A074036 -id:A074036 - cp's OEIS frontend

A169802 Numbers k equal to A074036(k) = sum of primes from least to largest prime factor.

2, 3, 5, 7, 10, 11, 13, 17, 19, 23, 29, 31, 37, 39, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 155, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

N. J. A. Sloane, May 24 2010

nonn

Union of primes (A000040) and A055233.

Erich Friedman, What's Special About This Number?

Cf. A055233 (this without primes), A074036 (sum of primes from spf(n) to gpf(n)).

Cf. A020639 (spf: smallest prime factor), A006530 (gpf: greatest prime factor).

upto=500;a={};Do[f=Map[First,FactorInteger[k]];If[k==Total[Select[Range[First[f],Last[f]],PrimeQ]],AppendTo[a,k]],{k,upto}]; a (* Paolo Xausa, Nov 27 2021 *)

select( {is_A169802(n,f=factor(n)[,1])=n>1&&n==vecsum(primes([f[1],f[#f]]))}, [1..222]) \\ M. F. Hasler, Nov 24 2021

A169804 Numbers k such that A074036(k) > k.

Original entry on oeis.org

14, 22, 26, 34, 38, 46, 51, 57, 58, 62, 69, 74, 76, 82, 86, 87, 92, 93, 94, 106, 111, 116, 118, 122, 123, 124, 129, 134, 141, 142, 146, 148, 158, 159, 164, 166, 172, 177, 178, 183, 185, 188, 194, 201, 202, 205, 206, 212, 213, 214, 215, 218, 219, 226, 235, 236, 237, 244, 249

Offset: 1

N. J. A. Sloane, May 24 2010

nonn

Cf. A074036, A169802.

A055233 Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.

Original entry on oeis.org

10, 39, 155, 371, 2935561623745, 454539357304421

Offset: 1

Carlos Rivera, Jun 21 2000

Composite n such that n = p_1 + p_2 + ... + p_k where the p_i are consecutive primes, p_1 is the smallest prime factor of n and p_k is the largest.
Concerning a(6): 454539357304421 is the product of two primes, 3536123 * 128541727 and also the sum of these two plus all the primes in between: 3536123 + 3536129 + 3536131 + ... + 128541719 + 128541727. I do not know if there are any terms in A055233 between 2935561623745 and 454539357304421. (I have searched for values of N satisfying N=Pa*Pb=Pa+...+Pb as far as 5.98*10^16, but this is not quite the same as A055233 or A055514.) - Robert Munafo, Nov 20 2002
This is a subsequence of A055514 where the sum must be divisible by the smallest and largest term, but they need not be its smallest and largest prime factor. Without restriction to composite numbers, all primes would be trivially included: see A169802. - M. F. Hasler, Nov 21 2021

			10 = 2*5 = 2 + 3 + 5;
39 = 3*13 = 3 + 5 + 7 + 11 + 13;
371 = 7*53 = 7 + 11 + 13 + ... + 53.

Erich Friedman, What's Special About This Number?
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 39
Miroslav Kureš, Straddled numbers: numbers equal to the sum of powers of consecutive primes from the least prime factor to the largest prime factor, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 2, 8-15.
Robert Munafo, Notable Properties of Specific Numbers
Carlos Rivera, Puzzle 98. Curio 39, The Prime Puzzles and Problems Connection.

Subsequence of A055514.
Cf. A074036 (sum of primes from sfp(n) to gpf(n)), A169802 (n = A074036(n)).
Cf. A020639 (spf: smallest prime factor), A006530 (gpf: greatest prime factor).

Select[Range[2, 10^3], And[CompositeQ@ #1, #1 == #2] & @@ {#, Total@ Prime[Range @@ PrimePi@ {#[[1, 1]], #[[-1, 1]]} &@ FactorInteger[#]]} &] (* Michael De Vlieger, Sep 04 2019 *)

a(5) found by Jud McCranie, Jul 03 2000
454539357304421 confirmed to be the 6th term by Donovan Johnson, Aug 23 2010
Example: removed last (see A055514). - Manuel Valdivia, Nov 19 2011

A074195 Sum of the primes from smallest prime factor of n to the largest prime factor of n = largest difference between consecutive divisors of n (ordered by size).

Original entry on oeis.org

4, 20, 1278, 1339, 11074308238, 19096347067

Offset: 1

Jason Earls, Sep 19 2002

10^11 < a(7) <= 8912510129422438. - Giovanni Resta, May 13 2016

Cf. A060681, A074036, A055233, A055514.

Select[Range[2, 2000], (p = First /@ FactorInteger[#]; #-#/p[[1]] == Sum[ Prime[i], {i, PrimePi@ p[[1]], PrimePi@ p[[-1]]}]) &] (* Giovanni Resta, May 13 2016 *)

isok(n) = {pf = factor(n)[,1]; my(pmin = vecmin(pf)); s = 0; forprime(p = pmin, vecmax(pf), s += p); s == n - n/pmin;} \\ Michel Marcus, Feb 03 2014

a(5)-a(6) from Giovanni Resta, May 13 2016

A074210 Numbers k such that the number of primes <= k is equal to the sum of primes from the smallest prime factor of k to the largest prime factor of k.

Original entry on oeis.org

1, 4, 12, 30, 272, 4717, 5402, 18487, 20115, 28372, 33998, 111040, 115170, 456975, 821586, 1874660, 4029676, 4060029, 59497900, 232668002, 313128068, 529436220

Offset: 1

Jason Earls, Sep 19 2002

No more terms through 20000000. - Ryan Propper, Jun 03 2006

			pi(272) = 58 and 272 = 2^4*17 and 2+3+5+7+11+13+17 = 58, so 272 is a term.

Cf. A000720, A074036.

Do[l = First /@ FactorInteger[n]; m = Range[First[l], Last[l]]; If[Plus @@ Select[m, PrimeQ] == PrimePi[n], Print[n]], {n, 2*10^7}] (* Ryan Propper, Jun 03 2006 *)

More terms from Ryan Propper, Jun 03 2006

a(19)-a(22) from Donovan Johnson, Oct 12 2008

A074256 Numbers k such that the sum of factorials of the digits of k equals the sum of the primes from the smallest prime factor of k to the largest prime factor of k.

Original entry on oeis.org

2, 242, 1323, 3200, 13050, 30000, 42432, 132300, 426205, 442244, 620425, 665353, 1261645, 1306254, 1453032, 1461363, 1523340, 1523466, 2025012, 2105334, 2134350, 2205102, 2613504, 2713421, 3005264, 3312400, 3314520, 3432000

Offset: 1

Jason Earls, Sep 20 2002

Numbers k such that A061602(k) = A074036(k). - Andrew Howroyd, Sep 18 2024

			242 = 2*11^2 and 2+3+5+7+11 = 28 and 2!+4!+2! = 28.

Cf. A061602, A074036.

okQ[n_]:=Module[{ifn=Transpose[FactorInteger[n]][[1]]}, Total[Prime[ Range[ PrimePi[ Min[ifn]], PrimePi[Max[ifn]]]]]==Total[IntegerDigits[n]!]]; Select[Range[ 2,3500000],okQ]  (* Harvey P. Dale, Apr 21 2011 *)

isok(n)={my(d=digits(n), s=sum(k=1, #d, d[k]!), f=factor(n)[,1]); if(#f, forprime(p=f[1], f[#f], s-=p)); s==0} \\ Andrew Howroyd, Sep 18 2024

More terms from Michel ten Voorde, Jun 20 2003
More terms from Sam Alexander, Feb 28 2005
Offset changed by Andrew Howroyd, Sep 18 2024

A076207 Triangular numbers whose sum of primes from smallest to largest prime factor is triangular and whose sum of composites from smallest to largest prime factor is triangular.

Original entry on oeis.org

3, 21, 105, 325, 7875, 9568125

Offset: 1

Jason Earls, Nov 02 2002

No more terms < 2.77*10^12. - David Wasserman, Jul 20 2005

a(7) > 2*10^15. - Donovan Johnson, Feb 03 2013

			325 = 5^2*13, sum of primes between: 5+7+11+13 = 36, triangular; sum of composites between: 6+8+9+10+12 = 45, triangular.

Cf. A000217, A074036, A074037.

Offset corrected by Donovan Johnson, Feb 03 2013

cp's OEIS Frontend

A169802 Numbers k equal to A074036(k) = sum of primes from least to largest prime factor.

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A169804 Numbers k such that A074036(k) > k.

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A055233 Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.

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A074195 Sum of the primes from smallest prime factor of n to the largest prime factor of n = largest difference between consecutive divisors of n (ordered by size).

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A074210 Numbers k such that the number of primes <= k is equal to the sum of primes from the smallest prime factor of k to the largest prime factor of k.

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A074256 Numbers k such that the sum of factorials of the digits of k equals the sum of the primes from the smallest prime factor of k to the largest prime factor of k.

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A076207 Triangular numbers whose sum of primes from smallest to largest prime factor is triangular and whose sum of composites from smallest to largest prime factor is triangular.

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