinyuan Wang, Aug 09 2023

A366066 a(n) is the largest positive integer k such that n can be expressed as the sum of k distinct positive integers that are coprime to each other.

Original entry on oeis.org

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

Offset: 0

Yifan Xie, Sep 28 2023

The indices at which k first appears, for k >= 0: 1, 3, 6, 11, 18, 29, 42, 59, 78 (A014284). Such n's are expressed as the sum of 1 and the first primes.
Runs with length >= 2 start at numbers k^2 - 1 (k >= 2).
If there are terms between runs of k and k+1, these two numbers occur alternately. Suppose that m is such a term that is b(m) terms after the first occurrence of k+1; if b(m) is odd, there are at least two even numbers in the expression of n as the sum of k+1 integers, which are not coprime to each other, so a(m) = k.

			For n = 11, 1+2+3+5=11; so a(11) = 4.
For n = 12, 1+4+7=12; so a(12) = 3.

Yifan Xie, Table of n, a(n) for n = 0..9999

Cf. A000040, A000290, A014284, A083375.

lista(nn) = v=[0]; f=[7, 12, 14, 19, 21, 23, 30, 32, 34, 43, 45, 47, 60, 62, 79]; for(n=1, nn, for(i=1, prime(n), v=concat(v, n))); for(n=1, 15, v[f[n]+1]=v[f[n]+1]-1); v;

a(n) = A083375(n) - 1 if and only if n = 7, 12, 14, 19, 21, 23, 30, 32, 34, 43, 45, 47, 60, 62, 79; otherwise, a(n) = A083375(n).

A118483 Partial sums of primes that are not Chen primes (starting with 1).

Original entry on oeis.org

1, 44, 105, 178, 257, 354, 457, 608, 771, 944, 1137, 1360, 1589, 1830, 2101, 2378, 2661, 2974, 3305, 3654, 4021, 4394, 4777, 5174, 5595, 6028, 6467, 6924, 7387, 7910, 8457, 9050, 9651, 10258, 10871, 11490, 12133, 12794, 13467, 14158, 14867, 15594

Offset: 0

Jani Melik, May 05 2006

nonn

Cf. A014284, A102540, A109611.

ischenprime:=proc(n); if (isprime(n) = 'true') then if (isprime(n+2) = 'true' or numtheory[bigomega](n+2) = 2) then RETURN('true') else RETURN('false') fi fi end: ts_partsum_notchenprime:=proc(n) local i,ans,tren; ans:=1: tren:=1: for i from 1 to n do if (ischenprime(i)='false') then tren := tren+i: ans:=[op(ans), tren]: fi od; RETURN(ans) end: ts_partsum_notchenprime(1000);

Accumulate[Join[{1},Select[Prime[Range[200]],PrimeOmega[#+2]>2&]]] (* Harvey P. Dale, Dec 14 2012 *)

A159074 Sum of the k in the range 1<=k<=n such that set of proper divisors of k is not a subset of the set of proper divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 11, 7, 19, 24, 38, 11, 50, 46, 55, 67, 95, 76, 113, 91, 124, 150, 176, 121, 200, 221, 242, 252, 306, 237

Offset: 1

Jaroslav Krizek, Apr 04 2009

nonn

The nomenclature of A159072 applies, where the terms in that sum are counted.

			a(8) = 7 adds k = 6, where {1, 2, 3} is not a subset of the divisor set {1, 2, 4} of n = 8, and k = 1, with an empty proper divisor set.

Cf.: A158976, A000217, A000040, A014284, A036234.

a(n) = A158976(n) + 1.
If p = prime, a(p) = A000217(p) - A158662(p) + 1 = A000217(p) - A014284[A036234(p)] + 1.
a(n)+A159073(n)=A000217(n). - R. J. Mathar, Apr 06 2009

Edited by R. J. Mathar, Apr 06 2009

A261033 Values of n such that A002110(n) - A007504(n) - 1 is a prime number.

Original entry on oeis.org

3, 5, 7, 9, 13

Offset: 1

Altug Alkan, Nov 18 2015

Values of n such that difference between product of first n+1 terms of A008578 and sum of first n+1 terms of A008578 is a prime number.

Initial primes of the form A002110(n) - A007504(n) - 1 are 19, 2281, 510451.

			a(1) = 3 because (2*3*5) - (2+3+5) - 1 = 19 is prime.
a(2) = 5 because (2*3*5*7*11) - (2+3+5+7+11) - 1 = 2281 is prime.
a(3) = 7 because (2*3*5*7*11*13*17) - (2+3+5+7+11+13+17) - 1 = 510451 is prime.

Cf. A002110, A007504, A008578, A014284.

for(n=1, 1e5, if(ispseudoprime(prod(k=1, n, prime(k)) - sum(k=1, n, prime(k)) - 1), print1(n, ", ")));

A383938 a(n) is the least positive integer k such that b(2*j) is prime for 1 <= j <= n but not prime for j = n+1, where b(1) = k and b(m+1) = b(m) + prime(m) for m >= 1.

Original entry on oeis.org

2, 5, 21, 129, 69, 1, 51, 23991, 171, 1371, 3, 322141431, 1431357020859

Offset: 0

Om S. M. Yadav, Aug 18 2025

Similar to A227547, primes are added in successive manner except that here the sequence breaks if an even-indexed term is not prime and considers preceding even-indexed prime as the last term of the sequence. For example, a(2) = 21 [21, 23, 26, 31, 38, 49] but since 49 is not prime, last two terms (38 and 49) are omitted leaving 31 as last term in the sequence.

a(12) is the last term, because b(j) is always divisible by 11 for some j in {2, 4, 6, 8, 10, 14, 16, 18, 22, 24, 26}. - Pontus von Brömssen, Aug 19 2025

			a(n) = k, b(m+1) = b(m) + prime(m); b(1) = k
For n = 0, a(0) = 2; b(m+1) = b(m) + prime(m): [2]
For n = 1, a(1) = 5; b(m+1) = b(m) + prime(m): [5, 7(5+2)]
For n = 2, a(2) = 21; b(m+1) = b(m) + prime(m): [21, 23(21+2), 26(23+3), 31(26+5)]
For n = 3, a(3) = 129; b(m+1) = b(m) + prime(m): [129, 131(129+2), 134(131+3), 139(134+5), 146(139+7), 157(146+11)]
For n = 4, a(4) = 69; b(m+1) = b(m) + prime(m): [69, 71(69+2), 74(71+3), 79(74+5), 86(79+7), 97(86+11), 110(97+13), 127(110+17)]
For n = 5, a(5) = 1; b(m+1) = b(m) + prime(m): [1, 3(1+2), 6(3+3), 11(6+5), 18(11+7), 29(18+11), 42(29+13), 59(42+17), 78(59+19), 101(78+23)]
For a(n), even-indexed term is prime. e.g. for a(3) = 129 [129, 131, 134, 139, 146, 157], even indexed terms 131, 139, 157 are primes.

Cf. A014284, A092163, A109723, A227547.

a(n) = my(vp=concat(2, vector(n+1, i, sum(k=1, 2*i+1, prime(k)))), v=concat(vector(n, i, 1), 0), k=1); while (apply(ispseudoprime, vector(n+1, i, vp[i]+k)) != v, k++); k; \\ Michel Marcus, Aug 19 2025

a(11) from Michel Marcus, Aug 19 2025

a(12) from Pontus von Brömssen, Aug 19 2025

cp's OEIS Frontend

A158975 a(n) = sum of numbers k <= n such that all proper divisors of k are divisors of n.

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A225854 Frequency of prime numbers between consecutive partial sums of primes.

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Mathematica

A264858 Integers k such that A007504(k) + 1 is a square.

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PARI

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A358223 Inverse Möbius transform of A181819, prime shadow function.

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A364798 Triangular numbers that for some k >= 1 are also the sum of the first k noncomposite numbers (1 together with the primes).

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A366066 a(n) is the largest positive integer k such that n can be expressed as the sum of k distinct positive integers that are coprime to each other.

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A118483 Partial sums of primes that are not Chen primes (starting with 1).

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Maple

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A159074 Sum of the k in the range 1<=k<=n such that set of proper divisors of k is not a subset of the set of proper divisors of n.

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